George Parsons Tibbets.

College requirements in algebra. A final review online

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University of California • Berkeley



The Theodore P, Hill Collection

of
Early American Mathematics Books



COLLEGE REQUIREMENTS



ALGEBRA



A FIMAL REVIEW



BY

GEORGE PARSONS TIBBETS, A.M.

Instructor in Mathematics, Williston Seminary



oJ«?o



BOSTON, U.S.A.
PUBLISHED BY GINN & COMPANY

1892



Copyright, 1892,
Bt GEORGE PARSONS TIBBETS, A.M.



Ttpography by J. S. Gushing & Co., Boston, U.S.A.



Presswork by Ginn & Co., Boston, U.S.A.



PREFACE.



The Williston students have found these reviews so
serviceable that a more convenient form has become nec-
essary. From a wide collection of college papers about
four hundred examples, illustrating nearly every prin-
ciple in Algebra, were selected and carefully arranged
by subjects. Whenever a suitable one could not be so
obtained, an original problem or one from foreign texts
was inserted. The parallel sections are for the use of
two divisions and for recitation -room drill.

Colleges will find the work useful as an initial review ;
while college candidates may be assured of entering if
they perform all the examples without aid.

Suggestions in regard to the work will be gladly

received.

G. P. T.

WiLLiSTOK Seminary,
Easthampton, Mass., Jak., 1892.



C0ITTENT8.



Section Page

Sight Problems I. 7

Parentheses and Evaluation . . . II., III. 8, 9

Factoring IV., V. 10,11

H.C. F., L.C. M., Evolution .... VI., VII. 12,13

Fractions VIII., IX. 14, 15

Simple Equations X., XI. 16, 17

Simultaneous Equations XII., XIII. 18, 19

Theory of Exponents XIV., XV. 20, 21

Radicals XVI., XVII. 22,23

Quadratics XVIII., XIX. 24, 25

Simultaneous Quadratics .... XX., XXI. 26, 27

Inequalities, Proportion, Variation . XXII., XXIII. 28, 29

Progressions XXIV., XXV. 30, 31

Binomial Theorem, Permutations . XXVL, XXVII. 32, 33

Undeter. Coef., Limits, Logarithms . XXVIII., XXIX. 34, 35

Specimen Paper, Advanced Problems. XXX., XXXI. 36, 37

Harvard, Yale XXXII., XXXIII. 38, 39

Vassar, Wellesley XXXIV., XXXV. 40, 41

Cornell, Bryn Mawr XXXVI., XXXVII. 42, 43

Technology, Sheffield XXXVIII., XXXIX. 44, 45

Princeton XL. 46

5



COLLEGE REQUIREMENTS IN ALGEBRA. 7

SECTION I.
SIGHT PROBLEMS.

1. Of 452 students who tried the examinations last June,

X were men, the rest were women. In all, z students
failed. How many of the failures were men ?

2. The sum of 3 consecutive numbers exceeds the middle

number by 10. What are the numbers ?

3. In how many weeks will x horses eat 50 bushels of oats,

if one horse eats y bushels in a week ?

4. A is 20 years old, and B is — 2 years older ; what is the

age of B ? [^Harvard.']

5. Two men working separately can do a piece of work in

X days and y days respectively ; find an expression for
the time in which both can do it working together.

\^IIarvard.^

6. A pole 100 feet high, standing on the side of a hill, breaks

off so as to form a right angle at the break, with the
top resting on the hill 75 feet from the foot of the
pole ; where did the pole break? \_Cornell.'\

7. Two steamers ply between the same two ports a distance

of 420 miles. One travels half a mile an hour faster
than the other, and is two hours less on the journey ;
find the rates. [^Vassar.^



8 COLLEGE REQUIREMENTS IN ALGEBRA.

SECTION 11.
PARENTHESES AND EVALUATION.

1. ?>x ~{bx — [^x — {y ■— x)]) ~{— X — Sy).

2. a - {2h + [?>c - ?>a - {a + h)] + 2a - {h + 3c)).

3. ax + h{x + c) + c^- [{a -b)x-{h- c)(b + c)].

4. ah —- c(x — b) — [(^ + c)(x — c) — c(b — \c — x\) — x"^].

