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    A REVIEW OF ALGEBRA


    BY
    ROMEYN HENRY RIVENBURG, A.M.

    HEAD OF THE DEPARTMENT OF MATHEMATICS
    THE PEDDIE INSTITUTE, HIGHTSTOWN, N.J.

    [Illustration]

    AMERICAN BOOK COMPANY
    NEW YORK    CINCINNATI    CHICAGO


    COPYRIGHT, 1914,
    BY ROMEYN H. RIVENBURG.


    COPYRIGHT, 1914, IN GREAT BRITAIN.


    A REVIEW OF ALGEBRA.

    E. P. 6




PREFACE


In most high schools the course in Elementary Algebra is finished by the
end of the second year. By the senior year, most students have forgotten
many of the principles, and a thorough review is necessary in order to
prepare college candidates for the entrance examinations and for
effective work in the freshman year in college. Recognizing this need,
many schools are devoting at least two periods a week for part of the
senior year to a review of algebra.

For such a review the regular textbook is inadequate. From an
embarrassment of riches the teacher finds it laborious to select the
proper examples, while the student wastes time in searching for
scattered assignments. The object of this book is to conserve the time
and effort of both teacher and student, by providing a thorough and
effective review that can readily be completed, if need be, in two
periods a week for a half year.

Each student is expected to use his regular textbook in algebra for
reference, as he would use a dictionary,--to recall a definition, a
rule, or a process that he has forgotten. He should be encouraged to
_think_ his way out wherever possible, however, and to refer to the
textbook only when _forced_ to do so as a last resort.

The definitions given in the General Outline should be reviewed as
occasion arises for their use. The whole Outline can be profitably
employed for rapid class reviews, by covering the part of the Outline
that indicates the answer, the method, the example, or the formula, as
the case may be.

The whole scheme of the book is ordinarily to have a page of problems
represent a day's work. This, of course, does not apply to the Outlines
or the few pages of theory, which can be covered more rapidly. By this
plan, making only a part of the omissions indicated in the next
paragraph, the essentials of the algebra can be readily covered, if need
be, in from thirty to thirty-two lessons, thus leaving time for tests,
even if only eighteen weeks, of two periods each, are allotted to the
course.

If a brief course is desired, the Miscellaneous Examples (pp. 31 to 35,
50 to 52), many of the problems at the end of the book, and the College
Entrance Examinations may be omitted without marring the continuity or
the comprehensiveness of the review.

                                                ROMEYN H. RIVENBURG.




CONTENTS


                                                       PAGES

    OUTLINE OF ELEMENTARY AND INTERMEDIATE ALGEBRA      7-13

    ORDER OF OPERATIONS, EVALUATION, PARENTHESES          14

    SPECIAL RULES OF MULTIPLICATION AND DIVISION          15

    CASES IN FACTORING                                16, 17

    FACTORING                                             18

    HIGHEST COMMON FACTOR AND LOWEST COMMON MULTIPLE      19

    FRACTIONS                                             20

    COMPLEX FRACTIONS AND FRACTIONAL EQUATIONS        21, 22

    SIMULTANEOUS EQUATIONS AND INVOLUTION             23, 24

    SQUARE ROOT                                           25

    THEORY OF EXPONENTS                                26-28

    RADICALS                                          29, 30

    MISCELLANEOUS EXAMPLES, ALGEBRA TO QUADRATICS      31-35

    QUADRATIC EQUATIONS                               36, 37

    THE THEORY OF QUADRATIC EQUATIONS                  38-41

    OUTLINE OF SIMULTANEOUS QUADRATICS                42, 43

    SIMULTANEOUS QUADRATICS                               44

    RATIO AND PROPORTION                              45, 46

    ARITHMETICAL PROGRESSION                              47

    GEOMETRICAL PROGRESSION                               48

    THE BINOMIAL THEOREM                                  49

    MISCELLANEOUS EXAMPLES, QUADRATICS AND BEYOND      50-52

    PROBLEMS--LINEAR EQUATIONS, SIMULTANEOUS EQUATIONS,
    QUADRATIC EQUATIONS, SIMULTANEOUS QUADRATICS       53-57

    COLLEGE ENTRANCE EXAMINATIONS                      58-80




OUTLINE OF ELEMENTARY AND INTERMEDIATE ALGEBRA


~Important Definitions~

    Factors; coefficient; exponent; power; base; term; algebraic
    sum; similar terms; degree; homogeneous expression; linear
    equation; root of an equation; root of an expression; identity;
    conditional equation; prime quantity; highest common factor (H.
    C. F.); lowest common multiple (L. C. M.); involution;
    evolution; imaginary number; real number; rational; similar
    radicals; binomial surd; pure quadratic equation; affected
    quadratic equation; equation in the quadratic form; simultaneous
    linear equations; simultaneous quadratic equations;
    discriminant; symmetrical expression; ratio; proportion; fourth
    proportional; third proportional; mean proportional; arithmetic
    progression; geometric progression; S [infinity]


~Special Rules for Multiplication and Division~

    1. Square of the sum of two quantities.
        (x + y)^2.

    2. Square of the difference of two quantities.
        (x - y)^2.

    3. Product of the sum and difference of two quantities.
        (s + t)(s - t).

    4. Product of two binomials having a common term.
        (x + r)(x + m).

    5. Product of two binomials whose corresponding terms are similar.
        (3x + 2t)(2x - 5t).

    6. Square of a polynomial.
        (m - n/3 + k)^2.

    7. Sum of two cubes.
        (x^3 + y^3)/(x + y) = x^2 - xy + y^2.

    8. Difference of two cubes.
        (x^3 - y^3)/(x - y) = x^2 + xy + y^2.

    9. Sum or difference of two like powers.
        (x^7 + y^7)/(x + y),
        (x^5 - y^5)/(x - y),
        (x^4 - y^4)/(x - y),
        (x^4 - y^4)/(x + y).

~Cases in Factoring~

    1. Common monomial factor.
        mx + my - mz = m(x + y - z).

    2. Trinomial that is a perfect square.
        x^2 ± 2xy + y^2 = (x ± y)^2.

    3. The difference of two squares.
        (a) Two terms.
            x^2 - y^2 = (x + y)(x - y).
        (b) Four terms.
            x^2 + 2xy + y^2 - m^2 = (x + y + m)(x + y - m).
        (c) Six terms.
            x^2 + 2xy + y^2 - p^2 - 2pq - q^2
              = [(x + y) + (p + q)][(x + y) - (p + q)].
        (d) Incomplete square.
            x^4 + x^2 y^2 + y^4
              = x^4 + 2x^2 y^2 + y^4 - x^2 y^2
              = (x^2 + y^2 + xy)(x^2 + y^2 - xy).

    4. Trinomial of the form x^2 + bx + c.
        x^2 - 5x + 6 = (x - 2)(x - 3).

    5. Trinomial of the form ax^2 + bx + c.
        20x^2 + 7x - 6 = (4x + 3)(5x - 2).

    6. Sum or difference of
        two cubes. See "Special Rules," 7 and 8.
        two like powers. See "Special Rules," 9.

    7. Common polynomial factor. Grouping.
        t^2 p + t^2 q - 2mp - 2mq
          = t^2(p + q) - 2m(p + q) = (p + q)(t^2 - 2m).

    8. Factor Theorem.
        x^3 + 17x - 18 = (x - 1)(x^2 + x + 18).

~H. C. F. and L. C. M.~

    a^2 + 2a - 3 = (a + 3)(a - 1).

    a^2 + 7a + 12 = (a + 3)(a + 4).

    a^4 + 27a = a(a + 3)(a^2 - 3a + 9).

    H. C. F. = a + 3.

    L. C. M. = (a + 3)(a - 1)(a + 4)a(a^2 - 3a + 9).

~Fractions~

    Reduction to lowest terms.

    Reduction of a mixed number to an improper fraction.

    Reduction of an improper fraction to a mixed number.

    Addition and subtraction of fractions.

    Multiplication and division of fractions.

    Law of signs in division, changing signs of factors, etc.

    Complex fractions.

~Simultaneous Equations~

    Solved by
        addition or subtraction.
        substitution.
        comparison.

    Graphical representation.

~Involution~

    Law of signs.

    Binomial theorem laws.

    Expansion of
        monomials and fractions.
        binomials.
        trinomials.

~Evolution~

    Law of signs.

    Evolution of monomials and fractions.

    Square root of algebraic expressions.

    Square root of arithmetical numbers.

    Optional
        Cube root of algebraic expressions.
        Cube root of arithmetical numbers.

~Theory of Exponents~

    Proofs: a^m × a^n = a^(m + n);
            (a^m)/(a^n) = a^(m - n);
            (a^m)^n = a^(mn);
            [a^(mn)]^(1/n) = a^m;
            (a/b)^n = (a^n)/(b^n);
            (abc)^n = a^n b^n c^n.

    Meaning of
        fractional exponent.
        zero exponent.
        negative exponent.

    Four rules
        To multiply quantities having the same base, add exponents.
        To divide quantities having the same base, subtract
            exponents.
        To raise to a power, multiply exponents.
        To extract a root, divide the exponent of the power by the
            index of the root.

~Radicals~

    Radical in its simplest form.

    Transformation of radicals
        Fraction under the radical sign.
        Reduction to an entire surd.
        Changing to surds of different order.
        Reduction to simplest form.

    Addition and subtraction of radicals.

    Multiplication and division of radicals
        a^(1/n) · b^(1/n) = [ab]^(1/n).
        ([ab]^(1/n))/(a^(1/n)) = b^(1/n).

    Rationalization
        Monomial denominator.
        Binomial denominator.
        Trinomial denominator.

    Square root of a binomial surd.

    Radical equations. _Always_ check results to avoid extraneous roots.

~Quadratic Equations~

    Pure. x^2 = a.

    Affected. ax^2 + bx + c = 0.

    Methods of solving
        Completing the square.
        Formula. Developed from ax^2 + bx + c = 0.
        Factoring.

    Equations in the quadratic form.

    Properties of quadratics
        r_1 = -b/2a + ([b^2 - 4ac]^(1/2))/(2a).
        r_2 = -b/2a - ([b^2 - 4ac]^(1/2))/(2a).
        Then
            r_1 + r_2 = -b/a.
            r_1 · r_2 = c/a.
        Discriminant, b^2 - 4ac, and its discussion.
        Nature or character of the roots.

~Simultaneous Quadratics~

    CASE I.

        One equation linear.

        The other quadratic.
            3x - y = 12,
            x^2 - y^2 = 16.

    CASE II.

        Both equations homogeneous and of the second degree.
            x^2 - xy + y^2 = 21,
            y^2 - 2xy = -15.

    CASE III.

        Any two of the quantities x + y, x^2 + y^2, xy, x^3 + y^3,
        x^3 - y^3, x - y, x^2 ± xy + y^2, etc., given.
            x^2 + y^2 = 41,
            x + y = 9.

    CASE IV.
        Both equations symmetrical or symmetrical except for sign.
        Usually one equation of high degree, the other of the first
        degree.
            x^5 + y^5 = 242,
            x + y = 2.

    CASE V. Special Devices

        I. Solve for a compound unknown, like xy, x + y, (1)/(xy),
        etc., first.
            x^2y^2 + xy = 6,
            x + 2y = -5.

        II. Divide the equations, member by member.
            x^4 - y^4 = 20,
            x^2 - y^2 = 5.

        III. Eliminate the quadratic terms.
            4x + 3y = 2xy,
            7x - 5y = 5xy.

~Ratio and Proportion~

    Proportionals
        mean,
        third,
        fourth.

    Theorems
        1. Product of means equals product of extremes.
        2. If the product of two numbers equals the product of two
           other numbers, either pair, etc.
        3. Alternation.
        4. Inversion.
        5. Composition.
        6. Division.
        7. Composition and division.
        8. In a series of equal ratios, the sum of the antecedents
           is to the sum of the consequents as any antecedent, etc.

    Special method of proving four quantities in proportion. Let
    a/b = x, a = bx, etc.

~Progressions~

    Development of formulas.
                                       { l = ar^(n - 1).
        { l = a + (n - 1)d.            { S = (ar^n - a)/(r - 1).
        { S = (n/2)(a + l).            { S = (rl - a)/(r - 1).
        { S = (n/2)[2a + (n - 1)d].    { S[infinity] = (a)/(1 - r).

    Insertion of means
        Arithmetical.
        Geometrical.

~Binomial Theorem~

    Review of binomial theorem laws. See Involution.

    Expansion of (a + b)^n.

    Finding any term by
        key number method.
        r^(th) or (r + 1)^(th) term method.




A REVIEW OF ALGEBRA




ORDER OF OPERATIONS, EVALUATION, PARENTHESES


    Order of operations:
      First of all, raising to a power and extracting a root.
      Next, multiplication and division.
      Last of all, addition and subtraction.


Find the value of:

  1. 5 · 2^2 - 25^(1/2) ÷ 5 + 2^2 · 8 ÷ 4 - 2.

  2. (3 × 6 ÷ 9)/2 - 2[100^(1/2)] ÷ 5 + 4 · 2^3 - (14 · 2)/28.

  3. 9 · 2 ÷ 6 + 3 - 2 · 4^2 ÷ 8^(1/3) - 4 + (3 · 2^2)/6.

Evaluate:

  4. (a^4 - a^3 + b^3)/([a^2 b^2]^(1/2)) + (c[a^(1/2)] + a^3bc)/(abc),
     if a = 1, b = 2, c = 3.

  5. t^(1/3) + [tm]^(1/3)  +  m^(1/3), if t = 8, m = 27.

  6. (2[3 + 2d + a]^(1/2))/(3[a + b - cx - c]^(1/2))
     + ((3c - d)x)/(7ad - [abc]^(1/2)), if
     a = 5, b = 3, c = -1, d = -2, x = 0.

  7. a - {5b - [a - (3c - 3b) + 2c - 3(a - 2b - c)]},
     if a = -3,  b = 4, c = -5.                          (_Yale._)

Simplify:

  8. m - [2m - {3r - (4r - 2m)}].

  9. 2a - [5d + {3c - (a + [2d - 3a + 4c])}].

  10. 3c^2 + c(2a - [6c - {3a + c - 4a}]).




SPECIAL RULES OF MULTIPLICATION AND DIVISION


Give results by inspection:

  1. (g + 1/2 k)^2.

  2. (s - (2m)/3)^2.

  3. (2v + 3w)(2v - 3w).

  4. (x + 3ts)(x - 7ts).

  5. (2l + 3g)(4l - 11g).

  6. (a - (2b)/3 + c - d)^2.

  7. (x^3 + 8m^3)/(x + 2m).

  8. (y^3 - 27k^(3m))/(y - 3k^m).

  9. (c^5 - d^5)/(c - d).

  10. (e^5 + d^5)/(e + d).

