20. The price of photographs is raised 50 cents per dozen; and, in consequence, six less than before are sold for $5. What was the original price?

21. What are eggs a dozen when two more for 30 cents would lower the price 2 cents a dozen?

22. A woman spends 75 cents in eggs; if she had bought a dozen less for the same money, they would have cost her 3 cents a dozen more. How many did she buy ?

23. The price of one kind of sugar per pound is 2 cents more than that of a second kind, and 10 pounds less of the first kind can be got for $2.40 than of the second. Find the price of each per pound.

24. One-half of the number of cents which a dozen apples cost is greater by 2 than twice the number of apples which can be bought for 30 cents. How many can be bought for $2.50 ?

25. Divide $ 3620 among A, B and C so that B shall receive $20 less than A, and C as many times B's share as there are dollars in A's share.

26. Find two fractions whose sum is §, and whose difference is equal to their product.

27. Two men start at the same time to meet each other from towns which are 25 miles apart. One takes 18 minutes longer than the other to walk a mile, and they meet in 5 hours. How fast does each walk?

28. The men in a regiment can be arranged in a column twice as deep as it is broad. If the number be diminished by 206, the men can be arranged in a hollow square three deep, having the same number of men in each outer side of the square as there were in the depth of the column. How many men were there at first in the regiment?

29. The area of a certain rectangle is equal to the area of a square whose side is six inches shorter than one of the sides of the rectangle. If the breadth of the rectangle be increased by one inch

and its length diminished by two inches, its area would be unaltered. Find the lengths of its sides.

30. The diagonal and the longer side of a rectangle are together five times the shorter side, and the longer side exceeds the shorter by 35 yards. What is the area of the rectangle?

31. If the greatest side of a rectangle be diminished by 3 yards and the less by 1 yard its area would be halved; and if the greater be increased by 9 yards and the less diminished by 2 yards its area would be unaltered. Find the sides.

32. Two trains A and B leave P for Q at the same time as two trains C and D leave Q for P. A passes C 120 miles from P, and D 140 miles from P. B passes C 126 miles from Q, and D half way between P and Q: find the distance from P to Q.

MISCELLANEOUS EXAMPLES

A. 1. Simplify 2x

IV.

[3x-9y - {2 x − 3 y − (x + 5 y)}].

2. Multiply a2+25 b2+4 c2+5 ab−2 ac+10 bc by a−5b+2c.

3. Divide x3+(4 ab − b2) x − ( a − 2 b) (a2+3 b2) by x−a+2b. 4. Find the factors of (i.) (2x + y − z)2 − (x + 2y + 4 z)2. (ii.) x2y2 — x2 — y2 + 1, and (iii.) x2y2x2 — x2z — y2z + 1.

8. The difference of the cubes of two consecutive integer numbers is 919: find the numbers.

B. 1. Simplify (x + 3)3 − 3(x + 2)3 + 3(x + 1)3 — x3.

2. Show that

(x − a)2+(y- b)2 + (a2 + b2 − 1) (x2 + y2 − y)
= (xa + by − 1)2 + (bx — ay)2.

=

3. Divide x - 2 a3x3 + as by x2 - 2 ax + a2.

4. Find the L. C. M. of

8x3 27, 16 x2 + 36 x2 + 81, and 6x2-5 x 6.

(i.) (6x)(1 + 2x) + 3x(x+5)= (x + 1)2 — X. (ii.) 5 x2+7x= 160.

(iii.) x2 + xy 10, y2 — xy = 3.

7. Show that the sum of the squares of the roots of the equation x2 - 5x + 2 = 0 is 21.

8. At a concert $b was received for reserved seats, and the same sum for unreserved seats. A reserved seat cost 50 cents more than an unreserved seat, but b more tickets for unreserved than for reserved seats were sold. How many tickets were sold altogether?

C. 1. Simplify 12 a

- 3{b − 2 (a – 3 b) — 2 a}.

2. Multiply a3 + 2 a2b — ab2 + 2 b3 by a3 — 2 a2b — ab2 — 2 b3.

3. Divide 24x2 - 10 x3y - 8 x2y2 + 10 xy3 — 4 y1 by 2 y2-xy-4x2.

4. Find the factors of

and of

9x2+9x+2,

4(ab — cd)2 — (a2 + b2 — c2 — d2)2.

7. Find the least value of x2 + 6x + 12, and the greatest value of 6 x - x2 - 4.

8. A and B have 45 coins between them, which are all dollars and dimes. A has four times as many dollars as dimes, and B has just as many dollars as dimes; also A has $9.50 more than B. How much money has each ?

D. 1. Show that

(b + c)2 — a2 + (c + a)2 − b2 + (a + b)2 − c2 = (a + b + c)2. 2. Arrange (1 + x)1 + 2(1 − x + x2) according to ascending powers of x.

3. Show that the difference between the squares of any two consecutive numbers is one more than double the smaller number.

For what value of x will all three expressions vanish?

7. If x1, x2 are the roots of ax2 + bx + c = 0, prove that

8. Out of a cask containing 60 gallons of alcohol a certain quantity is drawn off and replaced by water. Of the mixture a second quantity, 14 gallons more than the first, is drawn off and replaced by water. The cask then contains as much water as alcohol. How much was drawn off the first time?

2. Show that (a + b)1 − (a2 − b2)2 = 4 ab (a + b)2, and that

2(a - b) (a — c) + 2(b − c) (b − a) + 2(c − a) (c — b)

3. Divide (a + 2 b − 3 c + d) 2 − (2 a + b + 3 c − d)2 by a + b.

(ii.) (x − 1)(x − 2) + (x − 2) (x − 3)+(x − 3) (x − 1) = 11.

(iii.) x

7. For what values of x are

other?

+ and + equal to one an

с
α с

8. A tricyclist rode 180 miles at a uniform rate. If he had ridden 3 miles an hour slower than he did, it would have taken him 3 hours longer. How many miles an hour did he ride?