5. - [ISx-Uy - 3\2x - 2(-y + 2x)- 7y\l

6. a-\-b\x-3a(y-2x) + Sx[4:a + 2{4:b + 3)]].

8. Evaluate a + 2x — {b + y — [a — x — (b — 2y)]l.



3/-

X — -\ X



9. Evaluate x — ( V^ + 1 + 2) — - when x-

X — 4:



10. Substitute x + S for y in y^ + 2f - I5y — S6.



COLLEGE REQUIREMENTS IN ALGEBRA,

SECTION III.
PARENTHESES AND EVALUATION.

1. 2a-[bb + {'dc-{a + [2b-Za + ^c])\l



2. 3a - {3a - [3a - (3a - 3a - 3a) - 3a] — 3a} - 3a.

3. {a + h)x — (h — c) c — [{h — x)h — {b — c){a + c)] — ax.

4. ^x - 4:y + b[- 4:x -\?>y - {2+1 X -2y) - m



5. 3c' + c(2a- [5c-{3a + c-4a|]).

6. ^y'-(~xy' + x'-^xy~-x'[-\f-y{xy-x')\].

7. (a^ -b'')c-{a- b){a[b + c]-b[a- c]).

8. Evaluate Va + Vab + Vb when a == 8, b^ 64.



9. Evaluate V~a\^h'+ ^^l±l^±^ 1 ^hen a - 8, ^ = 1.



10. Substitute y - 3 for rr in x' + 2x''~15x- 36.



10 COLLEGE REQ UIREMENTS IN ALGEBRA.

SECTION IV.
FACTORING.

1. Difference of Squares : {x^ + 3/^ — 2:^)'^ — ^tx^yK

2. Cubes : x^ — 1.

3. Simple Trinomial : a;' - 13 ^y + 363/^

4. Complex Trinomial : Qx^ — bxy — Qy"^.
\ 5. Literal Trinomial : x'^ -{-(a-\ - \xy + y'^.

6. Four Terms : a' + a'h'' — ^c^ - c\

7. Five Terms : 6a^ + 5a^ — 6^'^ — 6ac-f- 4^>'(?.

8. Six Terms : 2d^ — ah — ac — ^IP' — bhc — c^

9. Perfect Square : X^x'' ^X'ox^ - ^x^ - \x^ ^ x^\
10. Imperfect Square : x^ — Tr^y + ?/.

V 11. Radicals: ^^4-1-

12. Separation : x^ — ^x^-\- \\x — 6.

13. Literal Exponents : x""^ + \x'^ + ^.

14. Parentheses : fl + yj - 2.r- fl - y^) + ^* (1 - .v)'-



COLLEGE REQUIREMENTS IN ALGEBRA. H

SECTION V.
FACTORING.

1. Difference of Squares : 1 — {x^ + y^) -f- ^xy.

2. Cabes: a}'' -'b'\

3. Simple Trinomial : x^ — 2x — ?>,

4. Complex Trinomial : Sa'"^ + 26i^ — 3.
6. Literal Trinomial : 1 — {jri^ + in!')x^ + m^n^x'^.

6. Four Terms : tyi^x + m^y — r^x — n^y.

7. Five Terms : ax^ — ?> ax^ + 2 ax^ + ax^ — ax.

8. Six Terms: (Sa^ - bah + l^ac + h'' - bhc + ^c\

9. Perfect Square : 67 a;' -f 49 + 9:?;^ - 70:r - 30rl

10. Imperfect Square : 9a' - 40a'Z)' + 16/A

11. Radicals: a + b.

12. Separation : x^ + 10a;' + 29a; + 20.

13. Literal Exponents : x^ — y".

14. Parentheses : a{a — 1) x"^ — {a — b ~ 1) xy — h (h -[- '[^ y'.



12 COLLEGE REQUIREMENTS IN ALGEBRA.

SECTION VI.
H.C. F., L. C. M., EVOLUTION.

Express both H. C. F. and L. C. M. in factors : —

1. ^x'+lZx-b,

Zx^ + 2x' + 2x-l, [Princeton.]

U'-I2h-^. [Yale.]

3. e>x'~-lSx^ + ^x'' + 2x,

6x' - 9x' + lbx'' - 27 X - 9. [Brown.]

