CHAPTER XXIV.

HARDER PROBLEMS.

184. IN previous chapters we have given collections of problems which lead to simple equations. We add here a few examples of somewhat greater difficulty.

Example 1. If the numerator of a fraction is increased by 2 and the denominator by 1, it becomes equal to ; and if the numerator and denominator are each diminished by 1, it becomes equal to : find the fraction.

Let x be the numerator of the fraction, y the denominator; then the fraction is

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y

Example 2. At what time between 4 and 5 o'clock will the minute-hand of a watch be 13 minutes in advance of the hour-hand?

Let x denote the required number of minutes after 4 o'clock; then, as the minute-hand travels twelve times as fast as the hourhand, the hour-hand will move over minute-divisions in x minutes.

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At 4 o'clock the minute-hand is 20 divisions behind the hour-hand, and finally the minute-hand is 13 divisions in advance; therefore the minute-hand moves over 20+13, or 33 divisions more than the hourhand.

Thus the time is 36 minutes past 4.

If the question be asked as follows: "At what times between 4 and 5 o'clock will there be 13 minutes between the two hands? we must also take into consideration the case when the minute-hand is 13 divisions behind the hour-hand. In this case the minute-hand gains 20 - 13, or 7 divisions.

Example 3. A grocer buys 15 lbs. of figs and 28 lbs. of currants for $2.60; by selling the figs at a loss of 10 per cent., and the currants at a gain of 30 per cent., he clears 30 cents on his outlay; how much per pound did he pay for each ?

Let x, y denote the number of cents in the price of a pound of figs and currants respectively; then the outlay is

1

10

.(1).

The loss upon the figs is x15x cents, and the gain upon the

From (1) and (2) we find that x=8, and y=5; that is, the figs cost 8 cents a pound, and the currants cost 5 cents a pound.

Example 4. Two persons A and B start simultaneously from two places, c miles apart, and walk in the same direction. A travels at the rate of p miles an hour, and B at the rate of q miles; how far will A have walked before he overtakes B?

Suppose A has walked x miles, then B has walked x- c miles.

A walking at the rate of p miles an hour will travel x miles in

Example 5. A train travelled a certain distance at a uniform rate. Had the speed been 6 miles an hour more, the journey would have occupied 4 hours less; and had the speed been 6 miles an hour less, the journey would have occupied 6 hours more. Find the distance.

Let the speed of the train be x miles per hour, and let the time occupied be y hours; then the distance traversed will be represented by xy miles.

On the first supposition the speed per hour is x+6 miles, and the time taken is y-4 hours. In this case the distance traversed will be represented by (x+6)(y – 4) miles.

On the second supposition the distance traversed will be represented by (x-6)(y+6) miles.

All these expressions for the distance must be equal;

:: xy = (x+6)(y −4) = (x − 6)(y+6).

From these equations we have

1. If the numerator of a fraction is increased by 5 it reduces to, and if the denominator is increased by 9 it reduces to find the fraction.

2. Find a fraction such that it reduces to if 7 be subtracted from its denominator, and reduces to on subtracting 3 from its numerator. 3. If unity is taken from the denominator of a fraction it reduces to; if 3 is added to the numerator it reduces to : required the fraction.

4. Find a fraction which becomes on adding 5 to the numerator and subtracting 1 from the denominator, and reduces to on subtracting 4 from the numerator and adding 7 to the denominator.

5. If 9 is added to the numerator a certain fraction will be increased by; if 6 is taken from the denominator the fraction reduces to required the fraction.

6. At what time between 9 and 10 o'clock are the hands of a watch together?

7. When are the hands of a clock 8 minutes apart between the hours of 5 and 6?

8. At what time between 10 and 11 o'clock is the hour-hand six minutes ahead of the minute-hand?

9. At what time between 1 and 2 o'clock are the hands of a watch in the same straight line?

10. When are the hands of a clock at right angles between the hours of 5 and 6?

11. At what times between 12 and 1 o'clock are the hands of a watch at right angles?

12. A person buys 20 yards of cloth and 25 yards of canvas for $35. By selling the cloth at a gain of 15 per cent. and the canvas at a gain of 20 per cent. he clears $5.75; find the price of each per yard.

13. A dealer spends $1445 in buying horses at $75 each and cows at $20 each; through disease he loses 20 per cent. of the horses and 25 per cent. of the cows. By selling the animals at the price he gave for them he receives $1140; find how many of each kind he bought.

14. The population of a certain district is 33000, of whom 835 can neither read nor write. These consist of 2 per cent. of all the males and 3 per cent. of all the females: find the number of males and females.

15. Two persons C and D start simultaneously from two places a miles apart, and walk to meet each other; if C walks p miles per hour, and D one mile per hour faster than C, how far will D have walked when they meet?

16. A can walk a miles per hour faster than B; supposing that he gives B a start of c miles, and that B walks n miles per hour, how far will A have walked when he overtakes B?

17. A, B, C start from the same place at the rates of a, a+b, a+26 miles an hour respectively. B starts n hours after A, how long after B must C start in order that they may overtake A at the same instant, and how far will they then have walked ?

18. Find the distance between two towns when by increasing the speed 7 miles per hour, a train can perform the journey in 1 hour less, and by reducing the speed 5 miles per hour can perform the journey in 1 hour more.

19. A person buys a certain quantity of land. If he had bought 7 acres more each acre would have cost $4 less, and if each acre had cost $18 more he would have obtained 15 acres less: how much did he pay for the land?

20. A can walk half a mile per hour faster than B, and threequarters of a mile per hour faster than C. To walk a certain distance C takes three-quarters of an hour more than B, and two hours more than A: find their rates of walking per hour.

21. A man pays $90 for coal; if each ton had cost 50 cents more he would have received 2 tons less, but if each ton had cost 75 cents less he would have received 4 tons more; how many tons did he buy?

22. A and B are playing for money; in the first game A loses one-half of his money, but in the second he wins one quarter of what B then has. When they cease playing, A has won $10, and B has still $25 more than A; with what amounts did they begin?

23. The area of three fields is 516 acres, and the area of the largest and smallest fields exceeds by 30 acres twice the area of the middle field. If the smallest field had been twice as large, and the other two fields half their actual size, the total area would have been 42 acres less than it is; find area of each of the fields.

24. A, B, C each spend the same amount in buying different qualities of cloth. B pays three-eighths of a dollar per yard less than A and obtains three-fourths of a yard more; C pays fiveeighths of a dollar per yard more than A and obtains one yard less; how much does each spend ?

25. B pays $28 more rent for a field than A; he has threefourths of an acre more and pays $1.75 per acre more. C pays $72.50 more than A; he has six and one-fourth acres more, but pays 25 cents per acre less; find the size of the fields.