  11. (x^4 - y^4)/(x - y).

  12. (x^4 - y^4)/(x + y).

  13. (a - .03)(a - .0007).

  14. (g^n - 1/2)(g^n + 3/4).

  15. (t^7 - v^(7/2))/(t - v^(1/2)).

  16. (k^32 + 1)(k^16 +1)(k^8 + 1)(k^4 +1)(k^2 +1)(k + 1)(k - 1).

  17. [(a + b) + (c + d)][(a + b) - (c + d)].

  18. (p - q + r - s)(p - q - r + s).

  19. (3m - n - l + 2r)(3m + n - l - 2r).

  20. (x + 5)(x - 2)(x - 5)(x + 2).

  21. (a^2 + b^2 - c - 2d + 3e)^2.

  22. (s + t - 2v/5 + 3w/6 + z^2)^2.

  23. (x^5 + 32)/(x + 2).


~References:~ The chapter on Special Rules of Multiplication and
                Division in any algebra.
              Special Rules of Multiplication and Division in the
                Outline in the front of the book.




CASES IN FACTORING


The number of terms in an expression usually gives the clue to the
possible cases under which it may come. By applying the _test_ for each
and eliminating the _possible_ cases one by one, the right case is
readily found. Hence, the number of terms in the expression and a ready
and accurate knowledge of the Cases in Factoring are the real keys to
success in this vitally important part of algebra.


CASE I. A common monomial factor. Applies to any number of terms.

    5cx - 5ct + 5cv - 15 c^2 m + 25 c^3 m^2
      = 5c(x - t + v - 3cm + 5c^2 m^2).

CASE II. A trinomial that is a perfect square. Three terms.

    x^2 ± 2xm + m^2 = (x ± m)^2.

CASE III. The difference of two squares.

    _A._ Two terms. x^2 - y^2 = (x + y)(x - y).

    _B._ Four terms.

         x^2 + 2xy + y^2 - m^2 = (x^2 + 2xy + y^2) - m^2
                               = (x + y + m)(x + y - m)

    _C._ Six terms. x^2 - 2xy + y^2 - m^2 - 2mn - n^2
            = (x^2 - 2xy + y^2) - (m^2 + 2mn + n^2)
            = (x - y)^2 - (m + n)^2
            = [(x - y) + (m + n)][(x - y) - (m + n)].

    _D._ An incomplete square. Three terms, and
         4th powers or multiples of 4.

         c^4 + c^2 d^2 + d^4 = c^4 + 2c^2 d^2 + d^4 - c^2 d^2
                             = (c^2 + d^2)^2 - c^2 d^2
                             = (c^2 + d^2 + cd)(c^2 + d^2 - cd).

CASE IV. A trinomial of the form x^2 + bx + c. Three terms.

    x^2 + x - 30 = (x + 6)(x - 5).


CASE V. A trinomial of the form ax^2 + bx + c. Three terms.

    20x^2 + 7x - 6 = (4x + 3)(5x - 2).

CASE VI.

    _A._ The sum or difference of two cubes. Two terms.

         x^3 + y^3 = (x + y)(x^2 - xy + y^2);
         x^3 - y^3 = (x - y)(x^2 + xy + y^2).

    _B._ The sum or difference of two like powers. Two terms.

         x^4 - y^4 = (x - y)(x^3 + x^2y + xy^2 + y^3);
         x^5 + y^5 = (x + y)(x^4 - x^3y + x^2 y^2 - xy^3 + y^4).

CASE VII. A common polynomial factor. Any _composite_ number of terms.

    t^2 p + t^2 q - t^2 r - g^2 p - g^2 q + g^2 r
    = t^2 (p + q - r) - g^2 (p + q -r)
    = (p + q - r)(t^2 - g^2)
    = (p + q - r)(t + g)(t - g).

CASE VIII. The Factor Theorem. Any number of terms.

    x^3 + 17x - 18 = (x - 1)(x^2 + x + 18).




FACTORING


Review the _Cases in Factoring_ (see Outline on preceding pages) and
write out the prime factors of the following:

  1. 8a^(13) + am^(12).

  2. x^7 + y^7.

  3. 4x^2 + 11x - 3.

  4. m^2 + n^2 - (1 + 2mn).

  5. -x^2 + 2x - 1 + x^4.

  6. x^(16) - y^(16).  (Five factors.)

  7. (x + 1)^2 - 5x - 29.

  8. x^4 + x^2 y^2 + y^4.

  9. x^4 - 11x^2 + 1.

  10. x^(2m) + 2 + 1/(x^(2m)).

  11. x^(6m) + 13x^(3m) + 12.

  12. 4a^2 b^2 - (a^2 + b^2 - c^2)^2.

  13. (x^2 - x - 6)(x^2 - x - 20).

  14. a^4 - 8a - a^3 + 8.

  15. p^3 + 7p^2 + 14p + 8.

  16. 18a^2 b + 60ab^2 + 50b^3.

  17. x^3 - 7x + 6.

  18. 24c^2 d^2 - 47cd - 75.

  19. (a^2 - b^2)^2 - (a^2 - ab)^2.

  20. a^2 x^3 - (8a^2)/(y^3) - x^3 + 8/(y^3).

  21. gt - gk + gl^2 + xt - xk + xl^2.

  22. (m - n)(2a^2 - 2ab) + (n - m)(2ab - 2b^2).

  23. a^2 - x^2 - y^2 + b^2 + 2ab + 2xy.

  24. (2c^2 + 3d^2)a + (2a^2 + 3c^2)d.

  25. (n(n - 1))/(1 · 2) a^(n - 2) b^2
        + (n(n - 1)(n - 2))/(1 · 2 · 3) a^(n - 3) b^3.

  26. (x - x^2)^3 + (x^2 - 1)^3 + (1 - x)^3.            (_M. I. T._)

  27. (27y^3)^2 - 2(27y^3)(8b^3) + (8b^3)^2.          (_Princeton._)

  28. (a^3 + 8b^3)(a + b) - 6ab(a^2 - 2ab + 4b^2).      (_M. I. T._)

Solve by factoring:

  29. x^3 = x.

  30. z^2 - 4z - 45 = 0.

  31. x^3 - x^2 = 4x - 4.

~Reference:~ The chapter on Factoring in any algebra.




HIGHEST COMMON FACTOR AND LOWEST COMMON MULTIPLE


Define H. C. F. and L. C. M.

Find by factoring the H. C. F. and L. C. M.:

  1. 3x^2 - 3x,
     12x^2 (x^2 - 1),
     18x^3 (x^3 - 1).

  2. (x^2 - 1)(x^2 + 5x + 6),
     (x^2 + 3x)(x^2 - x - 6).                           (_Harvard._)

  3. x^2 - y^2,
     x^2 + y^2,
     x^3 + y^3,
     x^6 + y^6,
     x^6 - y^6.                          (_College Entrance Board._)

  4. x^3 + x^2 - 2,
     x^3 + 2x^2 - 3.                                    (_Cornell._)

  5. x^5 - 2x^4 + x^2,
     2x^4 - 4x^3 - 4x + 6.                                 (_Yale._)

  6. x^2 + a^2 - b^2 + 2ax,
     x^2 - a^2 + b^2 + 2bx,
     x^2 - a^2 - b^2 - 2ab.                             (_Harvard._)

  7. 2x^2 - x - 15,
     3x^2 - 11x + 6,
     2x^3 - x^2 - 13x - 6.               (_College Entrance Board._)

  8. (tv - v^2)^3,
     v^3 - t^2v,
     t^3 - v^3,
     v^2 - 2vt + t^2.

Pick out the H. C. F. and the L. C. M. of the following:

  9. 8(x^2 + y)^(32) (t^2 + z)^(19) (m - n^3)^(14),
     12(x^2 + y)^(23) (t^2 + z)^(41) (m - n^3)^(17),
     18(m - n^3)^(11) (x^2 + y)^(39) (t^2 + z)^(37).

  10. 17ax^3 (y + z)^(10) (y - x)^(19) (x + z)^(27),
      34a^2 x^4 (y + z)^(11) (y - x)^(21) (x + z)^(13),
      51a^3 x^5 (y + z)^4 (x + z)^(32) (y - x)^(29).

~Reference:~ The chapter on H. C. F. and L. C. M. in any algebra.




FRACTIONS


Define: fraction, terms of a fraction, reciprocal of a number.

Look up _the law of signs_ as it applies to fractions. Except for this,
fractions in algebra are treated exactly the same as they are in
arithmetic.

  1. Reduce to lowest terms:

    (_a_) 32/24;

    (_b_) (a^6 - x^6)/(a^4 - x^4);

    (_c_) [(a + b)^2 - (c + d)^2]/[(a + c)^2 - (b + d)^2].  (_M. I. T._)

  2. Reduce to a mixed expression:

    (_a_) 756/11;

    (_b_) (a^3 + b^3)/(a - b).

  3. Reduce to an improper fraction:

    (_a_) 45-1/8;

    (_b_) 9-11/12 qt.;

    (_c_) a^2 - ab + b^2 - (b^3)/(a + b).

Add:

  4. 5/18 + 7/9 + 11/16 + 5/8.

  5. 5/(1 + 2x) - (3x)/(1 - 2x) + (4 - 13x)/(4x^2 - 1).

  6. 1/[x(x - a)(x - b)] + 1/[a(a - x)(a - b)] + 1/[b(b - x)(b - a)].

Multiply:

  7. 72/121 × 55/56 × 77/90.

  8. (b - y)/(a^3 + y^3) × (ca + cy)/(b^2 + by)
        × (b^6 + y^6)/(b^2 + y^2) × b/c.

Divide:

  9. (12/25) ÷ (6/50).

  10. [1 - (ab)/(a^2 - ab + b^2)] [1 - (ab)/(a^2 + 2ab + b^2)]
          ÷ (a^3 - b^3)/(a^3 + b^3).                           (_Yale._)

  11. [(x^4 - y^4)/(x^2 - y^2) ÷ (x + y)/(x^2 - xy)]
          ÷ [(x^2 + y^2)/(x - y) ÷ (x + y)/(xy - y^2)].   (_Sheffield._)

Simplify:

  12. [(4y)/x - (15y^2)/(x^2) + 4] ÷ [4 - (16y)/x + (15y^2)/(x^2)]
          × [3 - (4x + 20y)/(2x + 5y)].

~Reference:~ The chapter on Fractions in any algebra.




COMPLEX FRACTIONS AND FRACTIONAL EQUATIONS


Define a complex fraction.

Simplify:

  1. (3/7 + 4/5)/(2 - 3/7 · 4/5).

  2. (2 - 3/2 + 2/3)/(5 - 2/3 + 3/2).

  3. 2 - 2/(1 - 1/[1 - 1/(1 + 1/2)]).

  4. a/(b^2) - a/[b^2 + (cb)/(a - c/b)].                    (_Harvard._)

  5. If m = 1/(a + 1), n = 2/(a + 2), p = 3/(a + 3), what is the value
     of m/(1 - m) + n/(1 - n) + p/(1 - p)?            (_Univ. of Penn._)

  6. Simplify the expression
     {x + y - 1/[x + y - xy/(x + y)]}(x^3 - y^3)/(x^2 - y^2).
                                                            (_Cornell._)

  7. Simplify [1 - (2xy)/((x + y)^2)]/[1 + (2xy)/((x - y)^2)]
                  ÷ {(1 - y/x)/(1 + y/x)}^2.

  8. Solve (7y + 9)/4 - [y - (2y - 1)/9] = 7.

  9. Solve 2-1/3 - (2/5)(x^2 + 3) = (10x)/3 + 1 - (2x^2)/5.

  10. How much water must be added to 80 pounds of a 5 per cent salt
      solution to obtain a 4 per cent solution?                (_Yale._)

~Reference:~ See Complex Fractions, and the first part of the chapter on
               Fractional Equations in any algebra.




FRACTIONAL EQUATIONS


  1. Solve for each letter in turn 1/b = 1/p + 1/q.

  2. Solve and check:

     (5x + 2)/3 - (3 - (3x - 1)/2) = (3x + 19)/2 - ((x + 1)/6 + 3).

  3. Solve and check:

     (1/2)(x - a/3) - (1/3)(x - a/4) + (1/4)(x - a/5) = 0.

  4. Solve (after looking up the special _short_ method):

     (3x - 1)/30 + (4x - 7)/15
         = x/4 - (2x - 3)/(12x - 11) + (7x - 15)/60.

  5. Solve by the special _short_ method:

     1/(x - 2) - 1/(x - 3) = 1/(x - 4) - 1/(x - 5).

  6. At what time between 8 and 9 o'clock are the hands of a watch
     (_a_) opposite each other? (_b_) at right angles? (_c_) together?

     Work out (_a_) and state the equations for (_b_) and (_c_).

  7. The formula for converting a temperature of F degrees Fahrenheit
     into its equivalent temperature of C degrees Centigrade is
     C = (5/9)(F - 32). Express F in terms of C, and compute F for the
     values C = 30 and C = 28.         (_College Entrance Exam. Board._)

  8. What is the price of eggs when 2 less for 24 cents raises the price
     2 cents a dozen?                                          (_Yale._)

  9. Solve 2/(x - 2) + 2/(4 - x^2) = 5/(x + 2).

~Reference:~ The Chapter on Fractional Equations in any algebra. Note
               particularly the special _short_ methods, usually given
               about the middle of the chapter.




SIMULTANEOUS EQUATIONS


NOTE. Up to this point each topic presented has reviewed to some
extent the preceding topics. For example, factoring reviews the
special rules of multiplication and division; H. C. F. and L. C. M.
review factoring; addition and subtraction of fractions and
fractional equations review H. C. F. and L. C. M., etc. From this
point on, however, the interdependence is not so marked, and
miscellaneous examples illustrating the work already covered will
be given very frequently in order to keep the whole subject fresh
in mind.

  1. Solve by three methods--addition and subtraction, substitution,
     and comparison:   { 5x + y = 11,
                       { 3x + 2y = 1.

Solve and check:

  2. { 12R_1 - 11R_2 = b + 12c,
     { R_1 + R_2 = 2b + c.

  3. { (r - s)/2 = 25/6 - (r + s)/3,
     { (r + s - 9)/2 - (s - r - 6)/3 = 0.

  4. One half of A's marbles exceeds one half of B's and C's together by
     2; twice B's marbles falls short of A's and C's together by 16; if
     C had four more marbles, he would have one fourth as many as A and
     B together. How many has each?          (_College Entrance Board._)

  5. The sides of a triangle are a, b, c. Calculate the radii of the
     three circles having the vertices as centers, each being tangent
     externally to the other two.                           (_Harvard._)

  6. Solve { 2x + 3y = 7, x - y = 1 } graphically; then solve
     algebraically and compare results. (Use coördinate or squared
     paper.)

Factor:

  7. x^4 + 4.