4. 4:x'-9x' + 6x-l,

6x^ ~ 7x'^ -{- 1. [Johns Hopkins.]

5. ex'-bx'-lOx' + ^x-lO,

4:X^ — Ax"^ — 9x -{• b. [Harvard.]

6. 8:r' + 27,

16^-* + 36a;' + 81,

6x'' + 5x-6. [Vassar.]

7. Square Root of 4a*+ 12a' + 5a' - 6a + 1.

8. Cube Root of Sa^ + I2a' - 6a* - 11a' + Sa' + Sa -1.



COLLEGE REQUIREMENTS IN ALGEBRA. 13

SECTION VIL
H.C. F., L C. M., EVOLUTION.

Express both H. C. F. and L. 0. M. in factors : —

1. 2a;^-lla;2-9,

4 ;rH- 1 1 ^* + 81. [Johns Hopkins.],

12:^^ + 10:^' -4. [AmheTst]

3. 15 aV - 20 aV - 65 a^x - 30 a\

12bx' + 20 bx' -IQbx- 166. [Brown.]

4. x'-Sx-2,

x'-2x'-x + 2, [DartmoiM..]

5. x'-l,

x'-2x-^, -

6 o;^ — a; — 20. [ Technology.]

6. 2:^:* + .'^'- 8^'-a: + 6,
4:x'+ 12:r' -a:'^-275;- 18,

4^.4 + 4:^^ - 11 x' - 9a; + 18. [Harvard.]

7. Square Root of 8^« + — - 8a;* + 2^' - + 2.

Sx^ x^

'\ , N^ [B7yn Mawr.]

8. Cube Root of x^ - 6:^;^ + ^x' + 28^' - 9^;*^ - 54a; - 27.



14 COLLEGE REQUIREMENTS IN ALGEBRA.



SECTION VIII.
FRACTIONS.
1. Reduction: 4.^ + 3.-10



4:?;'+ 7a;' -3:^-15



2. Addition and Subtraction :



a^ —he , l)^ + ca , c^ -\- ah



(a - h)(a - c) (h-\- c){h — a) (c — a)(c + h)



3. Multiplication



4. Division :



'x + y_ _ x~y _ 4?/'
X — y X -{-y x^ — y^



by



x + y
. 2y



' x' + y' x""- yn ^^ [ x-y ^ x-^y



5. Complex :



12 + ^



1 + -



2 + ;



6. Evaluation



TP 4:ah P J , 1 1 c! x + 2a . X +2h

If x = -, find the value of — ^-^-H Htt*

a + b X ~2a X — 'Ab



1 — ^ V ^



7. Miscellaneous:



l + y\l-^x



x^ + y^ — X -\-y



\-y \-f



COLLEGE REQUIREMENTS IN ALGEBRA. 15

SECTION IX.
FRACTIONS.



1. Eeduction :



2. Addition and Subtraction ;



1 2^^+ 11^^ -43a; -24



' -+ . \ ,. +■ '



x(x — a){x — b) a{a — x){a — b) b(b — x){b — a)

3. Multiplication: ^-y ^ ca-h cy b^ ^- f b^

^ a^ + y' b'' + by h' + f c

4. Division : [1^11^ - 1^1 by [1+^ + i + ^l-

[l + x' l + x] ^ \l-x' ' 1-^J

5. Complex: i

^ + ^"-^

1. ^-2



2a;+l



6. Evaluation : If — ?— = a, -^ = 5, _1



y + 2; :r + 2; ^ + y



find value of h



a , b , c



l+« 1+^ l+c
b'



7. Miscellaneous: _Ji^ ^' + ^'^

a + <^ 2\ a + bj



16 COLLEGE REQUIREMENTS IN ALGEBRA.

SECTION X.
SIMPLE EQUATIONS.



1.



'-{'■



5.r + 2 /o ?>X'-V\ 3:?: +19 /^ + 1 , q\



20 3a,' + 4 5



4. ^^^^-A(y-4)-|(y-6) + A.



