NOTE.-Most of the above coins vary in value from time to time.

PROPORTION AND CANCELLATION.

Proportion, from proportio ;

Cancellation, from cancello, (Latin.)

PROPORTION, arithmetically considered, is the identity or similitude of two or more ratios. Four numbers are proportional when the fourth contains the third just as many times as the second contains the first: 2, 4, 8, 16 are proportional, because 16 contains 8 just as many times as 4 contains 2; i. e. 2 into 4 goes 2 times, and 8 into 16 goes 2 times.

How many times 2 in 6? What number will bear the same proportion to 6 that 6 bears to 2? Ans. 18. Why?

How many times 3 in 12? What number will bear the same proportion to 12 that 12 bears to 3?

How many times 5 in 10? What number will bear the same proportion to 10 that 10 bears to 5?

CANCELING or Cancellation, in Arithmetic, is a process of abridging or shortening arithmetical operations, by taking away those parts, before the work is begun, which have an equal and opposite effect upon the final result; thus: 8+2 2=8, or 82+2 = 8, or 8 × 2 ÷ 2 = 8, or 82 × 2: 8. Draw a short oblique line over the 2's, (thus, 2;) and in all these cases you have the true result, without any work.

Again, 12 x 69 8. Now divide the and 9 each by 3, and you have for their quotients 2 and 3, which may be used in place of the original numbers: 12 x 2 ÷ 3 = 8, the same as before.

What is meant by proportion? When are four numbers said to be propor tional? What is canceling, or cancellation? How is canceling done?

EXAMPLES.

1. Cancel 10+ 4 2+2 4+3 −2 −1.

2. Cancel 10 x 65 x 5 ÷ 6.

Ans. 10.

Ans. 10. Ans. 4. Ans. 2, or 13.

SINGLE PROPORTION,

OR

THE SINGLE RULE OF THREE.

SINGLE PROPORTION, or the Single Rule of Three, is the method of finding, by three given terms, or numbers, a fourth term, that shall bear the same proportion to the third that the second does to the first.

In the given question, the learner will always find two terms alike in kind; i. e. two terms will be money, weight, measure, etc., and one term will be like the answer.

I. Rule for placing the terms.

Place the three given numbers, or terms, in a line, with the points between them, as follows:

1. Place that number in the third place, which is like the required number or answer.

2. Then consider whether the required number, or answer, must be greater or less than the third term. If greater, set the greater of the two remaining terms in the second place, and the less in the first place; but if the answer is to be less than the third term, set the less in the second place, and the greater in the first.

II. Rule for working.

1.-Reduce the first and second terms to the lowest denomination in either, and the third to the lowest in it.

2.-Multiply the second and third terms together, and

What is Single Proportion, or The Single Rule of Three? What is said about two of the terms, etc.? What is the rule for placing the terms-1st? 2d? What is the rule for working-1st? 2d?

divide the product by the first term; the quotient will be the answer to the question, in the same denomination to which the third term was reduced.

Sometimes it will be best to reduce the terms before you make the statement. Canceling should be attended to; the operation may often be very much shortened by it.

PROOF.-Reverse the statement, i. e. make the answer the third term and the first term the second, etc.

EXAMPLES.

1. If 2 yds. 2 qrs. of cloth cost $7.50, how much for 12 yds

2. How much for 14 lbs. of silver at 5d. per dwt.?

Canceled.

$7.50

Ans. £70.
Proof.

dwts. dwt. £
3360: 1 :: 70

20

1400s.

12

2020

Idwt. 3360 :: 5d.

5

12)16800d.

3360)16800(5d. 16800

14X5=£70

2|0)1400s.

£70

3. At $4 for 2 yds. of cloth, how much for
By analysis, $4÷2 × 12 = 24.
4. How much for 9 yds. at $5 for 2 yds.?
5. How much for 16 yds. at $9 for 3 yds.?
6. How much for 7 yds. 2 qrs. at $5 for 1 yd.

7. How much for 15 yds. at $6 for 21 yds.?

12 yds. ?

Ans. $24. Ans. $22.50. Ans. $48.

What is said about reducing the terms? What about canceling? Proof?

