CHAPTER XXIV. HARDER PROBLEMS. 191. IN previous chapters we have problems which lead to simple equations. examples of somewhat greater difficulty. given collections of We add here a few Example 1. 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 Therefore 15x+28y cents. 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 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? x 12 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. 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 2013, or 7 divisions. Example 3. 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? - c miles. hours; and B will travel x- c miles in being equal, we have Suppose A has walked x miles, then B has walked x A walking at the rate of p miles an hour will travel x miles in х Ρ hours: these two times X x-c Example 4. 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 repre(x-6) (y+6) miles. sented by or and or All these expressions for the distance must be equal; .. xy=(x+6) (y − 4) = (x − 6) (y +6). From these equations we have xy=xy+by-4x-24, 6y-4x=24... xy=xy-6y+6x-36, From (1) and (2) we obtain x=30, y=24. .(1); .(2). Example 5. A person invests $3770, partly in 3 per Cent. Bonds at $102, and partly in Railway Stock at $84 which pays a dividend of 4 per cent.: if his income from these investments is $136.25 per annum, what sum does he invest in each ? Let x denote the number of dollars invested in Bonds, y the number of dollars invested in Railway Stock; then Therefore he invests $2720 in Bonds and $1050 in Railway Stock. EXAMPLES XXIV. 1. A sum of $100 is divided among a number of persons; if the number had been increased by one-fourth each would have received a half-dollar less: find the number of persons. 2. I bought a certain number of marbles at four for a cent; I kept one-fifth of them, and sold the rest at three for a cent, and gained a cent: how many did I buy? 3. I bought a certain number of articles at five for six cents; if they had been eleven for twelve cents, I should have spent six cents less how many did I buy? 4. A man at whist wins twice as much as he had to begin with, and then loses $ 16; he then loses four-fifths of what remained, and afterwards wins as much as he had at first: how much had he originally, if he leaves off with $80 ? 5. I spend $69.30 in buying 20 yards of calico and 30 yards of silk; the silk costs as many quarters per yard as the calico costs cents per yard: find the price of each. 6. A number of two digits exceeds five times the sum of its digits by 9, and its ten-digit exceeds its unit-digit by 1: find the number. 7. The sum of the digits of a number less than 100 is 6; if the digits be reversed the resulting number will be less by 18 than the original number: find it. 8. A man being asked his age replied, "If you take 2 years from my present age the result will be double my wife's age, and 3 years ago her age was one-third of what mine will be in 12 years. What were their ages? 9. At what time between one and two o'clock are the hands of a watch first at right angles? 10. At what time between 3 and 4 o'clock is the minute-hand one minute ahead of the hour-hand? 11. When are the hands of a clock together between the hours of 6 and 7? 12. It is between 2 and 3 o'clock, and in 10 minutes the minutehand will be as much before the hour-hand as it is now behind it: what is the time? 13. At an election the majority was 162, which was three-elevenths of the whole numbers of voters: what was the number of the votes on each side? 14. A certain number of persons paid a bill; if there had been 10 more each would have paid $2 less; if there had been 5 less each would have paid $2.50 more: find the number of persons, and what each had to pay. 15. A man spends $100 in buying two kinds of silk at $4.50 and $4 a yard; by selling it at $4.25 per yard he gains 2 per cent.: how much of each did he buy? H. A. 12 16. Ten years ago the sum of the ages of two sons was one-third of their father's age: one is two years older than the other, and the present sum of their ages is fourteen years less than their father's age: how old are they? 17. A and B start from the same place walking at different rates; when A has walked 15 miles B doubles his pace, and 6 hours later passes A: if A walks at the rate of 5 miles an hour, what is B's rate at first? 18. A basket of oranges is emptied by one person taking half of them and one more, a second person taking half of the remainder and one more, and a third person taking half of the remainder and six more. How many did the basket contain at first? 19. A person swimming in a stream which runs 13 miles per hour, finds that it takes him four times as long to swim a mile up the stream as it does to swim the same distance down: at what rate does he swim? 20. At what times between 7 and 8 o'clock will the hands of a watch be at right angles to each other? When will they be in the same straight line? 21. The denominator of a fraction exceeds the numerator by 4; and if 5 is taken from each, the sum of the reciprocal of the new fraction and four times the original fraction is 5: find the original fraction. 22. Two persons start at noon from towns 60 miles apart. One walks at the rate of four miles an hour, but stops 2 hours on the way; the other walks at the rate of 3 miles an hour without stopping: when and where will they meet? 23. A, B, and C travel from the same place at the rates of 4, 5, and 6 miles an hour respectively; and B starts 2 hours after A. How long after B must C start in order that they may overtake A at the same instant? 24. A dealer bought a horse, expecting to sell it again at a price that would have given him 10 per cent. profit on his purchase; but he had to sell it for $50 less than he expected, and he then found that he had lost 15 per cent. on what it cost him: what did he pay for the horse? 25. A man walking from a town, A, to another, B, at the rate of 4 miles an hour, starts one hour before a coach travelling 12 miles an hour, and is picked up by the coach. On arriving at B, he finds that his coach journey has lasted 2 hours: find the distance between A and B. 26. What is the property of a person whose income is $1140, when one-twelfth of it is invested at 2 per cent., one-half at 3 per cent., one-third at 4 per cent., and the remainder pays him no dividend? |