6. 12^3a;-.25(a;-4)-.3(5:r + 14)| = 47.
a: a — hex x ac ~ 4:bx



6



2 2Z^c 6c 3^c



7. _^_3_|__^ = JL._2a(2-3a).
2a 4a=^ 3a'^ ^ ^



8. At what time between 7 and 8 o'clock are tlie hands of
a watch (a) together ?
(5) opposite?
(c) at right angles ?



COLLEGE REQUIREMENTS IN ALGEBRA. 17

SECTION XL
SIMPLE EQUATIONS.



-e-^-f^)



1. 2a; . -. ^ ,

4



2. 3.-^-4 = 5l±li.



3. ^+ig-g(3.-4)+^^"-^X^"-^) = x'-g-.
3 5^^ 6 15



4.



K'-i)-i('-f)+K'-i)^»-



5. 3.3^_:22^Zl^=,la; + 9.9.



^ ax —h 1 — X . r,

6. — :; [-m^O.



7. dax ~2bx ~^c — \mx = ^c + ^7nx — n — bx + 2ax.

^. A man bought a horse, and expected to sell it at 10 %
profit ; but had to sell it for $50 less than he expected,
and then found he had lost 15 % on what it cost him.
What did he pay for the horse ?



18



COLLEGE REQUIREMENTS IN ALGEBRA.



SECTION XII.



SIMULTANEOUS EQUATIONS.



ax -\-hy = c ^
o!x + h^y = c' )



1_

3x



53/



bx 3y 4



1. Simple :

2. Reciprocals :

3. Parentheses :

4. Three Quantities :

5. Fractions:

6. Numerical: x — 2y + Sz = 2

2x-3y+ z^l
Sx- y + 2z = 9 ,

7. Problem. A firm has Java coffee at a cents per pound

and Mocha at b cents per pound. How much of each
is there in a mixture of a — b pounds if it can be sold
at c cents a pound without loss ?



(a + b)x — (a — b)y =■- 4:ab^
(a-b)x-(a-\-b)y = I

ax -\- b7/= I
cy -\- dz =771

ex -\- fz ^ n



771 71 _






a


X y






>


n _, 771 __


b


X y





COLLEGE REQUIREMENTS IN ALGEBRA.



19



SECTION XIII.



SIMULTANEOUS EQUATIONS.



1. Simple :



x + y



__2(aMi^ 1



x-y-



i^ - b'



2.


Keciprocals :


7 _^ 4 _
V^ Vy

^-+ ' ■■

■\Jx Vy


-4
= 1




3.


Parentheses :


7(^ + 3/) + 3(^-2/) =
n{x + y)-Z{x~y)^


4.


Three Quantities :


y + z=a-
x+z =b ►
x + y — c)


5.


Fractions :


a h








a c








b c j




6.


Numerical :


2^+ 4y + 272^-28
7:r- 32/ = 152= 3






^x — Y)y -


-33


z= 4



7. Problem. A and B walk in a circle whose circumference
is C. If they start from the same point and go in
opposite directions they meet in 4 hours. If they start
from opposite points and walk around the circle in the
same direction they meet in 8 hours. Find the rate
at which each one walks.



20 COLLEGE REQUIREMENTS IN ALGEBRA.

SECTION XIV.
THEORY OF EXPONENTS.



1. Discussion: — ; x^.

2. Proof: (a"*)*» = a""*.

3. Kemoval of negative exponents : ~ ^ ~ V

1 — x~^y~'^ + x~'^

4. Combination: (-}]^^y,(^\Km.



_3
C 4



%) \h)



5. Multiplication : (a* — a^-h^ + a'Z>* - ah + ah^ - b^

by (a* + 5*).

6. Division : x'^y~^ — 2 + x'^y'^ by x^y~^ — :?;"^3/^



7. Involution :



</xJj



8. Evolution :

1 + 4y'* - 2y"^ - 4^"^ + 25y"* - 24:y"i + 16y-l



9. Reduction :



^2_,_4_l_4^-|



10. Problem. Find the number whose cube root is one-fifth
of its square root.



COLLEGE REQUIREMENTS IN ALGEBRA. 21

SECTION XV.
THEORY OF EXPONENTS.

