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PROOF.

§ 143. The product of the two means is equal to the product of the extremes (see § 141). Hence, if either of these equal products be divided by one of the mean terms the quotient will be the other. Therefore,

Divide the product of the extremes by one of the mean terms, and if the work is right the quotient will be the other mean term.

EXAMPLES.

1. The 1st term is 4, the 2d 8, the 3d 12, and the answer 24 : is the answer true ?

The product of the extremes is 96. If this be divided by 8 OPERATION OF PROOF. the quotient is 12; if by 12 the

24 X4=96 quotient is 8: hence, the an

8)96(12; or swer was true.

12)96(8

RULE OF THREE BY CANCELLING.

If two numbers are to be multiplied together and their product divided by a third, the operations may be abridged by striking out or cancelling any factor which is common to the divisor and either of the other numbers. For example, if 6 is to be multiplied by 8 and the product divided by 4, we have 6X8 48

6X8

=12; or,

=6X2=12: 4 4

4 in the latter case we cancelled the factor 4 in the numerator and denominator, and multiplied 6 by the quotient 2. It is found most convenient to

OPERATION. draw a vertical line and to place

6 the numbers to be multiplied to

18 2 gether on the right, and the divisors on the left. Then,

Ans. 12. I. If there be two equal numbers, one on each side of the line, omit them.

JI. If any number on the left has a common divisor with a number on the right, divide those numbers by their

18....

greatest common divisor and write the quotients opposite the numbers. Then multiply together the remaining numbers and the quotients, and divide the product on the right of the line by the product on the left.

Note 1.-If the product on the left of the line is 1, no division will be necessary.

EXAMPLES.

12.

1. Multiply 24 by 16 and divide the product by 12.

The greatest common divisor between 12 and 24 is 12.

124 ... 2 1 16

Ans. 32.

Q. How do you write the numbers before cancelling? If there are equal numbers on each side of the line what do you do with them? If a number on the left has a common divisor with a number on the right, what is done? If the number on the left of the line is 1, will a division be necessary? Why not? 2. What is the 4th term of the proportion 16 : 15

Ans. Here 16 is the greatest com

OPERATION. mon divisor between 16 and 48,

15 and gives 3 for the quotient on the 16 right.

Ans. 45

:: 48

:

16 ... 1148... 15

3. If 4 pounds of tea cost $8, what will 12 pounds cost at the same rate ?

18

2

12 Ans. $24.

48 cost ,

48 1/144 ...3 cost?

Ans. $201,75. 5. If 25 yards of cloth cost £2 38 4d, what will 5 yards cost? In this example we have 5 lest

15 1 on the left of the line, by which 25...5 £2 3s 4d the product on the right must be

5)£2 3s 4d divided.

Ans. 8s 8d.

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APPLICATIONS.

1. If 4 hats cost $12, what will 55 cost at the same rate ?

Ans. $165. 2. What is the value of 2cwt. of sugar at 5d per pound?

Ans. £4 13s 4d. 3. If 40 yards of cloth cost $170, what will 325 yards cost?

Ans. $1381,25. 4. If 240 sheep yield 660 pounds of wool, how many pounds will be obtained from 1200 ? Ans. 33001).

5. If 2 gallons of molasses cost 65 cents, what will 3 hogsheads cost?

Ans. $61,421 6. If a man travels at the rate of 210 miles in 6 days, how far will he travel in a year supposing him not to travel on Sundays?

Ans. 10955 miles. 7. If 1 yard of cloth cost $3,25, what will be the cost of 3 pieces each containing 25 yards?

Ans. 243,75. 8. If 30 barrels of flour will support 100 men for 40 da how long would it subsist 25 men ? Ans. 10 days.

9. If 30 barrels of flour will support 100 men for 40 days, how long would it subsist 400 men ?

Ans. 160 days.