  8. 2d^(10) - 1024d.

  9. 2(x^3 - 1) - 7(x^2 - 1).

~References:~ The chapters on Simultaneous Equations and Graphs in any
                algebra.




SIMULTANEOUS EQUATIONS AND INVOLUTION


  1. Solve    (3/4)x - (5/3)y = 11-1/2,
              (5/8)x - (3/2)y = 10-1/4.

     Look up the method of solving when the unknowns are in the
     denominator. Should you clear of fractions?

  2. Solve    1/x - 1/y - 1/z = 1/a,
              1/y - 1/z - 1/x = 1/b,
              1/z - 1/x - 1/y = 1/c.

  3. Solve graphically and algebraically
              2x - y = 4,
              2x + 3y = 12.

  4. Solve graphically and algebraically
              3x + 7y = 5,
              8x + 3y = -18.

Review:

  5. The squares of the numbers from 1 to 25.

  6. The cubes of the numbers from 1 to 12.

  7. The fourth powers of the numbers from 1 to 5.

  8. The fifth powers of the numbers from 1 to 3.

  9. The binomial theorem laws. (See Involution.)

Expand: (Indicate first, then reduce.)

  10. (b + y)^7.

  11. [(2a)/3 - 1]^5.

  12. (x^2 + 2a)^5.

  13. (x - y + 2z)^3.

  14. A train lost one sixth of its passengers at the first stop, 25 at
      the second stop, 20% of the remainder at the third stop, three
      quarters of the remainder at the fourth stop; 25 remain. What was
      the original number?                                  (_M. I. T._)

~References:~ The chapter on Involution in any algebra. Also the
                references on the preceding page.




SQUARE ROOT


Find the square root of:

  1. 1 + 16m^6 - 40m^4 + 10m - 8m^3 + 25m^2.

  2. (a^2)/(x^2) + (6a)/x + 11 + (6x)/a + (x^2)/(a^2).

  3. Find the square root to three terms of x^2 + 5.

  4. Find the square root of 337,561.

  5. Find the square root of 1823.29.

  6. Find to four decimal places the square root of 1.672.
                                                          (_Princeton._)

  7. Add 2/[(x - 1)^3] + 1/[(1 - x)^2] - 2/(1 - x) - 1/x.

  8. Find the value of:
       (64^(1/3) · 12)/24 ÷ 2 × 3 - (2 · 7^2)/(14) ÷ 7
           × 1 + (1^(1/3) · 1^7)/(1 · 1^2) - 4 · 0.

  9. Simplify [(x + y)^5 + (x - y)^5][(x + y)^5 - (x - y)^5].

  10. Solve by the short method:
        5/(7 - x) - [(2-1/4)x - 3]/4 - (x + 11)/8 + (11x + 5)/16 = 0.

  11. It takes 3/4 of a second for a ball to go from the pitcher to the
      catcher, and 1/2 of a second for the catcher to handle it and get
      off a throw to second base. It is 90 feet from first base to
      second, and 130 feet from the catcher's position to second. A
      runner stealing second has a start of 13 feet when the ball leaves
      the pitcher's hand, and beats the throw to the base by 1/8 of a
      second. The next time he tries it, he gets a start of only 3-1/2
      feet, and is caught by 6 feet. What is his rate of running, and
      the velocity of the catcher's throw?                  (_Cornell._)

~Reference:~ The chapter on Square Root in any algebra.




THEORY OF EXPONENTS


Review the proofs, for positive integral exponents, of:

    I. a^m × a^n = a^(m + n).

   II. (a^m)/(a^n) = a^(m - n).

  III. (a^m)^n = a^(mn).

   IV. [a^(mn)]^(1/n) = a^m.

    V. [a/b]^n = (a^n)/(b^n).

   VI. (abc)^n = a^n b^n c^n.


~To find the meaning of a fractional exponent.~

Assume that Law I holds for _all_ exponents.

If so, a^(2/3) · a^(2/3) · a^(2/3) = a^(6/3) = a^2.

Hence, a^(2/3) is _one of the three equal factors_ (hence the cube root)
of a^2.

    Therefore a^(2/3) = [a^2]^(1/3).

In the same way,

    a^(4/5) · a^(4/5) · a^(4/5) · a^(4/5) · a^(4/5) = a^(20/5) = a^4.

Hence, a^(4/5) is _one of the five equal factors_ (hence the fifth root)
of a^4.

    Therefore a^(4/5) = [a^4]^(1/5).

In the same way, in general, a^(p/q) = [a^p]^(1/q).

Hence, _the numerator of a fractional exponent indicates the power, the
denominator indicates the root_.


~To find the meaning of a zero exponent.~

Assume that Law II holds for _all_ exponents.

If so, (a^m)/(a^m) = a^(m - m) = a^0. But by division, (a^m)/(a^m) = 1.

    Therefore a^0 = 1. Axiom I.


~To find the meaning of a negative exponent.~

Assume that Law I holds for _all_ exponents.

If so, a^m × a^(-m) = a^(m - m) = a^0 = 1.

Hence, a^m × a^(-m) = 1.

    Therefore a^(-m) = 1/(a^m).

Rules:

    _To multiply quantities having the same base, add exponents._

    _To divide quantities having the same base, subtract exponents._

    _To raise a quantity to a power, multiply exponents._

    _To extract a root, divide the exponent of the power by the
    index of the root._

  1. Find the value of 3^2 - 5 × 4^0 + 8^(-2/3) + 1^(2/5).

  2. Find the value of 8^(-2/3) + 9^(3/2) - 2^(-2) + 1^(-2/5) - 7^0.

Give the value of each of the following:

  3. (3^0)/5, 3/(5^0), (3^0)/(5^0),
     3^0 × 5, 3 × 5^0, 3^0 × 5^0, 3^0 + 5^0, 3^0 - 5^0.

  4. Express 7^0 as some power of 7 divided by itself.

Simplify:

  5. 16^(1/3) · 2^(1/2) · 32^(5/6).     (Change to the same base first.)

  6. [2/(8^(-3))]^(1/5).

  7. [(x^n)^(n + 2)]/[(x^(n + 1))(x^(n - 1))].

  8. (x + 3x^(2/3) - 2x^(1/3))(3 - 2x^(-1/3) + 4x^(-2/3)).

  9. [(a^2b)/(c^2d)]^(1/2) × [(c^3d)/(ab^3)]^(1/3)
         × [(a^(1/3)c)/(b^(1/4)d^(5/12))]^2.

  10. [(a^(-4))/(b^(-2)c)]^(-3/4)
          × [(a^(-1)b[c^(-3)]^(1/2))/(ab^(-1))]^(1/2).

  11. [([a^2]^(1/3))/([b^(-1)]^(1/4)) · ([c^(-3)]^(1/2))/(a^(1/3))
          · (b^(-1/4)a^(1/3))/(c^(-1))]^(-6).

~Reference:~ The chapter on Theory of Exponents in any algebra.


Solve for x:

  1. x^(2/3) = 4.

  2. x^(-3/4) = 8.

Factor:

  3. x^(2/3) - 9.

  4. x^(3/5) + 27.

  5. x^(2a) - y^(-6).

  6. a^(1/3) x^(1/2) - 3a^(1/3) + 5x^(1/2) - 15.

  7. Find the H. C. F. and L. C. M. of
     a^2 + a^(3/2) b^(1/2) + a^(1/2) b^(3/2) - b^2,
     a^2 - a^(3/2) b^(1/2) - a^(1/2) b^(3/2) - b^2.

  8. Simplify the product of:
     (ayx^(-1))^(1/2), (bxy^(-2))^(1/3),
     and (y^2a^(-2)b^(-2))^(1/4).                         (_Princeton._)

  9. Find the square root of:
     25a^(4/3)b^(-3) - 10a^(2/3)b^(-3/2) - 49
         + 10a^(-2/3)b^(3/2) + 25a^(-4/3)b^3.

  10. Simplify [(2^(n + 2))/(4^(-n)) ÷ (8^n)/(2^3)]^(1/5).

  11. Find the value of
      (7 · 13^0 ÷ 7)/(21^0) + 3^0 × (4^0 · 7^0)/[(7a + b)^0] + 8^(-2/3).

  12. Express as a power of 2: 8^3; 4^5; 4^3 · 8^(2/3) · 16^(3/4).

  13. Simplify
      {[(x^(a + 1))/(x^(1 - a))]^a
          ÷ [(x^a)/(x^(1 - a))]^(a - 1)}^(1/(3a - 1)).

  14. Simplify
      [(x^(5/2) y^(4/3))/(z^(-5/4)) · (z^4)/(x^(-3) y^(-5/3))
          ÷ (y^(-2) z^(1/4))/(x^(-1/2))]^(1/5).

  15. Expand (a^(1/2) + b^(1/3))^4, writing the result with fractional
      exponents.

~Reference:~ The chapter on Theory of Exponents in any algebra.




RADICALS


  1. Review all definitions in Radicals, also the methods of
     transforming and simplifying radicals. When is _a radical in its
     simplest form_?

  2. Simplify (to simplest form):
       [2/3]^(1/2);
       [1/11]^(1/2);
       [3/5]^(1/3);
       3[5/6]^(1/2);
       (2a/b)[(8b^2)/(27a)]^(1/2);
       [5/(x^n)]^(1/2n);
       (a + b)^2 [(-a^4)/((a + b)^5)]^(1/3);
       27^(1/2);
       [54]^(1/3);
       -5[125^(1/2)].

  3. Reduce to entire surds:
       2[3^(1/2)];
       2[3^(1/4)];
       6[2^(1/3)];
       a[[b^2]^(1/n)];
       -3[2^(1/3)];
       3a[[(a + 2)/(6a^2)]^(1/3)];
       (a + 2y)[(a - 2y)/(a + 2y)]^(1/2).

  4. Reduce to radicals of lower order (or simplify indices):
       [a^2]^(1/4);
       [a^3]^(1/6);
       [27a^3]^(1/6);
       [81 a^4 x^8]^(1/12);
       [9x^2 y^4 z^10]^(1/2n).

  5. Reduce to radicals of the same degree (order, or index):
       7^(1/2) and [11]^(1/3);
       5^(1/3) and 3^(1/4);
       7^(1/6) and 3^(1/2);
       [x^m]^(1/n) and [x^n]^(1/m);
       [c^y]^(1/x), [c^z]^(1/y), and [c^x]^(1/z).

  6. Which is greater, 3^(1/2) or 4^(1/3)? [23]^(1/3) or 2[2^(1/2)]?

  7. Which is greatest, 3^(1/2), 5^(1/3), or 7^(1/4)? Give work and
     arrange in descending order of magnitude.

Collect:

  8. 128^(1/2) - 2[50^(1/2)] + 72^(1/2) - 18^(1/2).

  9. 2[5/3]^(1/2) + (1/6)60^(1/2) + 15^(1/2) + [3/5]^(1/2).

  10. [(m - n)^2a]^(1/2) + [(m + n)^2a]^(1/2) - [am^2]^(1/2)
        + [a(n - m)^2]^(1/2) - a^(1/2).

  11. A and B each shoot thirty arrows at a target. B makes twice as
      many hits as A, and A makes three times as many misses as B.
      Find the number of hits and misses of each.      (_Univ. of Cal._)

~Reference:~ The chapter on Radicals in any algebra (first part of the
               chapter).


The most important principle in Radicals is the following:

    (ab)^(1/n) = a^(1/n) b^(1/n).

    Hence [ab]^(1/n) = a^(1/n) · b^(1/n).

    Or, a^(1/n) · b^(1/n) = [ab]^(1/n).

From this also ([ab]^(1/n))/(a^(1/n)) = b^(1/n).


Multiply:

  1. 2[4^(1/3)] by 3[6^(1/3)].

  2. 2^(1/2) by 3^(1/3).

  3. 2^(1/4) by 4^(1/6).

  4. [a + x^(1/2)]^(1/2) by [a - x^(1/2)]^(1/2).

  5. 2^(1/2) + 3^(1/2) - 5^(1/2) by 2^(1/2) - 3^(1/2) + 5^(1/2).

  6. -p/2 + ([p^2 - 4q]^(1/2))/2 by -p/2 - ([p^2 - 4q]^(1/2))/2.

Divide:

  7. 27^(1/2) by 3^(1/2).

  8. 4[18^(1/2)] by 5[32^(1/2)].

  9. 3[12]^(1/3) by 6^(1/2).

  10. 3^(1/2) by 3^(1/4).

  11. 6[105^(1/2)] + 18[40^(1/2)] - 45[12^(1/2)] by 3[15^(1/2)].
                                                     (_Short division._)

  12. 10[18]^(1/3) - 4[60]^(1/3) + 5[100]^(1/3) by 3[30]^(1/3).

Rationalize the denominator:

  13. 2/(3^(1/2));
      7/(7^(1/2));
      5/(2[5^(1/2)]);
      3/([a^2]^(1/5));
      4/([a^3]^(1/7)).

  14. 2/(2^(1/2)) + 3^(1/2));
      (a^(1/2) + b^(1/2))/(a^(1/2) - b^(1/2));
      3/(3 - 3^(1/2)).

  15. [3^(1/2) + 2^(1/2)]/[6^(1/2) + 3^(1/2) - 2^(1/2)].

Review the method of finding the square root of a binomial surd. (By
inspection preferably.) Then find square root of:

  16. 5 + 2[6^(1/2)].

  17. 17 - 12[2^(1/2)].

  18. 7 - 33^(1/2).

~Reference:~ The chapter on Radicals in any algebra, beginning at
               Addition and Subtraction of Radicals.




MISCELLANEOUS EXAMPLES, ALGEBRA TO QUADRATICS


Results by inspection, examples 1-10.

Divide:

  1. (x^(5/17) + y^(5/17))/(x^(1/17) + y^(1/17)).

  2. (x - y)/(x^(1/3) - y^(1/3)).

  3. (m^2 + n^2)/(m^(2/3) + n^(2/3)).

  4. (x - y^2)/(x^(1/3) - [y^2]^(1/3)).

Multiply:

  5. [a^(-3/4) + 2/(m^(1/2))]^2.

  6. (K^(-2/7) - g^(-11/25))^2.

  7. (r^(2s) + l^(-3m))(r^(2s) - l^(-3m)).

  8. [a^(-2) + b^(-3) - 1/(c^2)]^2.

  9. (3K^x + 4t^(-3))(3K^x - 7t^(-3)).

  10. (2y^(2/7) - 40K^3)(3y^(2/7) + 55K^3).

Factor:

  11. x^(2/3) - 64.

  12. y^(3/5) + 27.

  13. b^(3/2) - 8m^(-1).

  14. 3p - 8p^(1/2) - 35.

Factor, using radicals instead of exponents:

  15. 60 - 7[3b^(1/2)] - 6b.