1. Discussion: ]^ ^;..^n.-n J"'

2. Proof: a'=l, a-^ = ~

a**

3. Removal of negative exponents : — ^ ., ■

c~* — a~^

1 1 /'y^Y y"^

4. Combination : a^y^ ^ ( i ) ^ i '

5. Multiplication :

(a^ - a* + 1 - a"* + a"^) by (a* + 1 + a"*).

6. Division: ^'''~'^' '^ i^T' '



7. Involution :






8. Evolution: :x:^ + 2a;2 - 3a;' - 4a;2 + 4a:.



9. Reduction :



a;^ + 2A/a;y + 43/*
10. Equation : V^ z::^ 2V2 ; find a;,



22 COLLEGE REQUIREMENTS IN ALGEBRA.

SECTION XVI.
RADICALS.



x+\ \ x— 1



1. Eeduction ;

'. + :



2. Addition and Subtraction :



V9 ab' - Va'b + -J a^h - 6 a^h' + 9 ah\

3. Arrangement : \/2^ V3, ^^\.

4. Multiplication: \x-\{^-V^)\x -\{\ + -\f^)\

5. Division : ^ ., „ ^ bv ^- —

6. Involution and Evolution : [2V3 + 3 V2 + V6]l



7. Eationalizatidn : — :i z==l-

-\/x + V^ + y



8. Imaginaries : V— 9:r* + V— 16.r* — V— (x — ly.



9. Binomial Surds : V^l + 12V5.



10. Radical Equation : Vx + 2 — Vx — 2 = 2:r.



COLLEGE REQUIREMENTS IN ALGEBRA. 23

SECTION XVII.
RADICALS.



1. Reduction: V(a^ - I)(a - 1)1

2. Addition and Subtraction :

2 Vl25 - ■^^ + -v'8l ~ (- 512)* + v^i^.

3. Arrangement : V3, V6, VlO.

4. M.l.ip,io.«o., (.-1 - |1)(.-1J-|1)(. + J-),

5. Division: Vsl/l-VV^^.



6. Involution and Evolution :



7. Rationalization :



'x^/yV



I ^ xy)
V3 2 - V^^



2 - V3 2 + V- 2

8. Imaginaries : [2V— 3 — 5V— 2]l

9. Binomial Surds : Vl4 + 4V6.
10. Radical Equations : V^ + 5 + V^ — 8 = V3.



24 COLLEGE REQUIREMENTS IN ALGEBRA.

SECTION XVIII.
QUADRATICS.

1. Simple: . I2x' + x-l^0.

2. Literal: aix" -x) + b(x^ + x)=-^'

3/ Fractional: (4a^- 5^)(.^ + D ^ 2..

^d' + b''

4. Parentheses: {x -2){x + '^){x'^ + ?>x-4.) = 0.



3:r- V^"'-8



5. Rationalization : =z^zz- = x -{- -^Jx^ — 8.

^ - Va;^ ~ 8

6. Fractional Exponents : . x^ — x^ = 256.

7 . Quadratic Form : {ax — hf — ^a (ax — h) = ^ a^

8. Radicals: — J£:^=: - V 10^+2 ^



Vl0a;-9 Vl0a;-9

9. Cubic Equation : x^ — x"^ — x + 1 = 0.

10. Formation :

Find equation with roots (a + b — c) and (a~b + c).

11. Problem. Tristram is ten years younger than Launcelot ;

and the product of the ages they attained in 1890 is
96. Find the ages they attain in 1908. [^Harvard.']



COLLEGE REQUIREMENTS IN ALGEBRA. 25

SECTION XIX.
QUADRATICS.

1. Simple: 91a;*' - 2a7- 45 = 0.

2. Literal :

3. Fractional :



(a + 2b)x_




a'


W


a -lb


a-


-2b


X


.r + 1


2


_ ^


+ 2


c


ex


ax


-bx



4. Parentheses : (x - l)(a; - 2)(x' -6x+9) = 0,

5. Binomial Surds : (1 - WS)x' + (2- VS)x= 2.

6. Fractional Exponents : 4 ^/x + Vx ^ 21.

7. Quadratic Form : x^ — 2x + 6^/x^ ~ 2a; + 5 = 11.

8. Radicals: Vo; + a + Vo; + Vo; — a = 0.

9. Bi-quadratic : x* — 2o!^-{-x — 2 = 0.

10. Formation :

Find equation with the roots ^^"^ / and (b — a).

a — b

11. Problem. A cask F is filled with 60 gallons of water,

and a cask Q with 40 gallons of brandy, x gallons
are drawn from each cask, mixed and replaced, and
the same operation is repeated. After the second
replacement there are 8|- gallons of brandy in P.
Find X.



