5. At 15 pence a pound, what will 1 cwt. of loaf sugar cost in dollars, New England currency? Ans. $23.333.+

If it is preferred to solve sums of this kind without canceling, it may be done by the following rule:

RULE. Reduce the given quantity to that denomination, the price of which is given, and multiply it by the price; then divide by such numbers as are required to reduce the value obtained to the required denomination.

6. How many dollars, New York currency, will 6 cwt. of sugar cost at 10 d. a pound?

7. How many dollars will 53 ells English cost, at 8s. New York currency, per yard? Ans. $66.25.

8. If I purchase melasses at 1 s. 3 d. per quart, how much in pounds, shillings, and pence, will 12 hogsheads of the same kind cost? Ans. 189 £.

9. If I purchase 16 cwt. of steel for $156, what will 1 qr. of a cwt. cost, at the same rate?

10. What cost 9 cwt. of sugar at 10 pence per pound? Ans. 42 £.

11. What cost 12 cwt. of sugar at 9 pence per pound? Ans. 50 £. 8 s.

12. What cost 42 cwt. of sugar at 3 s. 8 d. per pound? Ans. 862 £. 8 s.

13. What would 480 yards cost, in federal money, at 2 pence, New York currency, per yard?

480. 2.

Statement, 12. 8 Ans. $10.+

For the solution of the following sums, see the table of Currencies, given in Reduction of Currencies.

14. What would 862 yards cost, in federal money, at 3 pence per yard, New England currency?

Statement,

862. 3.
12 6

Ans. $35.916.+

15. What would 920 yards cost, in federal money, at 4 pence per yard, New Jersey currency? Ans. $46.

16. What would 988 pounds of rice cost, in federal money, at 6 d. per pound, South Carolina currency? Ans. $105.857.+ 17. What would 899 gallons of vinegar cost, in federal money, at 8 d. per gallon, New York currency? Ans. $74.916.+

18. How much will 672 yards cost, in federal money, at 6s. 6 d. New York currency, per yard? Ans. $546.

19. What will 1000 yards of ribbon cost, in federal money, at 3 s. 4 d. New England currency, per yard? Ans. $555.

555.+

20. How many dollars will 123 yards of cloth cost, at 10 s. New Jersey currency, per yard? Ans. $164.

21. What will 687 yards of cloth cost in federal money, at 5 s. South Carolina currency, per yard? Ans. $736.071.+ 22. How many dollars and cents will 127 gallons of wine cost, at 3 s. 4 d. New England currency, per gallon? Ans. $70.555.+

23. How many dollars will pay for 14 cwt. 3 qr. of hay, at 15 s. 8 d. New York currency, per cwt.? Ans. $28.885.+ 24. How many dollars will pay for 12 cwt. 2 qr. of cheese, at 2 £. 15 s. New Jersey currency, per cwt.? Ans. $91.666.+

25. What will 12 cwt. 7 lb. of brown sugar cost, at 6d. New Jersey currency, per pound? Ans. $90.066.+

26. A. owes B. 1138 £. but can pay only 15 s. on a pound; what will B. receive? Ans. 853 £. 10 s.

27. C. has. a journey of 75 leagues to perform. In what time will he complete it, if he travel 30 miles a day. Ans. 7 days.

28. How long will it take to travel one fourth round the earth, at the rate of 36 miles a day; the whole circumference being 360 degrees, and one degree 69 miles? Ans. 173 days.

29. If 9 lb. of coffee cost 27 s. how many dollars, at 6 s. each, will 45 lb. cost? Ans. $22.50.

30. If I buy 20 pieces of cloth, each 20 ells, for 12 s. 6 d. per ell, how many dollars, at 8 s. will pay for the same? Ans. $625.

31. How many dollars will 7 casks of prunes cost, each weighing 4 cwt. 2 qr., at 2 £. 19 s. 8 d. New York currency, per cwt.? Ans. $234.937.+

32. A vessel at sea discharges a cannon, the report of which reaches me in 1 minute, 30 seconds. How far distant is she, allowing sound to travel 1142 feet in a second? Ans. 19 miles, 3 furlongs, and 291 rods.