  16. 15m - 2[[mn]^(1/2)] - 24n.

  17. a - b (factor as difference of two squares).

  18. a - b (factor as difference of two cubes).

  19. a - b (factor as difference of two fourth powers).

  20. Find the H. C. F. and L. C. M. of
        x^2 + xy^(1/2) - 2y,
        2x^2 + 5xy^(1/2) + 2y,
        2x^2 - xy^(1/2) - y.

  21. Solve (short method)
        (x - 7)/(x - 8) - (x - 8)/(x - 9)
          = (x - 4)/(x - 5) - (x - 5)/(x - 6).

  22. Simplify
        (ab/c + bc/a + ca/b)/(a/bc + b/ca + c/ab)
          × [((a + b + c)^2)/(ab + bc + ca) - 2].         (_Princeton._)


  1. Solve for p: 2^(p - 3) = 128.

  2. Solve for t: t^(3/2) = -27.

  3. Find the square root of 8114.4064. What, then, is the square root
     of .0081144064? of 811440.64? From any of the above can you
     determine the square root of .081144064?

  4. The H. C. F. of two expressions is a(a - b), and their L. C. M. is
     a^2b(a + b)(a - b). If one expression is ab(a^2 - b^2), what is the
     other?

  5. Solve (short method):
       5/(7 - x) - [(2-1/4)x - 3]/4 - (x + 11)/8 + (11x + 5)/16 = 0.

  6. Solve
       2/m - 3/n + 10/p = -3,
       4/m + 5/p + 6/n = 15,
       1/m - 1/n + 5/p = -1/2.

  7. Simplify
       21[2/3]^(1/2) - 5[4/5]^(1/2) + 6[4-1/6]^(1/2)
         - 10[3-1/5]^(1/2) + (40/3)[11-1/4]^(1/2).

  8. Does [16 × 25]^(1/2) = 4 × 5? Does [16 + 25]^(1/2) = 4 + 5?

  9. Write the fraction 5/(4 + 2[3^(1/2)]) with rational denominator,
     and find its value correct to two decimal places.

  10. Simplify
      [{([p + [p^2 - q]^(1/2)]/2)^(1/2)
          + ([p - [p^2 - q]^(1/2)]/2)^(1/2)}^2]/[p + q^(1/2)].
                                                          (_Princeton._)


  1. Rationalize the denominator of
     {6^(1/2) + 3^(1/2) - 3[2^(1/2)]}/{6^(1/2) - 3^(1/2) + 3[2^(1/2)]}.
                                                       (_Univ. of Cal._)

  2. Simplify [2^(n + 4) - 2(2^n)]/[2(2^(n + 3))].    (_Univ. of Penn._)

  3. Find the value of [1 + 8^(-x/3)]/[(8x)^(1/2) + 10^(x - 2)],
     when x = 2.                                            (_Cornell._)

  4. Find the value of x if
       x^(6/5) = y^4,
       y^(2/3) = 9.                                         (_M. I. T._)

  5. A fisherman told a yarn about a fish he had caught. If the fish
     were half as long as he said it was, it would be 10 inches more
     than twice as long as it is. If it were 4 inches longer than it
     is, and he had further exaggerated its length by adding 4 inches,
     it would be 1/5 as long as he now said it was. How long is the
     fish, and how long did he first say it was?            (_M. I. T._)

  6. The force _P_ necessary to lift a weight _W_ by means of a certain
     machine is given by the formula

       P = a + bW,

     where _a_ and _b_ are constants depending on the amount of friction
     in the machine. If a force of 7 pounds will raise a weight of 20
     pounds, and a force of 13 pounds will raise a weight of 50 pounds,
     what force is necessary to raise a weight of 40 pounds? (First
     determine the constants _a_ and _b_.)                  (_Harvard._)

  7. Reduce to the simplest form:
     [[4/[2^(n + 2)]]^(1/n);
     [ax(a^(-1)x - ax^(-1))]/[x^(2/3) - a^(2/3)].

  8. Determine the H. C. F. and L. C. M. of (xy - y^2)^3 and y^3 - x^2y.
                                             (_College Entrance Board._)


  1. Simplify (a - 8m)/(a^(1/3) - 2m^(1/3)) - 2a^(1/3)m^(1/3).

  2. Simplify, writing the result with rational denominator:
     ([a^(1/2) + (1)/(x^(-1/2))]^2 - [(1)/(a^(-1/2)) - x^(1/2)]^2)
         / (x + [a^2 + x^2]^(1/2)).                         (_M. I. T._)

  3. Find [7 - 48^(1/2)]^(1/2).

  4. Expand ([a^3]^(1/2) - [b^5]^(1/2))^5.

  5. Expand and simplify (1 - 2[3^(1/2)] + 3[2^(1/2)])^2.

  6. Solve the simultaneous equations
       x ^(-1/2) + 2y^(-1/2) = 7/6,
       2x^(-1/2) - y^(-1/2) = 2/3.                             (_Yale._)

  7. Find to three places of decimals the value of
         {[(a + b)^(-1/3)]/[(11a + b^2)^(1/6)]
             · [({a^3 - b^3)^(-1/2)]/[(a - b)^(1/2)]}^(1/2),
     when a = 5 and b = 3.                                 (_Columbia._)

  8. Show that (10 - 4[5^(1/2)])/(5 + 3[5^(1/2)]) is the negative of
     the reciprocal of (10 + 4[5^(1/2)])/(5 - 3[5^(1/2)]).
                                                           (_Columbia._)

  9. Solve and check {5}/{[3x + 2]^(1/2)} = [3x + 2]^(1/2) + [3x - 1]^(1/2).

  10. Assuming that when an apple falls from a tree the distance
      (S meters) through which it falls in any time (t seconds) is
      given by the formula S = (1/2)gt^2 (where g = 9.8), find to two
      decimal places the time taken by an apple in falling 15 meters.
                                             (_College Entrance Board._)


Excellent practice may be obtained by solving the ordinary formulas used
in arithmetic, geometry, and physics _orally, for each letter in turn_.

ARITHMETIC

    p = br
    i = prt
    a = p + prt

GEOMETRY

    K = (1/2) bh
    K = bh
    K = (a^2)/4 3^(1/2)
    K = (1/2) (b + b') h
    K = [pi] R^2
    C = 2 [pi] R
    K = [pi] R L
    S = 4 [pi] R^2
    V = [pi] R^2 H
    V = (1/3) [pi] R^2 H
    V = (4/3) [pi] R^3
    S = ([pi] R^2 E)/(180)
    C/(C') = R/(R')
    K/(K') = (R^2)/(R'^2)

PHYSICS

    v = gt
    s = (1/2) gt^2
    s = (v^2)/(2g)
    C = E/R
    E = (wv^2)/(2g)
    e = (4Pl^3)/(bh^3 m)
    E = (mv^2)/(2)
    t = [pi] [l/g]^(1/2)
    F = (mV^2)/(r)
    mh = (mv^2)/(2g)
    R = gs/(g + s)
    E = (4n^2l^2w)/(g)
    C = (5/9)(F - 32)




QUADRATIC EQUATIONS


  1. Define a quadratic equation; a pure quadratic; an affected (or
     complete) quadratic; an equation in the quadratic form.

  2. Solve the pure quadratic (7)/(3S^2) - (11)/(9S^2) = 5/6.

Review the first (or usual) method of completing the square. Solve by it
the following:

  3. x^2 + 10x = 24.

  4. 2x^2 - 5x = 7.

  5. (x - 1)/2 + 2/(x - 1) = 2-1/2.

  6. ax^2 + bx + c = 0.

Review the solution by factoring. Solve by it the following:

  7. x^2 + 8x + 7 = 0.

  8. 24x^2 = 2x + 15.

  9. 3 = 10x - 3x^2.

  10. -7 = 6x - x^2.

Solve, by factoring, these equations, which are not quadratics:

  11. x^4 = 16.

  12. x^3 = 8.

  13. x^3 = x.

Review the solution by formula. Solve by it the following:

  14. 5x^2 - 6x = 8.

  15. (1/2)(x + 1) - (x/3)(2x - 1) = -12.

  16. x^2 + 4ax = 12a^2.

  17. 3x^2 = 2rx + 2r^2.

Solve graphically:

  18. x^2 - 2x - 8 = 0.

  19. x^2 + x - 2 = 0.

~Reference:~ The chapter on Quadratic Equations in any algebra (first
               part of the chapter).


  1. Solve by three methods--formula, factoring, and completing the
     square: x^2 + 10x = 24.

Review equations in the quadratic form and solve:

  2. x^4 - 5x^2 = -4.

  3. 2[x^(-2)]^(1/3) - 3[x^(-1)]^(1/3) = 2.

  4. (x + 3)/(x - 3) + 6 = 5[(x + 3)/(x - 3)]^(1/2).
     (Let y = [(x + 3)/(x - 3)]^(1/2) and substitute.)

  5. 3x^2 - 4x + 2[3x^2 - 4x - 6]^(1/2) = 21.

  6. x^2 + 5x - 5 = (6)/(x^2 + 5x).

Solve and check:

  7. [x + 7]^(1/2) + [3x - 2]^(1/2) = (4x + 9)/([3x - 2]^(1/2)).

  8. [x^2 - 5]^(1/2) + 6/[[x^2 - 5]^(1/2)] = 5.

  9. (10w)/([10w - 9]^(1/2)) - [10w + 2]^(1/2) = 2/([10w - 9]^(1/2)).

Give results by inspection:

  10. (a^(1/2) + b^(1/2))(a^(1/2) - b^(1/2)).

  11. ([10 + 19^(1/2)]^(1/2))([10 - 19^(1/2)]^(1/2)).

  12. How many gallons each of cream containing 33% butter fat and
      milk containing 6% butter fat must be mixed to produce 10
      gallons of cream containing 25% butter fat?

  13. I have $6 in dimes, quarters, and half-dollars, there being 33
      coins in all. The number of dimes and quarters together is ten
      times the number of half-dollars. How many coins of each kind
      are there?                             (_College Entrance Board._)

~Reference:~ The last part of the chapter on Quadratic Equations in
               any algebra.




THE THEORY OF QUADRATIC EQUATIONS


~I. To find the sum and the product of the roots.~

The general quadratic equation is

        ax^2 + bx + c = 0.                                   (1)

    Or, x^2 + (b/a)x + c/a = 0.                              (2)

To derive the formula, we have by transposing

    x^2 + (b/a)x = -c/a.

Completing the square,

    x^2 + (b/a)x + [b/2a]^2 = (b^2)/(4a^2) - c/a = (b^2 - 4ac)/(4a^2).

Extracting square root, x + b/2a = [±[b^2 - 4ac]^(1/2)]/(2a).

Transposing, x = -b/2a ± [[b^2 - 4ac]^(1/2)]/(2a).

Hence, x = [-b ± [b^2 - 4ac]^(1/2)]/(2a).

These two values of x we call _roots_.

For convenience represent them by r_1 and r_2.

    Hence,        r_1 = -b/2a + [[b^2 - 4ac]^(1/2)]/(2a).
                  r_2 = -b/2a - [[b^2 - 4ac]^(1/2)]/(2a).
            ---------------------------------------------
    Adding, r_1 + r_2 = -(2b)/(2a) = -b/a.                   (3)

    Also,            r_1 = -b/2a + [[b^2 - 4ac]^(1/2)]/(2a).
                     r_2 = -b/2a - [[b^2 - 4ac]^(1/2)]/(2a).
                 -------------------------------------------
    Multiplying, r_1 r_2 = (b^2)/(4a^2) - (b^2 - 4ac)/(4a^2)
                         = (b^2 - b^2 + 4ac)/(4a^2)
                         = (4ac)/(4a^2) = c/a.               (4)

Hence we have shown that

        r_1 + r_2 = -b/a,
    and r_1 r_2 = c/a.

Or, referring to equation (2) above, we have the following rule:

    _When the coefficient of x^2 is unity, the sum of the roots is
    the coefficient of x with the sign changed; the product of the
    roots is the independent term._

EXAMPLES:

  1. x^2 - 9x + 21 = 0.
         Sum of the roots = 9.
         Products of the roots = 21.

  2. 3x^2 - 7x - 18 = 0.
         Sum of the roots = 7/3.
         Product of the roots = -6.

  3. -21x = 17 - 4x^2.
         Sum of the roots = 21/4.
         Product of the roots = -17/4.


~II. To find the nature or character of the roots.~

  As before, r_1 = -b/2a + [[b^2 - 4ac]^(1/2)]/(2a),
             r_2 = -b/2a - [[b^2 - 4ac]^(1/2)]/(2a).

The [b^2 - 4ac]^(1/2) determines the _nature_ or _character_ of the
roots; hence it is called the _discriminant_.

~If b^2 - 4ac is positive, the roots are real, unequal, and either
rational or irrational.~

~If b^2 - 4ac is negative, the roots are imaginary and unequal.~

~If b^2 - 4ac is zero, the roots are real, equal, and rational.~

EXAMPLES:

  1. x^2 - 4x + 2 = 0.

     [b^2 - 4ac]^(1/2) = [16 - 8]^(1/2) = 8^(1/2).
     Therefore: The roots are real, unequal, and irrational.

  2. x^2 - 4x + 6 = 0.

     [b^2 - 4ac]^(1/2) = [16 - 24]^(1/2) = -8^(1/2).
     Therefore: The roots are imaginary and unequal.

  3. x^2 - 4x + 4 = 0.

     [b^2 - 4ac]^(1/2) = [16 - 16]^(1/2) = 0^(1/2).
     Therefore: The roots are real, equal, and rational.


~III. To form the quadratic equation when the roots are given.~

Suppose the roots are 3, -7.

  Then, x = 3,                       Or,      x - 3 = 0,
        x = -7.                               x + 7 = 0.
                                     -------------------
  Multiplying to get a quadratic,    (x - 3)(x + 7) = 0.

Or, x^2 + 4x - 21 = 0.

_Or_, use the sum and product idea developed on the preceding page. The
coefficient of x^2 must be unity.

Add the roots and change the sign to get the coefficient of x.

Multiply the roots to get the independent term.

Therefore: The equation is x^2 + 4x - 21 = 0.

In the same way, if the roots are [2 + 3^(1/2)]/7, [2 - 3^(1/2)]/7, the
equation is

    x^2 - (4/7)x + 1/49 = 0.


Find the sum, the product, and the nature or character of the roots of
the following:

  1. x^2 - 7x + 12 = 0.

  2. 9x^2 - 6x + 1 = 0.

  3. x^2 + 2x + 9734 = 0.

  4. 16 + 5/x = 17/(x^2).

  5. (x - 8)/(x - 3) = x.

  6. (x + 7)(x - 6) = 70.

  7. x^2 - x(2)^(1/2) = 3.

  8. pr^2 + qr + s = 0.

Form the equations whose roots are:

  9. 5, -3.