26 COLLEGE REQUIREMENTS IN ALGEBRA.

SECTION XX.
SIMULTANEOUS QUADRATICS.

1. Substitute: 2x + ?>y =12|

?>x'-2xy = lb)

2. Obtain ^ry : x — y= 4]

a;2 + y2 _ 106 j

3. ljQiy = vx\ x^ + \Oxy=ll'\

bxy-Zy"-^ 2)



x" + xy + y'



= 66 I

= 28)



4. Divide : x^ — y^ = 66 "

X^ + X'^

5. Retain Fractions : =

X y

- + - =20
x^ y^

6. Yale: ^'^- 2/^ = 9

1^:?1::8:7



7. Wellesley : x"^ -\- xy =^lb

xy-y'



8. Micbigan University \ x -^y — V:ry = 7
x'-\-y''-^xy - 133



9. Columbia ; Mines : ^' + ?/ + 3:r = 73 — 2:?;3/ '
y2_|_^ =44 — 3?/



I



COLLEGE REQUIREMENTS IN ALGEBRA. 27

SECTION XXI.
SIMULTANEOUS QUADRATICS.

' = 17 I

!

3. Let y =^vx: x^ + ^xy = — \



1. Substitute : x — ?>y — l

x^ — 2xy-\-^y'

2. OhidJin xy \ ^ — y =10

:^2 _^ y2 _ 53



'-12 J



4. Divide : x^ — xy + y'^ =^

x' + xY + y'=^ 243 .



5. Retain Fraction :



X


1


= 7


1

x'


1


= 25



6. Vassar : :r^ + y^ = 56

^ + y -2



7. Princeton: x"^ + xy + y^ = 52^

xy — x^



}



8. Univ. Penn. : x^ — xy -ry'^ =^ |

x' + a;y + y* =- 133 J

9. Harvard: -^ ^^_^ _ ^±1^ V Q "

3/:(7a;-2y) = (Z)-a):(2a-9Z>) ■



28 COLLEGE REQUIREMENTS IN ALGEBRA.

SECTION XXII.
INEQUALITIES, PROPORTION. VARIATION.

1. Prove that the square of half the sum of any two quan-

tities < half the sum of their squares.

2. Prove a'>a + -~l if a>l.

a

3. What two numbers whose difference is d are to each other

as a is to 5 ?

4. If a' — 3/ is a mean proportional between y and {y+z — 2x),

show that :r is a mean proportional between y and 2.

5. Prove that if

2x\y\\a:h, \a~ x :\a + x \ :h — y \h -[-y.

6. Prove that a proportion taken by inversion is a true pro-

portion.

7. A varies jointly as B and C\ and A — ^ when B=^,

(7-2. Find ^ when ^-5, (7=7.

8. A varies as the square of B, and inversely as the square

of (7, and ^-4 when ^= - 1, 0=2. What is the

value of ^' + ^' when B = 2, (7=2?
B^ - A^

9. The volume of a sphere varies as the cube of the radius.

If 3 spheres with radii 9, 12, and 15 inches are melted
into a single sphere, find its radius.



COLLEGE REQUIREMENTS IN ALGEBRA. 29

SECTION XXIII.
INEQUALITIES, VARIATION, PROPORTION.



1. Prove that | — ^ ) < ah.



2. Find the limit of x in

(3x + 2)(x-S)> (x + 4:){Sx-l)-B.

3. What number added to 2, 20, 9, 34, will make the results

proportional ?

4. Find two numbers such that their sum, difference, and

the sum of their squares are to each other as 4, 1, 17.

5. If a:b : : c : d show that

a:b:: -VSa' + bc' : VSb' + bd\

6. Prove. that a proportion taken by division is a true pro-

portion.

7. The area of a circle varies as the square of its radius and

the area of a circle is 154 sq. ft. when the radius is
7 ft. Find the area of the circle whose radius is 10 ft.
6 in.