33. How many yards of flannel, 1 yard wide, will line 125 yards of broadcloth, 1 ell English wide? Ans. 156.

34. How many yards of cloth may be bought for $37.62, if of a yard cost 66 cents? Ans. $42 yd. 3 qr.

35. How many dollars will 28 yards of linen cost, at 5 s. 6 d. New England currency, per yard? Ans. $25.666.+

36. Bought 32 yards of muslin, at 6 s. 8 d. New York currency, per yard. What was the cost, in federal money? Ans. $26.666.+

37. How many gallons of wine may be purchased for $45, at 6 s. New York currency, per gallon? Ans. 60 gallons.

QUESTIONS.-What is ratio? What is the former of the two numbers between which the ratio exists called? What is the latter? How is the direct ratio of any two numbers obtained? How is the inverse ratio obtained? Between what quantities only, does ratio exist? How is simple ratio expressed? How is the ratio of any couplet affected by multiplying or dividing both the antecedent and the consequent by the same number? In what two ways may the ratio of any two numbers be multiplied? In what two ways is the ratio of any couplet divided by any number? When two or more ratios are multiplied together, what is the resulting ratio called? What does it equal?

What is Note 1st? What constitutes proportion? How many equal ratios are required for a statement of proportion? What terms must be of the same kind? When any four terms are proportional, what are the first and fourth called? And what are the second and third called? How does the product of the extremes compare with the product of the means? When any of the four terms are wanting, explain how it may be found. In what do operations in Simple Proportion

consist? What is the rule? What note follows the rule? What is the rule for canceling? What is Note 2d? What is the rule, when it is required to find the value of a quantity in one denomination, the price of some other denomination being given? What is Note 3d ? What is the rule for solving sums of this kind without canceling?

COMPOUND PROPORTION.

Simple Proportion consists of two equal ratios. Compound Proportion is that in which the relation of one of the given quantities to a required quantity of the same name, is traced through two or more simple proportions.

The smallest number of terms of which a statement in compound proportion can consist, is five. Of these terms, one is always of the same name as the answer required; and the others are always two of a kind. The following sum will serve as an illustration:

If 3 men, in 4 days, spend $5, how many dollars will 6 men spend in 12 days?

In the above sum, there are five terms given, viz. two of men, two of days, and one of dollars; and dollars are also required for the answer; so that when the sixth term is found, the sum may be resolved into three simple ratios, the third of which is a compound of the preceding two. These ratios are 3 men: 6 men, and 4 days: 12 days, and $5 : the required number of dollars. Now it is obvious, that the ratio of $5 to the required number of dollars, is not the same as the ratio of 3 men to 6 men; for then no regard would be paid to the time, and the solution would be effected on the supposition that one man or a number of men would spend as much in one day, as in any given number of days. Nor is it the same as the ratio of 4

days to 12 days, for then the supposition would be, that 3 men would spend as much as 6, or any number of men. But it is a ratio compounded of the two ratios, viz. the ratios of 3 men :6 men, and of 4 days: 12 days. The ratio of 3:6=2, and $5×2=$10. This would double the given quantity of money. Again, the ratio of 4: 12, is 3; this would treble the sum last obtained, viz. $10, and would give 10×3=$30, which is the answer to the above question. Now, it will be observed, that the $5 in the above operation was multiplied by the product of the two simple ratios; for 3:6=2, and 4:12=3; therefore, 2×3=6, the product of the simple ratios, and $5×6= $30, Ans.

The same result would have been obtained, by multiplying the $5 by the consequents, or latter terms of each ratio, and dividing their product by the product of the antecedents, or former terms of the same ratios. Thus, 3:6 and 4:12, are the given ratios of which 6 and 12 are consequents; therefore, 6 × 12×5=360, and 3×4=12, the product of the antecedents; hence, 360÷12=30 dollars, the same answer as before. Hence, we have the following rule:

RULE. Write the number which is of the same kind as the answer required, for the third term. Of the remaining terms, take any two of the same name, and arrange them as directed in Simple Proportion; then any other two of a kind, and so on till the terms are all taken. Lastly, multiply the product of the second terms by the third term, and divide the last product by the product of the first terms, and the quotient will be the required

term or answer.

Ex. 1. If 4 men build a wall ten feet long, 3 feet high, and 2 feet thick, in 6 days, in what time will 12 men build one 100 feet long, 4 feet high, and 3 feet thick? The question asked is, in what time will the work be done; therefore, by the rule, 6 days is the third term.

12 men: 4 men,

10 feet length: 100 feet length,
3 feet high: 4 feet high,

2 feet thick: 3 feet thick,

: 6 days.

It is obvious that 12 men would require less time to execute a given piece of work than 4 men, and also that a wall 100