  10. 2/3, 5/3.

  11. c + d, c - d.

  12. -3, -5.

  13. [2 ± -3^(1/2)]/5.

  14. 8/3 + (2/3)37^(1/2), 8/3 - (2/3)37^(1/2).

  15. [-2 ± -2^(1/2)]/2.

  16. Solve x^2 - 3x + 4 = 0. Check by substituting the values of x;
      then check by finding the sum and the product of the roots.
      Compare the amount of labor required in each case.

  17. Solve (x - 3)(x + 2)(x^2 + 3x - 4) = 0.

  18. Is e^(4z) + 2e^(3z) + e^(2z) + 2e^z + 2 + e^(-2z) a perfect
      square?

  19. Find the square root (short method):
      (x^2 - 1)(x^2 - 3x + 2)(x^2 - x - 2).

  20. Solve (1.2x - 1.5)/(1.5) + (.4x + 1)/(.2x - .2) = (.4x + 1)/(.5).

  21. The glass of a mirror is 18 inches by 12 inches, and it has a
      frame of uniform width whose area is equal to that of the glass.
      Find the width of the frame.




OUTLINE OF SIMULTANEOUS QUADRATICS


~Simultaneous Quadratics~

    CASE I.

        One equation linear.
        The other quadratic.
            2x + y = 7,
            x^2 + 2y^2 = 22.

        METHOD: Solve for x as in terms of y, or _vice versa_, in the
                linear and substitute in the quadratic.

    CASE II.

        Both equations homogeneous and of the second degree.
            x^2 - xy + y^2 = 39,
            2x^2 - 3xy + 2y^2 = 43.

        METHOD: Let y = vx, and substitute in both equations.

        ALTERNATE METHOD: Solve for x in terms of y in one equation and substitute in the other.

    CASE III.

        Any two of the quantities
             x + y
             x^2 + y^2
             xy
             x - y
             x^3 + y^3
             x^3 - y^3
             x^2 + xy + y^2
             x^2 - xy + y^2
        given.

            x + y = 5,
            x^2 - xy + y^2 = 7.

        METHOD: Solve for x + y and x - y; then add to get x, subtract
                to get y.

    CASE IV.

        Both equations symmetrical or symmetrical except for sign.
        Usually one equation of high degree, the other of the first
        degree.
            x^5 + y^5 = 242,
            x + y = 2.

        METHOD: Let x = u + v and y = u - v, and substitute in both
                equations.

    ~Special Devices~

          I. Consider some compound quantity like xy, [x - y]^(1/2),
             [xy]^(1/2), x/y, etc., as the unknown, at first. Solve
             for the compound unknown, and combine the resulting
             equation with the simpler original equation.

                x^2 y^2 + xy = 6,
                x + 2y = -5.

         II. Divide the equations member by member. Then solve by
             Case I, II, or III.

                x^3 - y^3 = 152,
                x - y = 2.

        III. Eliminate the quadratic terms. Then solve by Case I,
             II, or III.

                xy + x = 15,
                xy + y = 16.




SIMULTANEOUS QUADRATICS


Solve:

  1. x + y = 7,
     x^2 + 4xy = 57.

  2. 2x^2 = 46 + y^2,
     xy + y^2 = 14.

  3. x^2 + y^2 = 25,
     x + y = 1.

  4. x^4 + y^4 = 2,
     x - y = 2.

  5. x^3 + y^3 = 28,
     x + y = 4.

  6. x^2 y^2 + xy - 12 = 0,
     x + y = 4.

  7. 2xy - x + 2y = 16,
     3xy + 2x - 4y = 10.

  8. (3x - 2y)(2x - 3y) = 26,
     x + 1 = 2y.

  9. 4x^2 + 3xy + 2y^2 = 18,
     3x^2 + 2xy - y^2 = 3.

  10. x^5 + y^5 = 242,
      x + y = 2.

  11. x - y + [x - y]^(1/2) = 6,
      xy = 5.

  12. 4x^2 - x + y = 67,
      3x^2 - 3y = 27.

  13. x - y - [x - y]^(1/2) = 2,
      x^3 - y^3 = 2044.                                        (_Yale._)

  14. x^2 + xy + x = 14,
      y^2 + xy + y = 28.                                  (_Princeton._)

  15. x^2 + y^2 = 13,
      y^2 = 4(x - 2). Plot the graph of each equation.      (_Cornell._)

  16. x^2 + y^2 = xy + 37,
      x + y = xy - 17.                                     (_Columbia._)

_In grouping the answers, be sure to associate each value of x with the
corresponding value of y._

  17. The course of a yacht is 30 miles in length and is in the shape of
      a right triangle one arm of which is 2 miles longer than the
      other. What is the distance along each side?

~Reference:~ The chapter on Simultaneous Quadratics in any algebra.




RATIO AND PROPORTION


  1. Define ratio, proportion, mean proportional, third proportional,
     fourth proportional.

  2. Find a mean proportional between 4 and 16; 18 and 50; 12m^2n
     and 3mn^2.

  3. Find a third proportional to 4 and 7; 5 and 10; a^2 - 9 and a - 3.

  4. Find a fourth proportional to 2, 5, and 4; 35, 20, and 14.

  5. Write out the proofs for the following, stating the theorem in
     full in each case:

     (_a_) The product of the extremes equals etc.

     (_b_) If the product of two numbers equals the product of two other
           numbers, either pair etc.

     (_c_) Alternation.

     (_d_) Inversion.

     (_e_) Composition.

     (_f_) Division.

     (_g_) Composition and division.

     (_h_) In a series of equal ratios, the sum of the antecedents is to
           the sum of the consequents etc.

     (_i_) Like powers or like roots of the terms of a proportion etc.

  6. If x : m :: 13 : 7, write all the possible proportions that can be
     derived from it. [See (5) above.]

  7. Given rs = 161m; write the eight proportions that may be derived
     from it, and quote your authority.

  8. (_a_) What theorem allows you to change any proportion into an
           equation?

     (_b_) What theorem allows you to change any equation into a
           proportion?

  9. If xy = rg, what is the ratio of x to g? of y to r? of y to g?

  10. Find two numbers such that their sum, difference, and the sum of
      their squares are in the ratio 5 : 3 : 51.               (_Yale._)

~Reference:~ The chapter on Ratio and Proportion in any algebra.


An easy and powerful method of proving four expressions in proportion is
illustrated by the following example:

Given a : b = c : d;

prove that 3a^3 + 5ab^2 : 3a^3 - 5ab^2 = 3c^3 + 5cd^2 : 3c^3 - 5cd^2.

Let a/b = r. Therefore a = br.

Also c/d = r. Therefore c = dr.

Substitute the value of a in the first ratio, and c in the second:

Then

  (3a^3 + 5ab^2)/(3a^3 - 5ab^2) = (3b^3r^3 + 5b^3r)/(3b^3r^3 - 5b^3r)
    = [b^3r(3r^2 + 5)]/[b^3r(3r^2 - 5)] = (3r^2 + 5)/(3r^2 - 5).

Also

  (3c^3 + 5cd^2)/(3c^3 - 5cd^2) = (3d^3r^3 + 5d^3r)/(3d^3r^3 - 5d^3r)
    = [d^3r(3r^2 + 5)]/[d^3r(3r^2 - 5)] = (3r^2 + 5)/(3r^2 - 5).

Therefore (3a^3 + 5ab^2)/(3a^3 - 5ab^2) = (3c^3 + 5cd^2)/(3c^3 - 5cd^2).

                                                                Axiom 1.

Or, 3a^3 + 5ab^2 : 3a^3 - 5ab^2 = 3c^3 + 5cd^2 : 3c^3 - 5cd^2.

If a : b = c : d, prove:

  1. a^2 + b^2 : a^2 = c^2 + d^2 : c^2.

  2. a^2 + 3b^2 : a^2 - 3b^2 = c^2 + 3d^2 : c^2 - 3d^2.

  3. a^2 + 2b^2 : 2b^2 = ac + 2bd : 2bd.

  4. 2a + 3c : 2a - 3c = 8b + 12d : 8b - 12d.

  5. a^2 - ab + b^2 : (a^3 - b^3)/a = c^2 - cd + d^2 : (c^3 - d^3)/c.

  6. The second of three numbers is a mean proportional between the
     other two. The third number exceeds the sum of the other two by 20;
     and the sum of the first and third exceeds three times the second
     by 4. Find the numbers.

  7. Three numbers are proportional to 5, 7, and 9; and their sum is 14.
     Find the numbers.                       (_College Entrance Board._)

  8. A triangular field has the sides 15, 18, and 27 rods, respectively.
     Find the dimensions of a similar field having 4 times the area.




~ARITHMETICAL PROGRESSION~


  1. Define an arithmetical progression.

Learn to derive the three formulas in arithmetical progression:

    l = a + (n - 1)d,
    S = (n/2)(a + l),
    S = (n/2)[2a + (n - 1)d].

  2. Find the sum of the first 50 odd numbers.

  3. In the series 2, 5, 8, ···, which term is 92?

  4. How many terms must be taken from the series 3, 5, 7, ···, to
     make a total of 255?

  5. Insert 5 arithmetical means between 11 and 32.

  6. Insert 9 arithmetical means between 7-1/2 and 30.

  7. Find x, if 3 + 2x, 5 + 6x, 9 + 5x are in A. P.

  8. The 7th term of an arithmetical progression is 17, and the 13th
     term is 59. Find the 4th term.

  9. How can you turn an A. P. into an equation?

  10. Given a = -5/3, n = 20, S = -5/3, find d and l.

  11. Find the sum of the first n odd numbers.

  12. An arithmetical progression consists of 21 terms. The sum of the
      three terms in the middle is 129; the sum of the last three terms
      is 237. Find the series. (Look up the short method for such
      problems.)                          (_Mass. Inst. of Technology._)

  13. B travels 3 miles the first day, 7 miles the second day, 11 miles
      the third day, etc. In how many days will B overtake A who started
      from the same point 8 days in advance and who travels uniformly 15
      miles a day?

~Reference:~ The chapter on Arithmetical Progression in any algebra.




~GEOMETRICAL PROGRESSION~

  1. Define a geometrical progression.

Learn to derive the four formulas in geometrical progression:

    { I. l = ar^(n - 1).
    {II. S = (ar^n - a)/(r - 1).

    {III. S = (rl - a)/(r - 1).
    { IV. S_{[infinity]} = (a)/(1 - r).

  2. How many terms must be taken from the series 9, 18, 36, ··· to
     make a total of 567?

  3. In the G. P. 2, 6, 18, ···, which term is 486?

  4. Find x, if 2x - 4, 5x - 7, 10x + 4 are in geometrical progression.

  5. How can you turn a G. P. into an equation?

  6. Insert 4 geometrical means between 4 and 972.

  7. Insert 6 geometrical means between 5/16 and 5120.

  8. Given a = -2, n = 5, l = -32; find r and S.

  9. If the first term of a geometrical progression is 12 and the sum to
     infinity is 36, find the 4th term.

  10. If the series 3-1/3, 2-1/2, ··· be an A. P., find the 97th term.
      If a G. P., find the sum to infinity.

  11. The third term of a geometrical progression is 36; the 6th term is
      972. Find the first and second terms.

  12. Insert between 6 and 16 two numbers, such that the first three of
      the four shall be in arithmetical progression, and the last three
      in geometrical progression.

  13. A rubber ball falls from a height of 40 inches and on each rebound
      rises 40% of the previous height. Find by formula how far it falls
      on its eighth descent.                                   (_Yale._)

~Reference:~ The chapter on Geometrical Progression in any algebra.




~THE BINOMIAL THEOREM~


  1. Review the Binomial Theorem laws. (See Involution.)

Expand:

  2. (b - n)^7.

  3. (x + x^(-1))^5.

  4. [a/x - x/a]^6.

  5. [x/2y - [xy]^(1/2)]^5.

  6. (x^2 - x + 2)^3.

  7. [(2[b^2]^(1/3))/(y) + (3[y^(1/2)])/(b^3)]^4.

  8. (a + b)^n = a^n + na^(n - 1)b + [n(n - 1)]/(1·2) a^(n - 2)b^2
        + [n(n - 1)(n - 2)]/(1·2·3) a^(n - 3)b^3
        + [n(n - 1)(n - 2)(n - 3)]/(1·2·3·4) a^(n - 4) b^4 + ···.

     Show by observation that the formula for the

     (r + 1)th term
        = [n(n - 1)(n - 2)···(n - r + 1)]/[1·2·3·4 ··· r] a^(n - r)b^r.

  9. Indicate what the 97th term of (a + b)^n would be.

  10. Using the expansion of (a + b)^n in (8), derive a formula for the
      rth term by observing how each term is made up, then generalizing.

Using either the formula in (8) or (10), whichever you are familiar
with, find:

  11. The 4th term of [a + 1/a]^(30).

  12. The 8th term of (1 + x[y^(1/2)])^(13).

  13. The middle term of (2a^(3/4) - y[a^(1/3)])^(10).

  14. The term not containing x in [x^3 - 2/x]^(12).

  15. The term containing x^(18) in [x^2 - a/x]^(15).

~Reference:~ The chapter on The Binomial Theorem in any algebra.




~MISCELLANEOUS EXAMPLES, QUADRATICS AND BEYOND~


  1. Solve the equation x^2 - 1.6x - .23 = 0, obtaining the values of
     the roots correct to three significant figures.        (_Harvard._)

  2. Write the roots of (x^2 + 2x)(x^2 - 2x - 3)(x^2 - x + 1) = 0.
                                        (_Sheffield Scientific School._)

  3. Solve
     2[2x + 2]^(1/2) + [2x + 1]^(1/2) = (12x + 4)/([8x + 8]^{1/2}).
                                                               (_Yale._)

  4. Solve the equation V = (H/3)(B + x + [Bx]^(1/2)) for x, taking
     H = 6, B = 8, and V = 28; and verify your result.      (_Harvard._)

  5. Solve  { x : y = 2 : 3,
            { x^2 + y^2 = 5(x + y) + 2.

  6. Solve 2x^2 - 4x + 3[x^2 - 2x + 6]^(1/2) = 15. (_Coll. Ent. Board._)

  7. Find all values of x and y which satisfy the equations:
      { x^(1/2) + y^(1/2) = 4,
      { 1/[[x + 1]^(1/2) - x^(1/2)] - 1/[[x + 1]^(1/2) + x^(1/2)] = y.
                                          (_Mass. Inst. of Technology._)

  8. If [alpha] and [beta] represent the roots of px^2 + qx + r = 0,
     find [alpha] + [beta], [alpha] - [beta], and [alpha][beta] in terms
     of p, q, and r.                                      (_Princeton._)

  9. Form the equation whose roots are 2 + [3]^(1/2) and 2 - [-3]^(1/2).

  10. Determine, without solving, the character of the roots of
      9x^2 - 24x + 16 = 0.                   (_College Entrance Board._)

  11. If a : b = c : d, prove that
      a + b : c + d = [a^2 + b^2]^(1/2) : [c^2 + d^2]^(1/2).
                                             (_College Entrance Board._)

  12. Given a : b = c : d. Prove that
      a^2 + b^2 : (a^3)/(a + b) = c^2 + d^2 : (c^3)/(c + d).
                                                          (_Sheffield._)

  13. The 9th term of an arithmetical progression is 1/6; the 16th term
      is 5/2. Find the first term.                          (_Regents._)


Solve graphically:

  1. x^2 - x - 6 = 0.