10. A owes B £679 6s, but compounds with him by paying 3s 4d on the pound: how much does B receive of his debt?

Ans. £113 4s 4d. 11. If 90 bushels of oats will feed 40 horses for 6 days, how long would 450 bushels last them? Ans. 30 days.

12. If 5cwt. 3qr. 141b. of sugar cost £6 ls 8d, what will 35cwt. 281b. cost?

Ans. £36 10s. 13. What is the cost of 3cwt. of coffee at 150 per pound?

Ans. £21. 14. If 3 quarters of a yard of velvet cost 7s 3d, how many yards can be bought for £13 158 6d?

Ans. 28yd. 2qr. 15. If an ingot of gold weighing 916. 9oz. 12 prwt. be worth £470 8s, what is that per grain?

Ans. 2d. 16. Bought 4 bales of cloth, each containing 6 pieces, and each piece 27 yards, at £16 4s per piece; what is the value of the whole, and the cost per yard ?

Ans. £388 16s 12s per yard. 17. What will be the cost of 72 yards of cloth, at the rate of £5 12s for 9 yards?

Ans. £44 16s. 18. A person's annual income is £146: how much is that per day?

Ans. 8s. 19. If 3 paces or common steps of a person, be equal to 2 yards, how many yards will 160 paces makes?

Ans. 106yd. 2ft. 20. What length must be cut off from a board that is 9 inches wide, to make a square foot, that is, as much as is contained in 12 inches in length and 12 in breadth?

Ans. 16 inches. 21. If 750 men require 22500 rations of bread for a month, how many rations will a garrison of 1200 men require?

Ans. 36000. 22. If 7cwt. lqr. of sugar cost $64,96, what will be the price of 4cwt. 2gr.?

Ans. $40,32. 23. The clothing of a regiment of foot of 750 men amounts to £2831 58: what will it cost to clothe a body of 3500 men?

Ans. £13212 10s. 24. How many yards of carpeting, that is 3 feet wide, will cover a floor that is 27 feet long and 20 feet broad?

Ans. 60 yards.

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25. What is the cost of 6 bushels of coal at the rate of £1 14s 6d the chaldron?

Ans. 5s 9d. 26. If 6352 stones of 3 feet long will complete a certain quantity of wall, how many stones of 2 feet long will raise the like quantity ?

Ans. 9528. 27. If a person can count 300 in two minutes, how many can he count in a day?

Ans. 216000. 28. A garrison of 536 men have provisions for 365 days: how long will those provisions last if the garrison be increased to 1124 men ? Ans. 174 days and 6476

29. What will be the tax upon £763 15s at the rate of 38 6d per pound sterling?

Ans. £133 13s 1 d. 30. What will be the tax on $3758, at the rate of 4 mills on the dollar ?

Ans. $15,032. 31. A certain work can be raised in 12 days by working 4 hours each day: how long would it require to raise the work by working 6 hours per day? Ans. 8 days.

32. What quantity of corn can I buy for 90 guineas, at the rate of 6 shillings a bushel ? Ans. 315 bushels.

33. A person failing in trade owes £977, at which time he has in money, goods, and recoverable debts £420 6s 31d: now, supposing an equal division among his creditors, how much will they get on the pound ?

Ans. 8s 71d. 34. A pasture of a certain extent having supplied a body of horse, consisting of 3000, with forage for 18 days, how many days would the same pasture have supplied a body of 2000 horse ?

Ans. 27 days. 35. Suppose a gentleman's income to be 600 guineas a year, and that he spends 25s 6d per day, one day with another: how much will he have at the end of the year ?

Ans. £164 12s 6d. 36. What is the cost of 30 pieces of lead, each weighing 1cwt. 121b. at the rate of 16s 4d the cwt. ?

Ans. £27 2s 6d. 37. The governor of a besieged place has provisions for 54 days at the rate of 21b, of bread per ration, but is desirous to prolong the siege to 80 days, in expectation of succor: in that case what must be the ration of bread ?

Ans. 12816.

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