8. How much for 17 yds. at $6 for 21 yds.? Ans. $42. 9. How much for 3 pieces of goods, each 21 yds., at 5d. per nail? (Cancel.) Ans. £21. 10. How much for 1 lb. of gold, at 7s. 2d. per dwt.? (Cancel.) Ans. £86. 11. How much for 2 lbs. of gold, at 7s. 3d. per dwt.? (Cancel.) Ans. £174.

12. Sold 3 hhds. of wine at 20d. per quart; how much money must I receive? Ans. £63. 13. If 2 cwt. 1 qr. 14 lbs. of sugar cost $7.50, how much for 21 cwt. 1 qr. 14 lbs. ? 14. If a dozen chairs cost $75, how much for 4 chairs?

Ans. $67.50.

Ans. $25. 15. How much for 8 chairs, when the price is £7 10s. per dozen?

Ans. £5.

16. If 12 bushels of rye be worth £3 12s., how much for 120 bushels? Ans. £36. 17. If 6 persons require 32 lbs. of flour in a week, how many pounds will be sufficient for 12 families of 8 persons each, for the same length of time? Ans. 512 lbs. 18. If I pay 75 cts. for riding 12 miles, how much ought I to pay for riding 144 miles?

Ans. $9. 19. How many yards of carpet, of a yard wide, will it require to cover a floor 18 feet long and 12 feet wide? 12 ft. = 4 yds.

Ans. 32 yds.

20. If the interest of $100 for one year be $6, what will be the interest of $500 for the same length of time? Ans. $30. If the scholar does not know the meaning of interest, he had better turn over to Interest, and get the meaning.

21. If I pay $30 for the use of a sum of money for 3 years, how long ought I to have the use of it for $120 ?

Ans. 12 years.

22. How much Connecticut cur. is equal to £120 New York cur.? Ans. £90.

23. How much New York cur. is equal to £90 Conn. cur. ?

Ans. £120.

24. How much New Hampshire cur. is equal to £240 12s. 6d. New York cur.? Ans. £180 9s. 41d.

It will not be necessary to reduce the third term in such case: work

by the compound rules.

25. How much New York cur. is equal to £160 10s. New Jersey cur.? Ans. £171 4s. 26. How much New Jersey cur. is equal to £156 12s. Rhode Island cur.? Ans. £195 15s. 27. How much Georgia cur. is equal to £140 Massachusetts Ans. £108 17s. 91d.+

cur.? 28. How much N. York cur. is equal to £236 10s. Georgia cur.? Ans. £405 8s. 7d. 29. Find the value of 480 sovereigns, at $4.83 each.

Ans. $2318.40.

30. Find the value of 42 doubloons, at $15.60 each.

Ans. $655.20.

31. Find the value of 480 English crowns, at $1.12 each.

Ans. $537.60.

32. If 5 francs be worth 93 cts., how much for 4760 francs? Ans. $885.36.

33. If 40 livres be worth $7.66, how much for 640 livres ? Ans. $122.56. 34. If £9 Sterling is equal to $40 Federal money, how much Federal money in £180 Sterling? Ans. $800. 35. When the English guinea passes for $5, how many Eng. guineas in $950 ? Ans. 190 guineas. 36. If $40 make £9 Sterling, how many pounds Sterling in $480? Ans. £108. 37. How many 5 franc pieces, each 93 cts., in $133.92 ?

Ans. 144.

38. How many ells Flemish will equal 60 ells English?

Ans. 100 E. Fl.

39. How many yards are equal to 80 ells French?

Ans. 120 yds.

40. How much for 40 yds. of stuff, at $1.25 per ell Flemish? Ans. $66.663.

41. If I buy 240 lbs. of butter for 20 cts. per lb., and 324 lbs. at 22 cts., how much for both lots? Ans. $119.28.

42. How much for 3421 bush. of corn at 621 cts. per bush.? Ans. $214.06. 43. How many tons of coal can be purchased for $71.25, at $4.75 per ton? Ans. 15 tons. 44. If 1 cwt. of sugar cost $6.72, how much for 14 lbs. ? Ans. 84 cts.

45. How much for 4784 ft. of boards, at $2.75 per 100 ft.? Ans. $131.56.