  2. x^2 + 3x - 10 = 0.

  3. Find four numbers in arithmetical progression, such that the sum of
     the first two is 1, and the sum of the last two is -19.

  4. What number added to 2, 20, 9, 34, will make the results
     proportional?

  5. Find the middle term of [3a^5 + (b^(3/4))/(2)]^8.

  6. Solve (x + 1)/(3x + 2) = (2x - 3)/(3x - 2) - 1 - 36/(4 - 9x^2).
                                                          (_Princeton._)

  7. A strip of carpet one half inch thick and 29-6/7 feet long is
     rolled on a roller four inches in diameter. Find how many turns
     there will be, remembering that each turn increases the diameter
     by one inch, and that the circumference of a circle equals
     (approximately) 22/7 times the diameter.               (_Harvard._)

  8. The sum of the first three terms of a geometrical progression is
     21, and the sum of their squares is 189. What is the first term?
                                                               (_Yale._)

  9. Find the geometrical progression whose sum to infinity is 4, and
     whose second term is 3/4.

  10. Solve 4x + 4[3x^2 - 7x + 3]^(1/2) = 3x^2 - 3x + 6.

  11. Solve { 2x^2 + 3xy - 5y^2 = 4,
            { 2xy + 3y^2 = -3.

  12. Two hundred stones are placed on the ground 3 feet apart, the
      first being 3 feet from a basket. If the basket and all the stones
      are in a straight line, how far does a person travel who starts
      from the basket and brings the stones to it one by one?


Solve graphically; and check by solving algebraically:

  1. { x^2 + y^2 = 25,
     { x + y = 1.

  2. x^2 - 3x - 18 = 0.

  3. x^2 + 3x - 10 = 0.

Determine the value of m for which the roots of the equation will be
equal: (HINT: See page 40. To have the roots equal, b^2 - 4ac must
equal 0.)

  4. 2x^2 - mx + 12-1/2 = 0.

  5. (m - 1)x^2 + mx + 2m - 3 = 0.

  6. If 2a + 3b is a root of x^2 - 6bx - 4a^2 + 9b^2 = 0, find the other
     root without solving the equation.               (_Univ. of Penn._)

  7. How many times does a common clock strike in 12 hours?

  8. Find the sum to infinity of
     2/(2^(1/2)), 1/(2^(1/2)), 1/(2[2]^(1/2)), ···.

  9. Solve [x/2 + 6/x]^2 - 6[x/2 + 6/x] + 8 = 0.

  10. Find the value of the recurring decimal 2.214214···.

  11. A man purchases a $500 piano by paying monthly installments of $10
      and interest on the debt. If the yearly rate is 6%, what is the
      total amount of interest?

  12. The arithmetical mean between two numbers is 42-1/2, and their
      geometrical mean is 42. Find the numbers.
                                       (_College Entrance Exam. Board._)

  13. If the middle term of [3x - (1)/(2[x^(1/2)])]^4 is equal to the
      fourth term of [2[x^(1/2)] + 1/2x]^7, find the value of x.
                                                            (_M. I. T._)




~PROBLEMS~

~Linear Equations, One Unknown~


  1. A train running 30 miles an hour requires 21 minutes longer to go
     a certain distance than does a train running 36 miles an hour. How
     great is the distance?                                 (_Cornell._)

  2. A man can walk 2-1/2 miles an hour up hill and 3-1/2 miles an hour
     down hill. He walks 56 miles in 20 hours on a road no part of which
     is level. How much of it is up hill?                      (_Yale._)

  3. A physician having 100 cubic centimeters of a 6% solution of a
     certain medicine wishes to dilute it to a 3-1/2% solution. How much
     water must he add? (A 6% solution contains 6% of medicine and 94%
     of water.)                                                (_Case._)

  4. A clerk earned $504 in a certain number of months. His salary was
     increased 25%, and he then earned $450 in two months less time than
     it had previously taken him to earn $504. What was his original
     salary per month?                       (_College Entrance Board._)

  5. A person who possesses $15,000 employs a part of the money in
     building a house. He invests one third of the money which remains
     at 6%, and the other two thirds at 9%, and from these investments
     he obtains an annual income of $500. What was the cost of the
     house?                                                 (_M. I. T._)

  6. Two travelers have together 400 pounds of baggage. One pays $1.20
     and the other $1.80 for excess above the weight carried free. If
     all had belonged to one person, he would have had to pay $4.50. How
     much baggage is allowed to go free?                       (_Yale._)

  7. A man who can row 4-1/3 miles an hour in still water rows
     downstream and returns. The rate of the current is 2-1/4 miles per
     hour, and the time required for the trip is 13 hours. How many
     hours does he require to return?


~Simultaneous Equations, Two and Three Unknowns~

  1. A manual training student in making a bookcase finds that the
     distance from the top of the lowest shelf to the under side of
     the top shelf is 4 ft. 6 in. He desires to put between these four
     other shelves of inch boards in such a way that the book space
     will diminish one inch for each shelf from the bottom to the top.
     What will be the several spaces between the shelves?

  2. A quantity of water, sufficient to fill three jars of different
     sizes, will fill the smallest jar 4 times, or the largest jar twice
     with 4 gallons to spare, or the second jar three times with 2
     gallons to spare. What is the capacity of each jar?       (_Case._)

  3. A policeman is chasing a pickpocket. When the policeman is 80 yards
     behind him, the pickpocket turns up an alley; but coming to the
     end, he finds there is no outlet, turns back, and is caught just as
     he comes out of the alley. If he had discovered that the alley had
     no outlet when he had run halfway up and had then turned back, the
     policeman would have had to pursue the thief 120 yards beyond the
     alley before catching him. How long is the alley?      (_Harvard._)

  4. A and B together can do a piece of work in 14 days. After they have
     worked 6 days on it, they are joined by C who works twice as fast
     as A. The three finish the work in 4 days. How long would it take
     each man alone to do it?                              (_Columbia._)

  5. In a certain mill some of the workmen receive $1.50 a day, others
     more. The total paid in wages each day is $350. An assessment made
     by a labor union to raise $200 requires $1.00 from each man
     receiving $1.50 a day, and half of one day's pay from every man
     receiving more. How many men receive $1.50 a day?      (_Harvard._)

  6. There are two alloys of silver and copper, of which one contains
     twice as much copper as silver, and the other three times as much
     silver as copper. How much must be taken from each to obtain a
     kilogram of an alloy to contain equal quantities of silver and
     copper?                                                (_M. I. T._)

  7. Two automobiles travel toward each other over a distance of 120
     miles. A leaves at 9 A.M., 1 hour before B starts to meet him, and
     they meet at 12:00 M. If each had started at 9:15 A.M., they would
     have met at 12:00 M. also. Find the rate at which each traveled.
                                                            (_M. I. T._)


~Quadratic Equations~

  1. Telegraph poles are set at equal distances apart. In order to have
     two less to the mile, it will be necessary to set them 20 feet
     farther apart. Find how far apart they are now.           (_Yale._)

  2. The distance S that a body falls from rest in t seconds is given by
     the formula S = 16t^2. A man drops a stone into a well and hears
     the splash after 3 seconds. If the velocity of sound in air is 1086
     feet a second, what is the depth of the well?             (_Yale._)

  3. It requires 2000 square tiles of a certain size to pave a hall, or
     3125 square tiles whose dimensions are one inch less. Find the area
     of the hall. How many solutions has the equation of this problem?
     How many has the problem itself? Explain the apparent discrepancy.
                                                            (_Cornell._)

  4. A rectangular tract of land, 800 feet long by 600 feet broad, is
     divided into four rectangular blocks by two streets of equal width
     running through it at right angles. Find the width of the streets,
     if together they cover an area of 77,500 square feet.  (_M. I. T._)

  5. (_a_) The height y to which a ball thrown vertically upward with a
           velocity of 100 feet per second rises in x seconds is given
           by the formula, y = 100x - 16x^2. In how many seconds will
           the ball rise to a height of 144 feet?

     (_b_) Draw the graph of the equation y = 100x - 16x^2.
                                             (_College Entrance Board._)

  6. Two launches race over a course of 12 miles. The first steams 7-1/2
     miles an hour. The other has a start of 10 minutes, runs over the
     first half of the course with a certain speed, but increases its
     speed over the second half of the course by 2 miles per hour,
     winning the race by a minute. What is the speed of the second
     launch? Explain the meaning of the negative answer.
                                        (_Sheffield Scientific School._)

  7. The circumference of a rear wheel of a certain wagon is 3 feet more
     than the circumference of a front wheel. The rear wheel performs
     100 fewer revolutions than the front wheel in traveling a distance
     of 6000 feet. How large are the wheels?                (_Harvard._)

  8. A man starts from home to catch a train, walking at the rate of 1
     yard in 1 second, and arrives 2 minutes late. If he had walked at
     the rate of 4 yards in 3 seconds, he would have arrived 2-1/2
     minutes early. Find the distance from his home to the station.
                                             (_College Entrance Board._)


~Simultaneous Quadratics~

  1. Two cubical coal bins together hold 280 cubic feet of coal, and
     the sum of their lengths is 10 feet. Find the length of each bin.

  2. The sum of the radii of two circles is 25 inches, and the
     difference of their areas is 125[pi] square inches. Find the radii.

  3. The area of a right triangle is 150 square feet, and its hypotenuse
     is 25 feet. Find the arms of the triangle.

  4. The combined capacity of two cubical tanks is 637 cubic feet, and
     the sum of an edge of one and an edge of the other is 13 feet.

     (_a_) Find the length of a diagonal of any face of each cube.

     (_b_) Find the distance from upper left-hand corner to lower
           right-hand corner in either cube.

  5. A and B run a mile. In the first heat A gives B a start of 20 yards
     and beats him by 30 seconds. In the second heat A gives B a start
     of 32 seconds and beats him by 9-5/11 yards. Find the rate at which
     each runs. (_Sheffield._)

  6. After street improvement it is found that a certain corner
     rectangular lot has lost 1/10 of its length and 1/15 of its width.
     Its perimeter has been decreased by 28 feet, and the new area is
     3024 square feet. Find the reduced dimensions of the lot. (_College
     Entrance Board._)

  7. A man spends $539 for sheep. He keeps 14 of the flock that he buys,
     and sells the remainder at an advance of $2 per head, gaining $28
     by the transaction. How many sheep did he buy, and what was the
     cost of each? (_Yale._)

  8. A boat's crew, rowing at half their usual speed, row 3 miles
     downstream and back again in 2 hours and 40 minutes. At full speed
     they can go over the same course in 1 hour and 4 minutes. Find the
     rate of the crew, and the rate of the current in miles per hour.
     (_College Entrance Board._)

  9. Find the sides of a rectangle whose area is unchanged if its length
     is increased by 4 feet and its breadth decreased by 3 feet, but
     which loses one third of its area if the length is increased by 16
     feet and the breadth decreased by 10 feet. (_M. I. T._)




COLLEGE ENTRANCE EXAMINATIONS




~UNIVERSITY OF CALIFORNIA~

ELEMENTARY ALGEBRA


  1. If a = 4, b = -3, c = 2, and d = -4, find the value of:
       (_a_) ab^3 - 3cd^2 + 2(3a - b)(c - 2d).
       (_b_) 2a^3 - 3b^4 + (4c^3 + d^3)(4c^2 + d^2).

  2. Reduce to a mixed number:
       (3a^4 - 4a^3 - 10a^2 + 41a - 28)/(a^2 - 3a + 4).

Simplify:

  3. (a + 2)/(a^2 + 3a - 40) - (b - 2)/(ab - 5b + 3a - 15).

  4. [1 - (2 - 3b - 2c)/(a + 2)]
       ÷ (a^2 - 4c^2 + 9b^2 + 6ab)/(2a^2 + a - 6).

  5. A's age 10 years hence will be 4 times what B's age was 11 years
     ago, and the amount that A's age exceeds B's age is one third of
     the sum of their ages 8 years ago. Find their present ages.

  6. Draw the lines represented by the equations
       3x - 2y = 13 and 2x + 5y = -4,
     and find by algebra the coördinates of the point where they
     intersect.

  7. Solve the equations { bx - ay = b^2 - ab,
                         { y - b = 2(x - 2a).

  8. Solve (2x + 1)(3x - 2) - (5x - 7)(x - 2) = 41.




~COLORADO SCHOOL OF MINES~

ELEMENTARY ALGEBRA


  1. Solve by factoring: x^3 + 30x = 11x^2.

  2. Show that 1 - [(a^2 + b^2 - c^2)/(2ab)]^2 = (a + b + c)(a + b - c)(a - b + c)(b + c - a) ÷ 4a^2b^2.

  3. How many pairs of numbers will satisfy simultaneously the two
     equations
      { 3x + 2y = 7,
      { x + y = 3?

Show by means of a graph that your answer is correct.

What is meant by eliminating x in the above equations by substitution?
by comparison? by subtraction?

  4. Find the square root of 223,728.

  5. Simplify: (_a_) [1/3]^(1/2) + [12]^(1/2) - [3/4]^(1/2).
               (_b_) (-[-3[-4]^(1/2)]^(1/2))^4.

  6. Solve the equation .03x^2 - 2.23x + 1.1075 = 0.

  7. How far must a boy run in a potato race if there are n potatoes in
     a straight line at a distance d feet apart, the first being at a
     distance a feet from the basket?




~COLUMBIA UNIVERSITY~

ELEMENTARY ALGEBRA COMPLETE


TIME: THREE HOURS

Six questions are required; two from Group _A_, two from Group _B_, and
both questions of Group _C_. No extra credit will be given for more than
six questions.


_Group A_

  1. (_a_) Resolve the following into their prime factors:
             (1) (x^2 - y^2)^2 - y^4.
             (2) 10x^2 - 7x - 6.

     (_b_) Find the H. C. F. and the L. C. M. of
             x^3 - 3x^2 + x - 3,
             x^3 - 3x^2 - x + 3.

  2. (_a_) Simplify
             [x/y + y/x - 2]/[1/x + 1/y] + [x/y + y/x + 2]/[1/x - 1/y].

     (_b_) If x : y = (x - z)^2 : (y - z)^2, prove that z is a mean
           proportional between x and y.