8. The offing at sea varies as the square root of the height

of the eye above sea-level, and the distance is 3 miles
when the height is 6 ft. Find the distance when the
height is 50 yds.



30 COLLEGE REQUIREMENTS IN ALGEBRA.

SECTION XXIV.
PROGRESSIONS.

1. Given a, d, and n in an arithmetical progression, to find I.

2. Given a, I, and r in a geometrical progression, to find the

sum s.

[Note. — The above problems are given frequently.]

3. The first and ninth terms of an arithmetical progression

are 5 and 22. Find the sum of 21 terms.

4. Find the nth. term of the series 2, 2-|-, 2|-.

5. Find d and I when a = 3, n~lb, s = — 165.

6. Insert 3 arithmetical means between — 9 and 18.

7. Find the twelfth term of V2, -2, 2V2, -4, etc.



8. Sum the infinite series q + ^ "I" 03 + ^^^•

o o o



9. The sum of 3 numbers in arithmetical progression is 12,
and the sum of their squares is 50. Find the num-
bers.

10. If a clock is constructed so as to strike up to 24, how
many strokes will it make in the revolution of the
index?



COLLEGE REQULREMENTS IN ALGEBRA. 31

SECTION XXV.
PROGRESSIONS.

1. Given a, I, and n, in an arithmetical progression, to find s.

2. Given a, n, and r, in a geometrical progression, to find L
[Note. — The above problems are given frequently.]

3. In an arithmetical progression, 5 = — ^-, n = 20, a = ^.

Find d.

4. Find the (2w)th term of 1, 3, 5, 7, etc.

5. Find a and n when Z = — 47, c? = — 1, s = — 1118.

6. Insert 3 geometrical means between ^ and 128.

7. Find the seventh term of — -J. i, — f , etc.
. 8. Sum the infinite series i + 2t + tts

9. A traveller has a journey of 132 miles. He goes 27
miles the first day, 24 the second, and so on, travelling
3 miles less each day. In how many days will he
complete his journey ?

10. Find the sum of all the numbers which are less than 500
and are divisible by 11 without a remainder. [FaZe.]



32 COLLEGE REQUIREMENTS IN ALGEBRA.

SECTION XXVI.

BINOMIAL THEOREM. PERMUTATIONS AND
COMBINATIONS.

1. Expand by the binomial theorem {^a^ — f ^Z-

2. Expand to 4 terms — - [Yale.]

Vl + x^

3. Expand (2a-Sb)-\

/ 3 h^ \^^

4. Obtain 4 terms (a^-l ) .

6. Find the fifth term of (x~'' - 2y^y\

V a^ -— ^ ] [ Harvard. ]

7. Find the term independent ofa;in (Zx ]•

8. Expand (1 + 2x^y to 4 terms.

9. Expand {a^ + l + a-J,

10. How many different amounts can be made up from 5

different coins ?

11. In how many ways can 7 children form a ring?

12. I have 5 single volumes and a set of 3 volumes. In how

many ways can I arrange these 8 books on a shelf,
keeping the set together and in order?



COLLEGE REQUIREMENTS IN ALGEBRA. 33

SECTION XXVII.

BINOMIAL THEOREM. PERMUTATIONS AND
COMBINATIONS.

1. Expand by the binomial theorem ( a ) •

V a;

2. Expand to 4 terms (a + x)'^ .

3. Expand (V3 — 8 \/(2)*. {^Technology.']

4. Obtain the first 8 and last 8 terms of {x — yf^.
■ 5. Find the fourth term of (2a; - ?>y)-\

6. Fourth term of I -y7=: — - a3^~M • {Harvard.]

7. Find the terms without radicals [ 2Va ~ a/- ] •

8. Expand to 5 terms (1 + of.

9. Expand {f — e'^^y.

10. How many different signals can be made with 12 different

flags by hoisting 4 at a time above each other ?

11. How many combinations can be made from the word

Payson taken 8 letters at a time ?

12. At a whist party there are 6 ladies and 6 gentlemen.

The host is to play with the most honored guest, and
the hostess with the poorest player. In how many
ways can the players be arranged if each man has a
lady partner ?



