CHAPTER XXIV. HARDER PROBLEMS. 191. 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. 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 15x+28y cents. .(1). 10 3 currants is x 28y cents; therefore the total gain is 10 42y 3x cents ; 3x (2). 5 2 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-band? 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. 12 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 hour. hand. 11 x=33; .. X=36. 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. = +7, Hence 12 7 which gives x=7 11 7' Therefore the times are 7 11 past 4, and 36' past 4. Example 3. Two persons A and B start simultaneously froni 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 a - c miles. P hours; and B will travel » – c miles in hours: these two times 9 being equal, we have X-C 9 P qu=px - pc; whence pc P-9° Therefore A has travelled pc miles. P-9 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 specd 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. or or 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; .. ay=(x+6) (y – 4)=(x – 6) (y+6). From these equations we have xy=xy + 6y – 4.30 – 24, (1); and xy=xy – 6y + 6x – 36, .(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 x+y=3770 ....(1). 3x The income from Bonds is $ or $ ; and that from Rail. 102' 34 way Stock is $ 434, or $3y 84' 56 2 Зу Therefore + = 1364 (2) 34 56 51 From (2) 2+ y=4632); 28 therefore by subtracting (1) 23 y=862); 28 whence y=28 x 371=1050; and from (1) X=2720. 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 fór 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 1} 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 be 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 21 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 cento 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 41 per cent., and the remainder pays him no dividend? |