13. What two numbers are as 4 to 5, from each of which if 10 be subtracted, the remainders will be as 2 to 3?

Let 4x and 5x denote the numbers required; then 4x-10, and 5x-10 will denote the remainders; by the question, 4x-10; 5x-10 :: 2 : 3, and (by Theo. 2.) 12x-30=10x-20; by transposition, 12x-10x=30-20, or 2x=10; .. x=5;

whence 4x=20, and 5x=25; .. the numbers are 20 and 25.

5x
4

then 4'; 5:x: the greater number;

also, by the question, x-10=the first remainder,

by transposition, 6x-5x-60-40'; .. x=20;

whence, the numbers are 20 and 25.

14. Two merchants, A and B, entered in partnership; A advanced £50 more than B; they each gained £100, which made their respective shares in the proportion of 13 to 12. How much did each person advance?

Let x = the sum A advanced; then x 50 the sum B advanced;

.. by the question, x+100 A's share after gain

ing £100,

and x-50+100, or x+50 B's share after

gaining £100;

..x+100 x +50 :: 13; 12;

.. 12x+1200=13x+650;

by transposition, x=550;

whence, A advanced £550, and B £500.

15. Between 9 and 10 o'clock in the evening, when the hour and minute hands are exactly together. Required the time.

Let x the time past 9;

then, since the minute hand goes 12 times round whilst the other goes once,

1219+x:x;

12x=9+x;

by transposition, 11x=9; .. X=

the time past 9 o'clock.

16. Bought two pieces of cloth, which together measured 70 yards, for £49. 10s.; I gave 15s. per yard for one piece, and 12s. per yard for the other; this latter piece being damaged I returned it. How much had I to pay?

Let x the number of yards at 15s. per yard; then 70—x = the number of yards at 12s. per yard; .. 15x= the cost of the piece at 15s. per yard, and 840-12 the cost of the piece at 12s. per yard;

.. by the question, 15x+840-12x=990;

by transposition, 3x=990-840-150;

whence, 50 yards at 15s. = £37. 10s. the sum to pay.

17. Four men, A, B, C, and D, bought a hive of bees for £1; A paid one third, B one fourth, C one fifth, and D one sixth. How much did each pay of the 20s.?

clearing the equation of fractions,

20x+15x+12x+10x=400, or 57x=400;

whence, A 7s., B 515s., C 417s., and D 339s.

18. A person bought a quantity of eggs at two a penny, and as many at three a penny, and sold them all out at five for two-pence, and by so doing lost 4d. What number of eggs did he buy?

the number of eggs at each price, or 2x = the eggs bought;

the price of the first quantity in pence,

and the price of the 2nd quantity in pence;

4.x 5

and 52: 2x: the selling price, which, by

the question, is less by 4d. than the cost price,

clearing the equation of fractions,

15x+10x-24x=120; .. x=120.

18. A person at play lost 10s. more than one half of his money; he afterwards lost 10s. more than half the sum he had left on going home, he was met by a robber, who took from him half of what he then had, and 10s. more; when he arrived at home, he found that he had only £5 left. What sum had he at first? Let x = the sum he had at first in pounds;

clearing this equation of fractions,

2x-6-x+3+4=40;'

.. by transposition, x=47.

EXAMPLES FOR PRACTICE

In Simple Equations of one unknown quantity.

1. What number is that, from the treble of which if 25 be subtracted, the remainder is 74 ?

Ans. 33.

2. Add 40 to A's age, and multiply the sum by 5; this product will be equal to 365. Required his age. Ans. 33.

3. Bought four pieces of cloth, which together measured 100 yards; the second was twice, the third three times, and the fourth four times as long as the first. Required the length of each piece.

Ans. 10, 20, 30, and 40 yards.

4. A Butcher bought two flocks of sheep, at a certain price per head; the first flock contained 12 sheep, and the second 17; he gave £5. 10s. more for one flock than for the other. Required the price of each sheep. Ans. 22s.

5. Lent my friend a certain sum of money at 4 per cent. per annum, which amounted to £560 in 3 years, at simple interest. Required the sum lent.

Ans. £500.

6. A person had 5 children, the eldest of which earned five times, the second four times, the third three times, and the fourth twice as much as the youngest; they altogether earned 6s. 3d. How much did each earn?

Ans. 5d., 10d., 15d., 20d., and 25d.