  3. A crew can row 10 miles in 50 minutes downstream, and 12 miles in
     an hour and a half upstream. Find the rate of the current and of
     the crew in still water.


_Group B_

  4. (_a_) Determine the values of k so that the equation
             (2 + k)x^2 + 2kx + 1 = 0
           shall have equal roots.

     (_b_) Solve the equations
             x^2 - xy + y^2 = 7,
             2x - 3y = 0.

     (_c_) Plot the following two equations, and find from the graphs
           the approximate values of their common solutions:
             x^2 + y^2 = 25,
             4x^2 + 9y^2 = 144.

  5. Two integers are in the ratio 4 : 5. Increase each by 15, and the
     difference of their squares is 999. What are the integers?

  6. A man has $539 to spend for sheep. He wishes to keep 14 of the
     flock that he buys, but to sell the remainder at a gain of $2 per
     head. This he does and gains $28. How many sheep did he buy, and at
     what price each?


_Group C_

  7. (_a_) Find the seventh term of [a + 1/a]^(13).

     (_b_) Derive the formula for the sum of n terms of an arithmetic
           progression.

  8. A ball falling from a height of 60 feet rebounds after each fall
     one third of its last descent. What distance has it passed over
     when it strikes the ground for the eighth time?




~CORNELL UNIVERSITY~

ELEMENTARY ALGEBRA


  1. Find the H. C. F.:
       x^4 - y^4,
       x^3 - xy^2 + x^2y - y^3,
       x^4 + 2x^2y^2 - 3y^4.

  2. Solve the following set of equations:
       x + y = -1,
       x + 3y + 2z = -4,
       x - y + 4z = 5.

  3. Expand and simplify:
       [2x^3 - 1/x]^7.

  4. An automobile goes 80 miles and back in 9 hours. The rate of speed
     returning was 4 miles per hour faster than the rate going. Find the
     rate each way.

  5. Simplify:
       {[(x + 1)/(x - 1)]^2 - 2 + [(x - 1)/(x + 1)]^2}
         /{[(x + 1)/(x - 1)]^2 - [(x - 1)/(x + 1)]^2}.

  6. Solve for x:
       (2x + 3)/(x - 1) - 6 = 5/(x^2 + 2x - 3).

  7. A, B, and C, all working together, can do a piece of work in
     2-2/3 days. A works twice as fast as C, and A and C together could
     do the work in 4 days. How long would it take each one of the three
     to do the work alone?




~CORNELL UNIVERSITY~

INTERMEDIATE ALGEBRA


  1. Solve the following set of equations:
       x + y = -1,
       2z + 5w = 1,
       x + 3y + 2z = -4,
       x - y + 4z + 4w = 5.

  2. Simplify:
       (_a_) [6 - 20^(1/2)]^(1/2).
       (_b_) [1 + [x^2 + 1]^(1/2)]/[1 + [x^2 + 1]^(1/2) + x^2].

  3. Find, and simplify, the 23d term in the expansion of
       [(2x^2)/(3) - 3/4]^(28).

  4. The weight of an object varies directly as its distance from the
     center of the earth when it is below the earth's surface, and
     inversely as the square of its distance from the center when it is
     above the surface. If an object weighs 10 pounds at the surface,
     how far above, and how far below the surface will it weigh 9
     pounds? (The radius of the earth may be taken as 4000 miles.)

  5. Solve the following pair of equations for x and y:
       x^2 + y^2 = 4,
       x = (1 + 2^(1/2))y - 2.

  6. Find the value of [1 + 8^(-x/3)]/[(8x)^(1/2) + 10^(x - 2)], when
     x = 2.

  7. From a square of pasteboard, 12 inches on a side, square corners
     are cut, and the sides are turned up to form a rectangular box. If
     the squares cut out from the corners had been 1 inch larger on a
     side, the volume of the box would have been increased 28 cubic
     inches. What is the size of the square corners cut out? (See the
     figure on the blackboard.)




~HARVARD UNIVERSITY~

ELEMENTARY ALGEBRA


TIME: ONE HOUR AND A HALF

Arrange your work neatly and clearly, beginning each question on a
separate page.

  1. Simplify the following expression:
     [[1/a + 1/(b + c)]/[1/a - 1/(b + c)] [1 + (b^2 + c^2 - a^2)/(2bc)].

  2. (_a_) Write the middle term of the expansion of (a - b)^14 by the
           binomial theorem.

     (_b_) Find the value of a^7b^7, if
             a = x^(2/7)y^(-3/2) and b = (1/2) x^(-1/7)y^(1/2),
           and reduce the result to a form having only positive
           exponents.

  3. Find correct to three significant figures the negative root of the
     equation
       1 - 2/(x + 1) + 4x/{(x + 1)^2} = 0.

  4. Prove the rule for finding the sum of n terms of a geometrical
     progression of which the first term is a and the constant ratio
     is r.

     Find the sum of 8 terms of the progression 5 + 3-1/3 + 2-2/9 + ···.

  5. A goldsmith has two alloys of gold, the first being 3/4 pure gold,
     the second 5/12 pure gold. How much of each must he take to produce
     100 ounces of an alloy which shall be 2/3 pure gold?




~HARVARD UNIVERSITY~

ELEMENTARY ALGEBRA


TIME: ONE HOUR AND A HALF

  1. Solve the simultaneous equations
       x + y = a + b,
       (y + b)/(x + a) = a/b,
     and verify your results.

  2. Solve the equation x^2 - 1.6x - 0.23 = 0, obtaining the values of
     the roots correct to three significant figures.

  3. Write out the first four terms of (a - b)^7. Find the fourth term
     of this expansion when
       a = [x^(-1) y^(1/2)]^(1/3), b = [9xy^(-4)]^(1/6),
     expressing the result in terms of a single radical, and without
     fractional or negative exponents.

  4. Reduce the following expression to a polynomial in a and b:
       (6a^3 + 7ab^2 + 12b^3)/(3a^2 - 5ab - 4b^2)
         - 1/[3/19b - (5a + 4b)/(19a^2)].

  5. The cost of publishing a book consists of two main items: first,
     the fixed expense of setting up the type; and, second, the running
     expenses of presswork, binding, etc., which may be assumed to be
     proportional to the number of copies. A certain book costs 35 cents
     a copy if 1000 copies are published at one time, but only 19 cents
     a copy if 5000 copies are published at one time. Find (_a_) the
     cost of setting up the type for the book, and (_b_) the cost of
     presswork, binding, etc., per thousand copies.




~HARVARD UNIVERSITY~

ELEMENTARY ALGEBRA


TIME: ONE HOUR AND A HALF

  1. Find the highest common factor and the lowest common multiple of
     the three expressions
       a^4 - b^4; a^3 + b^3; a^3 + 2a^2 b + 2ab^2 + b^3.

  2. Solve the quadratic equation
       x^2 - 1.6x + 0.3 = 0,
     computing the value of the larger root correct to three significant
     figures.

  3. In the expression
       x^2 - 2xy + y^2 - 4[2^(1/2)](x + y) + 8,
     substitute for x and y the values
       x = (u + v + 1)/[2^(1/2)], y = (u - v + 1)/[2^(1/2)],
     and reduce the resulting expression to its simplest form.

  4. State and prove the formula for the sum of the first n terms of a
     geometric progression in which a is the first term and r the
     constant ratio.

  5. A state legislature is to elect a United States senator, a majority
     of all the votes cast being necessary for a choice. There are three
     candidates, A, B, and C, and 100 members vote. On the first ballot
     A has the largest number of votes, receiving 9 more votes than his
     nearest competitor, B; but he fails of the necessary majority. On
     the second ballot C's name is withdrawn, and all the members who
     voted for C now vote for B, whereupon B is elected by a majority of
     2. How many votes were cast for each candidate on the first ballot?




~MASSACHUSETTS INSTITUTE OF TECHNOLOGY~

ALGEBRA A


TIME: ONE HOUR AND THREE QUARTERS

  1. Factor the expressions:
       x^3 + x^2 = 2x. x^3 + x^2 - 4x - 4.

  2. Simplify the expression:
       [1 - (b^2)/(a^2)][1 - (ab - b^2)/(a^2)](a^4)/(a^3 + b^3)
         · (a - b)/(a^2 + b^2).

  3. Find the value of x + [1 + x^2]^(1/2), when
       x = (1/2)[[a/b]^(1/2) - [b/a]^(1/2)].

  4. Solve the equations:
       (7x + 6)/11 + y - 16 = (5x - 13)/2 - (8y - x)/5,
       3(3x + 4) = 10y - 15.

  5. Solve the equations:
        A     +  C     = 2,
       -A + B +  C + D = 1,
       2A - B + 2C + D = 5,
            B      + D = 1.

  6. Two squares are formed with a combined perimeter of 16 inches. One
     square contains 4 square inches more than the other. Find the area
     of each.

  7. A man walked to a railway station at the rate of 4 miles an hour
     and traveled by train at the rate of 30 miles an hour, reaching his
     destination in 20 hours. If he had walked 3 miles an hour and
     ridden 35 miles an hour, he would have made the journey in 18
     hours. Required the total distance traveled.




~MASSACHUSETTS INSTITUTE OF TECHNOLOGY~

ALGEBRA B


TIME: ONE HOUR AND THREE QUARTERS

  1. How many terms must be taken in the series 2, 5, 8, 11, ··· so
     that the sum shall be 345?

  2. Prove the formula x = [-b ± [b^2 - 4ac]^(1/2)]/(2a) for solving the
     quadratic equation ax^2 + bx + c = 0.

  3. Find all values of a for which [\sq]a is a root of x^2 + x + 20 =
     2a, and check your results.

  4. Solve {x^2 + 3y^2 = 10, x - y = 2,} and sketch the graphs.

  5. The sum of two numbers x and y is 5, and the sum of the two middle
     terms in the expansion of (x + y)^3 is equal to the sum of the
     first and last terms. Find the numbers.

  6. Solve x^4 - 2x^3 + 3x^2 - 2x + 1 = 0.

     (HINT: Divide by x^2 and substitute x + 1/x = z.)

  7. In anticipation of a holiday a merchant makes an outlay of $50,
     which will be a total loss in case of rain, but which will bring
     him a clear profit of $150 above the outlay if the day is pleasant.
     To insure against loss he takes out an insurance policy against
     rain for a certain sum of money for which he has to pay a certain
     percentage. He then finds that whether the day be rainy or pleasant
     he will make $80 clear. What is the amount of the policy, and what
     rate did the company charge him?




~MASSACHUSETTS INSTITUTE OF TECHNOLOGY~

ALGEBRA A


TIME: TWO HOURS

  1. Simplify
       [m + 1/m]^2 + [n + 1/n]^2 + [mn + 1/mn]^2
         - [m + 1/m][n + 1/n][mn + 1/mn].

  2. Find the prime factors of
     (_a_) (x - x^2)^3 + (x^2 - 1)^3 + (1 - x)^3.
     (_b_) (2x + a - b)^4 - (x - a + b)^4.

  3. (_a_) Simplify
     [(x^q)/(x^r)]^(q + r) [(x^r)/(x^p)]^(r + p)[{x^p/{x^q}]^(p + q).

     (_b_) Show that
     ([[x]^[1/(n+1)]]^(1/n))/([[x]^[1/(n+2)]]^[1/(n+1)]) =
       {x^(1/n) · [x]^[1/(n+2)]}/{[x^2]^[1/(n+1)]}.

  4. Define _homogeneous terms_.

     For what value of n is x^n y^(5 - n/2) + x^(n + 1) y^(2n - 6) a
     homogeneous binomial?

  5. Extract the square root of
       x(x - 2^(1/2))(x - 8^(1/2))(x - 18^(1/2)) + 4.

  6. Two vessels contain each a mixture of wine and water. In the first
     vessel the quantity of wine is to the quantity of water as 1 : 3,
     and in the second as 3 : 5. What quantity must be taken from each,
     so as to form a third mixture which shall contain 5 gallons of wine
     and 9 gallons of water?

  7. Find a quantity such that by adding it to each of the quantities a,
     b, c, d, we obtain four quantities in proportion.

  8. What values must be given to a and b, so that (3a + 2b + 17)/2,
     (2a - 3b + 25)/3, 4 - 5a - 13b may be equal?




~MOUNT HOLYOKE COLLEGE~

ELEMENTARY ALGEBRA


TIME: TWO HOURS

  1. Factor the following expressions:

     (_a_) a^(3/4) - b^(3/4).

     (_b_) x^2 y^2 z^2 - x^2 z - y^2 z + 1.

     (_c_) 16(x + y)^4 - (2x - y)^4.

  2. (_a_) Simplify

       (a^2 + b^2){(b^4)/(b^2 - a^2) - a^2}/{a/(a + b) + b/(a - b)}}.

     (_b_) Extract the square root of x^4 - 2x^3 + 5x^2 - 4x + 4.

  3. Solve the following equations:

     (_a_) 1/x + 1/y = 5, 1/(x^2) + 1/(y^2) = 13.

     (_b_) x^2 - 5x + 2 = 0.

     (_c_) [27x + 1]^(1/2) = 2 - 3[3x^(1/2)].

  4. Simplify:

     (_a_) 7[54]^(1/3) + 256^(1/6) + [432/(-250)]^(1/3).

     (_b_) 1/[(a - b)(b - c)] + 1/[(c - a)(b - a)].

     (_c_) Find [19 - 8[3^(1/2)]]^(1/2).

  5. Plot the graphs of the following system, and determine the solution
     from the point of intersection:
       { x - 2y = 0,
       { 2x - 3y = 4.

  6. (_a_) Derive the formula for the solution of ax^2 + bx + c = 0.

     (_b_) Determine the value of m for which the roots of
           2x^2 + 4x + m = 0 are (i) equal, (ii) real, (iii) imaginary.

     (_c_) Form the quadratic equation whose roots are
           2 + 3^(1/2) and 2 - 3^(1/2).

  7. A page is to have a margin of 1 inch, and is to contain 35 square
     inches of printing. How large must the page be, if the length is to
     exceed the width by 2 inches?

  8. (_a_) In an arithmetical progression the sum of the first six terms
           is 261, and the sum of the first nine terms is 297. Find the
           common difference.

     (_b_) Three numbers whose sum is 27 are in arithmetical
           progression. If 1 is added to the first, 3 to the second, and
           11 to the third, the sums will be in geometrical progression.
           Find the numbers.

     (_c_) Derive the formula for the sum of _n_ terms of a geometrical
           progression.

  9. (_a_) Expand and simplify (2a^2 - 3x^3)^7.

     (_b_) For what value of x will the ratio 7 + x : 12 + x be equal to
           the ratio 5 : 6?




