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 7s. 6d. in money were given for nine bushels of oats. What was the price of a bushel of each ?

Let

Then

8 x 5

x= the price of a bushel of oats in pence.

3 the price of a bushel of barley, &c.

Ans. Barley 45d., oats 30d. per bushel.

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?

And

3

= the price of a eggs at 3 for a cent.

These added together make what the eggs cost.

The whole number is 2x; these at 5 for two cents

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 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

+ 5

conditions it was to be filled in a hours.

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. would it take the three together to do it?

How long

Ans. 2 days.

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

will fill

12

will fill 1⁄2

the time in which B will fill it alone. Both

of it in an hour, A alone

of it in an hour, &c.

of it, and B Ans. 30 hours. 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?

Ans. 20 days. 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 8 times in the 12 minutes, &c. Ans. The less 13, and the greater 26 gallons. 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? Ans. Their annual income is £125; A spends £100, and B £180.

38. After paying one of my money, and of the remainder, I had 72 guineas left. How much had 1 at first? Ans. 120 guineas.

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? Ans. 50 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. Ans. The parts are 12, 8, and 6. Let x the first part. The second part must be

41. It is required to divide the number 54 into three such parts, that the first, of the second, and of the third, may be all equal to each other.

42. A person has

the second part, &c.

Ans. The parts are 12, 18, and 24. 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?

Ans. £15, and £20, respectively. 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 eighths the value of the first. Required the value of each horse.

Ans. $384, and $441, respectively.

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

the price of an orange,

and y the price of an apple.

3x+2y= 13,

5x + 3y = 21.

Multiply the first equation by 3, and the second by 2,

1.

2.

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, 9+2y= 13

6.

7.

x

y = 2 cents, the price of an apple. Note. In this example I observed, that the coefficient of y 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 a would have been alike, and x instead of y would have been made to disappear by subtraction, and the same result would have been finally obtained. It is evident, that the coefficients of either of the unknown quantities may always be rendered alike in the two equations, by multiplying the first equation by the coefficient which the quantity that you wish to make disappear has in the second