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x = the price of an orange. 58 5* Then
= the price of a lemon by the first condi
17 tion, &c.
19. A footman, who contracted for $72 a year and a livery suit, was turned away at the end of 9 months, and received only $32 and the livery. What was the value of the livery?
20. A landlord let his farm for £10 a year in money and a certain number of bushels of corn. When corn sold at 10s. a bushel, he received at the rate of 10s. an acre for his land; but when it sold for 13s. 6d. a bushel, he received 13s. an acre. How many
bushels of corn did he receive ? Let
X = the number of bushels. Then 10x + 200 = the year's rent in shillings ; 10 x + 200
= x + 20 = the number of acres. 10 27 x + 400 = the year's rent at the second rate in sixpences. 27 x + 400
= the number of acres, which must be equal to 26 the other, &c.
21. A man commenced trade with a certain sum of money, which he improved so well, that at the year's end he found he had doubled his first stock wanting $1000; and so he went on every year doubling the last year's stock wanting $1000; at the end of the third year he found that he had just three times as much money as he commenced with. What was his first stock ?
22. A man, having a certain sum of money, went to a tavern, where he borrowed as much money as he then had, and then spent a shilling ; with the remainder he went to another tavern, where he borrowed as much as he then had, and then spent a shilling, and so he went to a third and a fourth tavern, borrowing and spending as before; after which he had nothing left. How much money had he at first ?
23. It is required to divide the number 60 into two such parts, that one seventh of the one may be equal to one eighth of the other.
24. It is required to divide the number 85 into two such parts that of the one added to of the other may make 60.
25. It is required to divide the number 100 into two such parts, that if one third of one part be subtracted from one fourth of the other, the remainder may be 11.
26. It is required to divide the number 48 into two such parts, that one part may be three times as much above 20, as the other wants of 20.
27. A man distributed 20 shillings among 20 people, giving 6 pence apiece to some, and 16 pence apiece to the rest. What number of persons were there of each kind ?
28. A man paid £100 with 208 pieces of money, a part guineas at 21s. each, and a part crowns at 5s. each. How many pieces were there of each sort?
29. A countryman had two flocks of sheep, the smaller consisting entirely of ewes, each of which brought him 2 lambs. On counting them he found that the number of lambs was equal to the difference between the two flocks. If all his sheep had been ewes, and brought forth three lambs apiece, his stock would have been 432. Required the number in each flock.
Let x = the number in the less.
3x = the number in the larger.
4x = the number in both, &c. 30. When the price of a bushel of barley wanted but 3d. to be to the price of a bushel of oats as 8 to 5, four bushels of barley and 75. 6d. in money were given for nine bushels of oats. What was the price of a bushel of each ? Let
x = the price of a bushel of oats in pence. Then 3= the price of a bushel of barley, &c.
5 31. A market-woman bought a certain number of eggs at the rate of 2 for a cent, and as many at 3 for a cent, and sold them out at the rate of 5 for two cents ; after which she observed, that she had lost four cents by them. How many eggs of each sort had she ?
= the number of each sort.
the price of x eggs at 2 for a cent.
And = the price of x eggs at 3 for a cent.
3 These added together make what the eggs cost. The whole number is 2 x ; these at 5 for two cents come to
2 3 5 32. A cistern has two fountains to fill it ; the first will fill it alone in 7 hours, and the second in 5 hours. In what time will the cistern be filled, if both run together?
Let x = the number of hours required to fill it.
The first would fill of it in an hour, and the second would fill of it in an hour.
Both together then would fill + } in an hour; and in * hours both would fill *
of it. But by the conditions it
ng 5 was to be filled in a hours. Therefore, + =1 cistern.
5 33. A gentleman, having a piece of work to do, hired two men and a boy to do it; one man could do it alone in 5 days, the other could do it alone in 8 days, and the boy could do it alone in 10 days. How long would it take the three together to do it?
34. A cistern, into which the water runs by two cocks, A and B, will be filled by them both running together in 12 hours; and by the cock A alone in 20 hours. In what time will it be filled by the cock B alone ?
Let X = the time in which В will fill it alone. Both will fill iof it in an hour, A alone a' of it, and B will fill 11-7 of it in an hour, &c.
35. A man and his wife usually drank out a vessel of beer in 12 days : but when the man was from home it would usually last the wife alone 30 days. In how many days would the man alone drink it out ?
36. The hold of a ship contained 442 gallons of water. This was emptied out by two buckets, the greater of which, holding twice as much as the other, was emptied twice in three minutes, but the less three times in two minutes; and the whole time of emptying was 12 minutes. Required the size of each.
The greater was emptied 3 times in the 12 minutes, &c. 37. Two persons, A and B, have the same income. A saves
of his ; but B, by spending £80 a year more than A, at the end of 4 years finds himself £220 in debt. What did each receive and expend annually ?
38. After paying i of my money, and of the remainder, I had 72 guineas left. How much had I at first ?
39. A bill of £120 was paid in guineas and moidores, the guineas at 21s., and the moidores at 27s. each ; the number of pieces of both sorts was just 100. How many were there of each ?
40. It is required to divide the number 26 into three such parts, that if the first be multiplied by 2, the second by 3, and third by 4, the products shall all be equal.
2 x Let x = the first part. The second part must be and
4 2 41. It is required to divide the number 54 into three such parts, that 4 of the first, f of the second, and 1 of the third, may be all equal to each other. Let
2 x = the first part. Then 3x = the second part, &c.
42. A person has two horses and a saddle, which of itself is worth £25. Now if the saddle be put upon the back of the first horse, it will make his value double that of the second; but if it be put upon the back of the second, it will make his value triple that of the first. What is the value of each horse ?
43. A man has two horses and a chaise, which is worth $183. Now if the first horse be harnessed to the chaise, the horse and chaise together will be worth once and two sevenths the value of the other ; but the other horse being harnessed, the horse and chaise together will be worth once and five
the third part 2 or
eighths the value of the first. Required the value of each horse.
Equations with two Unknown Quantities. VIII. Many examples involve two or more unknown quantities. In fact, many of the examples already given involve several unknown quantities, but they were such, that they could all be derived from one. When it is necessary to use two unknown quantities in the solution, the question must always contain two conditions, from which two equations may be derived. When this is not the case the question cannot be solved.
1. A boy bought 2 apples and 3 oranges for 13 cents ; he afterwards bought, at the same rate, 3 apples and 5 oranges for 21 cents. How much were the apples and oranges apiece?
Let x= the price of an orange,
and y=the price of an apple. 1.
3x + 2y= 13,
5x + 3y= 21. Multiply the first equation by 3, and the second by 2, 3.
9x + 6y= 39 4.
10x + 6y=42. Subtract the first from the second, because the y's being alike in each, the difference between the numbers 39 and 42 must depend upon the x's. 5. x = 3 cents, the price of an orange. Putting this value of x into the first equation,
y : 2 cents, the price of an apple. Proof. 2 apples at 2 cents each come to 4 cents, and 3 oranges at 3 cents come to 9 cents. 9 + 4
9+4= 13. So 3 apples and 5 oranges come to 21 cents.
Note. In this example I observed, that the coefficient of in the first equation is 2, and in the second, the coefficient of y is 3. I multiplied the whole of the first equation by 3, and the whole of the second by 2; this formed two new equations in which the coefficients of y are alike. If the first equation had been multiplied by 5 and the second by 3, the coefficients of x would have been alike, and instead of y would have been