~UNIVERSITY OF PENNSYLVANIA~

ELEMENTARY ALGEBRA


TIME: THREE HOURS

  1. Simplify: [(a + x)/(a - x) - (a - x)/(a + x)] ÷ (4ax)/(a^2 - x^2).

  2. Find the H. C. F. and L. C. M. of
     10ab^2(x^2 - 2ax), 15a^3b(x^2 - ax - 2a^2), 25b^3(x^2 - a^2)^2.

  3. A grocer buys eggs at 4 for 7¢. He sells 1/4 of them at 5 for 12¢,
     and the rest at 6 for 11¢, making 27¢ by the transaction. How many
     eggs does he buy?

  4. Solve for t:
     (t + 4a + b)/(t + a + b) - (4t - a - 2b)/(t + a - b) = -3.

  5. Find the square root of
     a^2 - (3/2)a^(3/2) - (3/2)a^(1/2) + (41/16)a + 1.

  6. (_a_) For what values of m will the roots of 2x^2 + 3mx = -2 be
           equal?

     (_b_) If 2a + 3b is a root of x^2 - 6bx - 4a^2 + 9b^2 = 0, find the
           other root without solving the equation.

  7. (_a_) Solve for x: [2x - 3a]^(1/2) + [3x - 2a]^(1/2) = 3[a^(1/2)].

     (_b_) Solve for m: 1 - (1)/(2 - m) = 1/(m + 2) + (m - 6)/(4 - m^2).

  8. Solve the system: x^2 + 2y^2 = 17; xy - y^2 = 2.

  9. Two boats leave simultaneously opposite shores of a river 2-1/4 mi.
     wide and pass each other in 15 min. The faster boat completes the
     trip 6-3/4 min. before the other reaches the opposite shore. Find
     the rates of the boats in miles per hour.

  10. Write the sixth term of [x/(2[y^2]^(1/3)) - (y^(1/2))/x]^9
      without writing the preceding terms.

  11. The sum of the 2d and 20th terms of an A. P. is 10, and their
      product is 23-47/64. What is the sum of sixteen terms?




~PRINCETON UNIVERSITY~

ALGEBRA A


TIME: TWO HOURS

Candidates who are at this time taking _both_ Algebra A and Algebra B
may omit from Algebra A questions 4, 5, and 6, and from Algebra B
questions 1 (_a_), 3, and 4.

  1. Simplify
     (a^3 + a^2b + ab^2)/(a^2 - 3ab - 4b^2) ÷ {(a^2 + 6ab - 7b^2)/(a^2 +
     8ab - 9b^2) · (a^3 - b^3)/(a^2 - 7ab + 12b^2)}.

  2. (_a_) Divide a^(5/2) + ab^(3/2) + b^(5/2) - 2a^(1/2)b^2 - a^(3/2)b
           by a^(3/2) - b^(3/2) + a^(1/2)b - ab^(1/2).

     (_b_) Simplify (1)/(x^(-1) + y^(-1)} · (x^(1/4)y^(1/2))^3 + 1.

  3. Factor: (_a_) (x^2 + 3x)^2 - (2x - 6)^2.

     (_b_) a^2 + ac - 4b^2 - 2bc.

  4. Solve 1/(x + 1) - (1)/(x - 1) - (1)/(x - 3) + (1)/(x - 5) = 0.

  5. Solve for x and y: mx + ax = my - by, x - y = a + b.

  6. The road from A to B is uphill for 5 mi., level for 4 mi., and then
     downhill for 6 mi. A man walks from B to A in 4 hr.; later he walks
     halfway from A to B and back again to A in 3 hr. and 55 min.; and
     later he walks from A to B in 3 hr. and 52 min. What are his rates
     of walking uphill, downhill, and on the level, if these do not
     vary?


ALGEBRA B

  1. Solve

     (_a_) (x + 1)/(x - 2) + (2x + 1)/(x + 1) + (3x + 3)/(1 - x) = 0.

     (_b_) [2x + 7]^(1/2) + [3x - 18]^(1/2) - [7x + 1]^(1/2) = 0.

     (_c_) 6/(x^2 + 2x) = 5 - 2x - x^2.

  2. Solve for x and y, checking one solution in each problem:

     (_a_) 2x + 3y = 1, 6/x + 1/y = 2.

     (_b_) x^2 = x + y, y^2 = 3y - x.

  3. A man arranges to pay a debt of $3600 in 40 monthly payments which
     form an A. P. After paying 30 of them he still owes 1/3 of his
     debt. What was his first payment?

  4. If 4 quantities are in proportion and the second is a mean
     proportional between the third and fourth, prove that the third
     will be a mean prop. between the first and second.

  5. In the expansion of [2x + 1/3x]^6 the ratio of the fourth term to
     the fifth is 2 : 1. Find x.

  6. Two men A and B can together do a piece of work in 12 days; B would
     need 10 days more than A to do the whole work. How many days would
     it take A alone to do the work?


ALGEBRA TO QUADRATICS

  1. Simplify
     (ab^(-2)c^2)^(1/2) · (a^3b^2c^(-3))^(1/3) + [(a^6)/(b)]^(1/3).

  2. Simplify
     a/[(a - b)(a - c)] + b/[(b - c)(b - a)] + c/[(c - a)(c - b)].

  3. Factor (_a_) x^4 - 10x^2 + 9.

     (_b_) x^2 + 2xy - a^2 - 2ay.

     (_c_) (a + b)^2 + (a + c)^2 - (c + d)^2 - (b + d)^2.

  4. Find H. C. F. of x^4 - x^3 + 2x^2 + x + 3 and (x + 2)(x^3 - 1).

  5. Solve
     x/(x - 2) + (x - 9)/(x - 7) = (x + 1)/(x - 1) + (x - 8)/(x - 6).

  6. The sum of three numbers is 51; if the first number be divided by
     the second, the quotient is 2 and the remainder 5; if the second
     number be divided by the third, the quotient is 3 and the remainder
     2. What are the numbers?




~SMITH COLLEGE~

ELEMENTARY ALGEBRA


  1. Factor e^(2x) - 2 + e^(-2x), x^(12) - 8, x^2 - x - y^2 - y,
     18a^2x^2 -24axy - 10y^2.

  2. Solve [7 + 4x + 3[2x^2 + 5x + 7]^(1/2)]^(1/2) - 3 = 0.

  3. The second term of a geometrical progression is 3[2^(1/2)], and the
     fifth term is 3/16. Find the first term and the ratio.

  4. Solve the following equations and check your results by plotting:

       { x^2 + y^2 - xy = 7,
       { x + y = 4.

  5. Solve

       1/(x^3) + 1/(y^3) = 243/8,
       1/x + 1/y = 9/2.

  6. In an arithmetical progression d = -11, n = 13, s = 0. Find a
     and l.

  7. Expand by the binomial theorem and simplify:

       [(2x)/(y^3) - (y^4)/(x^5 [-6]^(1/2))]^5.

  8. The diagonal of a rectangle is 13 ft. long. If each side were
     longer by 2 ft., the area would be increased by 38 sq. ft. Find the
     lengths of the sides.




~SMITH COLLEGE~

ELEMENTARY ALGEBRA


  1. Find the H. C. F. of 8x^3 - 27, 32x^5 - 243, and
     6x^3 - 9x^2 + 4x - 6.

  2. Solve:

     (_a_) (2x + 5)^(-5) + 31(2x + 5)^(-5/2) = 32.

     (_b_) (x - 1)^(1/2) + (3x + 1)^(1/2) = 4.

  3. A farmer sold a horse at $75 for which he had paid x dollars. He
     realized x per cent profit by his sale. Find x.

  4. Find the 13th term and the sum of 13 terms of the arithmetical
     progression

       (2^(1/2) - 1)/2, (2^(1/2))/2, (1)/[2([2]^(1/2) - 1)], ···.

  5. The difference between two numbers is 48. Their arithmetical mean
     exceeds their geometrical mean by 18. Find the numbers.

  6. Expand by the binomial theorem and simplify

     [3a^(-2) - a/[-2]^(1/2)]^5.

  7. Solve:

       1/x + 1/y = 3/2,
       1/(x^2) + 1/(y^2) = 5/4.

  8. Solve the following equations and check the results by finding the
     intersections of the graphs of the two equations:

       { x^2 = 4y,
       { x + 2y = 4.




~VASSAR COLLEGE~

ELEMENTARY AND INTERMEDIATE ALGEBRA


Answer any six questions.

  1. Find the product of

     [1 + 2a/3 - (5a^2)/(6)] and [2 - 3a/4 + (a^2)/(3)].

  2. Resolve into linear factors:

     (_a_) 4x^2 - 25;

     (_b_) 6x^2 - x - 12;

     (_c_) a^2b^2 + 1 - a^2 - b^2;

     (_d_) y^3 + (x - 3)y^2 - (3x - 2)y + 2x.

  3. Reduce to simplest form:

     (_a_) z/(1/x - 1/y) + y/(1 - y/x) - x/(1 - x/y).

     (_b_) [-(x^3)^(1/2)]^(1/3) × (4y^(-3))^(1/2).

  4. (_a_) Divide x^(3/2) - x^(-3/2) by x^(1/2) - x^(-1/2).

     (_b_) Find correct to one place of decimals the value of
           [5^(1/2) + 7^(1/2)]/[2 - 3^(1/2)].

  5. (_a_) If a/b = c/d, show that (a^2 + c^2)/(b^2 + d^2) = ac/bd.

     (_b_) Two numbers are in the ratio 3 : 4, and if 7 be subtracted
           from each the remainders are in the ratio 2 : 3. Find the
           numbers.

  6. Solve the equations:

     (_a_) (x + 1)/(2) - 3/x = x/3 - (5 - x)/(6).

     (_b_) 11x^2 - 11-1/4 = 9x.

     (_c_) { x^2 - 2y^2 = 71,
           { x + y = 20.

  7. A field could be made into a square by diminishing the length by 10
     feet and increasing the breadth by 5 feet, but its area would then
     be diminished by 210 square feet. Find the length and the breadth
     of the field.




~VASSAR COLLEGE~

ELEMENTARY AND INTERMEDIATE ALGEBRA


Answer six questions, including No. 5 and No. 7 or 8. Candidates in
Intermediate Algebra will answer Nos. 5-9.

  1. Find two numbers whose ratio is 3 and such that two sevenths of
     the larger is 15 more than one half the smaller.

  2. Determine the factors of the lowest common multiple of
     3x^4 (x^3 - y^3), 15 (x^4 - 2x^2y^2 + y^4), and
     10y (x^4 + x^2y^2 + y^4).

  3. Find to two decimal places the value of
     [4a^(-2/5) + b^0[ab^(-1)]^(1/2)]^(1/2), when a = -32 and b = -8.

  4. Solve the equations: 2x + 5y = 85, 2y + 5z = 103, 2z + 5x = 57.

  5. Solve any 3 of these equations:

     (_a_) x^2 + 44 - 15x = 0.

     (_b_) 2/x - x/5 = x/20 - 223/30.

     (_c_) x^2 + 8x - [4x^2 + 32x + 12]^(1/2) = 21.

     (_d_) 5/(x + 1) + 8/(x - 2) = 12/(40 - 2x).

  6. The sum of two numbers is 13, and the sum of their cubes is 910.
     Find the smaller number, correct to the second decimal place.

  7. The sum of 9 terms of an arithmetical progression is 46; the sum of
     the first 5 terms is 25. Find the common difference.

  8. Explain the terms, and prove that if four numbers are in
     proportion, they are in proportion by _alternation_, by
     _inversion_, and by _composition_. Find x when
     (3 + x)/(3 - x) = (40 + x^3)/(40 - x^3).

  9. Find the value of x in each of these equations:

     (_a_) 7x^(1/4) - 3x^(1/2) = 2.

     (_b_) (x^2 + 2)^(5/2) + 3/{[x^2 + 2]^(1/2)} = 4x^2 + 8.




~YALE UNIVERSITY~

ALGEBRA A


TIME: ONE HOUR

Omit one question in Group II and one in Group III. Credit will be given
for _six_ questions only.


_Group I_

  1. Resolve into prime factors: (_a_) 6x^2 - 7x - 20;
     (_b_) (x^2 - 5x)^2 - 2(x^2 - 5x) - 24; (_c_) a^4 + 4a^2 + 16.

  2. Simplify
     [5 - (a^2 - 19x^2)/(a^2 - 4x^2)] ÷ [3 - (a - 5x)/(a - 2x)].

  3. Solve
     [2(x - 7)]/(x^2 + 3x - 28) + (2 - x)/(4 - x) - (x + 3)/(x + 7) = 0.


_Group II_

  4. Simplify [2^(1/2) + 2[3^(1/2)]]/[2^(1/2) - 12^(1/2)], and compute
     the value of the fraction to two decimal places.

  5. Solve the simultaneous equations
       { x^(-1/2) + 2y^(-1/2) = 7/6,
       { 2x^(-1/2) - y^(-1/2) = 2/3.


_Group III_

  6. Two numbers are in the ratio of c : d. If a be added to the first
     and subtracted from the second, the results will be in the ratio
     of 3 : 2. Find the numbers.

  7. A dealer has two kinds of coffee, worth 30 and 40 cents per pound.
     How many pounds of each must be taken to make a mixture of 70
     pounds, worth 36 cents per pound?

  8. A, B, and C can do a piece of work in 30 hours. A can do half as
     much again as B, and B two thirds as much again as C. How long
     would each require to do the work alone?




~YALE UNIVERSITY~

ALGEBRA B


TIME: ONE HOUR

Omit one question in Group I and one in Group II. Credit will be given
for _five_ questions only.


_Group I_

  1. Solve (x + a)/(x + b) + (x + b)/(x + a) = 5/2.

  2. Solve the simultaneous equations
       { x^2y^2 + 28xy - 480 = 0,
       { 2x + y = 11.
     Arrange the roots in corresponding pairs.

  3. Solve 3x^(-3/2) + 20x^(-3/4) = 32.


_Group II_

  4. In going 7500 yd. a front wheel of a wagon makes 1000 more
     revolutions than a rear one. If the wheels were each 1 yd. greater
     in circumference, a front wheel would make 625 more revolutions
     than a rear one. Find the circumference of each.

  5. Two cars of equal speed leave A and B, 20 mi. apart, at different
     times. Just as the cars pass each other an accident reduces the
     power and their speed is decreased 10 mi. per hour. One car makes
     the journey from A to B in 56 min., and the other from B to A in 72
     min. What is their common speed?


_Group III_

  6. Write in the simplest form the last three terms of the expansion
     of (4a^(3/2) - a^(1/2) x^(1/3))^8.

  7. (_a_) Derive the formula for the sum of an A. P.

     (_b_) Find the sum to infinity of the series 1, -1/2, 1/4,
           -1/8, ···. Also find the sum of the positive terms.





End of Project Gutenberg's A Review of Algebra, by Romeyn Henry Rivenburg