34 COLLEGE REQUIREMENTS IN ALGEBRA.

SECTION XXVIII.

UNDETERMINED COEFFICIENTS, LIMITS,
LOGARITHMS.



2 + a; .

1. Expand into a series of four terms by unde-

1 + x — x^ "^

termined coefficients.



2. Develop — t — into a series.
^ 3 + 4^



X

3. Separate — — — into partial fractions.

\X — Jl K ^ — A)



3-^2

4. Separate into partial fractions.

^ {x-l)\x+\f ^



5. Find the limit, when x increases without limit, of

{x 4- l)(x'' - 3)
x'-Zx



6. Prove log^m^ Q^a^



7. Simplify



2.372 X 7232 x .003722



(- 22.37)(72230000)

2

8. Find the value of x in the equation 5* = 30.



COLLEGE REQUIREMENTS IN ALGEBRA. 35

SECTION XXIX.

UNDETERMINED COEFFICIENTS, LIMITS,
LOGARITHMS.

1. Expand into a series by undetermined coefficients

1 + 23;



2> -A- X
2. Develop to four terms - — — ^^

A — X — X



3. Separate -^ ^ into partial fractions.



X" — a;



4. Separate ^ , — — - into partial fractions.

^ {x-'l)\\-2x) ^



5. Evaluate fa' - 1)(^' + 2) ^^^ ^ ^ ^^



6. Prove that logja X log„6 = 1.

7. Simplify ^l26VT08^ v'1008a/T62.

8. Given the amount of a given principal for a given num-

ber of years, to find the rate per cent.



36 COLLEGE REQUIREMENTS IN ALGEBRA.

SECTION XXX.
SPECIMEN PAPER.

1. Evaluation: The score of the Amherst-Technology game

on Saturday was

V A
Find numerical value if e ~f — g = A = 4.

2. Expression : A man's monthly salary is %x. His weekly

expenses are %w, besides annual tax of %t, and semi-
annual insurance of $5. How much does he save
yearly ?



3. Parentheses: c-[2a-5— (3a-2Z>— 4a-3^)]. [Sheffield.']

4. Multiplication: {\(f^ + ^0^-^ + \){\a~ \).

5. Division: {a^ -^h" ~^c^ + ah ~ ac + lbc)

by {a-h + 2c). [Wor. Tech.]

6. Formulas : Square [(b — 2)x + (1 — h)].

7. Inspection: Divide ( a^ ) by la ) [Iliiies.]

8. Factoring : x"^ —y"^ — z^ + 2yz. [Yale.]

r. r. ^r ( 12 x' - 29 X + U,

9. G. C. Measure : < Harvara.

{l8x''+ 307-10. ^ ^

10. Miscellaneous : H. C. F. by factoring or division



■x'-(a' + b')x' + a'b\

.x'-(a+ by x' + 2ab(a + b)x - a^h\



French
Collection.



COLLEGE REQUIREMENTS IN ALGEBRA. 37

SECTION XXXI.
ADVANCED PROBLEMS.

1. A certain librarian spends every year a fixed sum for

books. In 1886, the cost of his purchases averaged
two dollars per volume ; in 1887, he bought 300 more
volumes than in 1886 ; and in 1888, 300 more volumes
than in 1887. The average cost per volume was thirty
cents lower in 1888 than in 1887. Find the number
of volumes bought each year, and the fixed price paid
for them. (Obtain two solutions.) [^Harvard.^

2. A and B start at the same time from two towns and

travel towards each other. When they meet B has
travelled a miles more than A ; it will take A h days
longer to reach the town B left, and B c days longer
to reach the town A left. Find the distance between
the towns. \^IIaTvard.']

3. Three students A, B, and C, agree to work out a series

of difficult problems in preparation for an examination ;
and each student determines to solve a fixed number
every day. A solves 9 problems per day, and finishes
the series 4 days before B ; B solves 2 more problems
per day than C, and finishes the series 6 days before
C. Find the number of problems and the number of
days given to them by each student. \^IIarvaTd.'\



38 COLLEGE REQUIREMENTS IN ALGEBRA.

SECTION XXXII.
HARVARD.



[Write legibly and without crowding ; give the work clearly, and


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