5. At $2.5625 per yard, how much cloth can you buy for $98.4? Ans. 38.4yd.

6. What cost 6 wt. 2qr. of hops, at $3.25 per cwt.? SOLU.-Reduce 2qr. to the decimal of a cwt.; then, 6cwt. 2qr. 6.5cwt.; $3.25×6.5= Ans. $21.125

7. What will be the cost of 7hhd. 23 gal. of wine at $49 per hhd.? Ans. $360.8888+ 8. Of 343 yd. 3 qr. linen, at $.16 per yd.? Ans. $55. 9. What cost 14 bu. 3pk. 4qt. of corn, at $0.625 per bushel? Ans. $9.296875 10. What will 13 A. 2R. 35 P. of land cost, at $17.28 per acre? Ans. $237.06 11. At $1.24 per yard, how much cloth can be bought for $19.065? Ans. 15.375 yd.=15 yd. 1 qr. 2 na. 12. At $0.3125 per bu., how much corn can be bought for $9.296875? Ans. 29.75 bu.=29 bu. 3 pk. 13. At $4.32 per A., how much land can you buy for $59.265? Ans. 13.71875 A.=13 A. 2 R. 35 P.

14. Add .34yd. .325 qr. .4na. Ans. 1qr. 3.14 na.

To add or subtract decimals of different denominations, first reduce them to the same denomination. In this example, reduce the dec. of a yd. to qr., then add the dec. of a qr.; next reduce this result to na., and add the dec. of a na.

15. From 1.53 yards take 1.32 qr. Ans. 1yd 3.2 na. 16. .05 of a year, (365.25 days,) take .5 of an hour. Ans. 18 da. 5 hr. 48 min.

17. .41 of a da. take .16 of an hr.

Ans. 9hr. 40 min. 48 sec.

18. In .4T.3hhd .8gal. how many pt.? Ans. 964 pt.

19. Find the value of .3 of a year, (365.25 days,) in integers. Ans. 109 da. 13 hr. 48 min.

Ans. 157788.

20. In .005 of a year how many sec.? 21. What decimal of a C. is 1 cu. in? Ans. .000004+

22. At $690.35 per mile, what cost a road 17 mi. 3 fur 15rd. long? Ans. $12027.19140625

In practice, only 3 or 4 places of decimals are generally used For additional problems, see Ray's Test Examples.

XIII. RATIO.

ART. 191. Ratio is the quotient arising from dividing one quantity by another of the same denomination;

Thus, the ratio of 2 to 6 is 3; as, 6÷÷2 gives the quotient 3. The ratio of 2 to 8 is 4; of 2 yd. to 10 yds., 5.

ILLUSTRATIONS.-1. Two quantities to be compared, or to have a ratio to each other, must be of the same kind, and in the same denomination, that the one may be some part of the other.

Thus, 2 yards has a ratio to 6 yards. But, 2 yards has no ratio to 6 dollars, the one being no part of the other.

2. Since ratio is the relation of two numbers expressed by their quotient; and since the quotient of 2 and 6 may be 6 divided by 2, or 2 divided by 6, either may be used to express their ratio,

Thus, in comparing two lines, one of which is 2, the other 6 inches long, if the first is taken as the unit (1) or standard of comparison, the second is three, that is, it is 3 times the first. If 6 is taken as the unit of comparison, 2 is one-third.

In finding the ratio between two numbers, the French take the first as the divisor, the English the last. The French method being regarded the most simple, is now generally used.

3. Finding the ratio of two numbers, is finding what part, or what multiple one is of the other.

The following are equivalent expressions: What is the ratio of 2 to 6? What part of 2 is 6? What multiple of 2 is 6?

4. The ratio of $2 to $10 is 5; of $2000 to $10000 is also 5: hence, ratio shows only the relative magnitude of two quantities.

ILLUS. 1. Can

REVIEW.-190. What is ratio? Give examples. quantities, not of the same kind, have a ratio? Why not? 2. What is the ratio of 2 to 6? 3. What are equivalent expressions?

ILLUS. 4. What does ratio show? In finding the ratio between two quan tities of the same kind, but of different denominations, what is required?

When the quantities are of the same kind, but of different denominations, reduce them to the same denomination.

of 4 in. to 3 yd.? Ans. .

of 25 to 15?

8. What is the ratio of 3 in. to 2 ft.? 9. Of 15 to 25? Ans. 13. 10. Of 4 to 10? of 10 to 4?

of 6 to 16? of 16 to 6?

ART. 192. A ratio is formed by two numbers, each of which is called a term, and both together, a couplet.

Thus, 2 and 6 together form a couplet of which 2 is the first term, and 6 the second.

The first term of a ratio is called the antecedent; the second, the consequent.

ART. 193. RATIO IS EXPRESSED IN TWO WAYS:

1st. In the form of a fraction, of which the antecedent is the denominator, and the consequent the numerator. The ratio of 2 to 6 is expressed by §; of 3 to 12, by 12. 2d. By a colon (:) between the terms of the ratio. Thus, the ratio of 2 to 6 is written 2:6; of 3 to 8, 3:8.

ART. 194. Since the ratio of two numbers is expressed by a fraction, of which the antecedent is the denominator, and the consequent the numerator, whatever is true of a. fraction, is true of a ratio. Hence,

REVIEW.-192. By what is a ratio formed? What is each number called? What both together? What is the first term called? The 2n? 193. In how many ways is ratio expressed? What is the first method? The second? Give examples of each.

1st. To multiply the consequent, or divide the antecedent, multiplies the ratio. Arts. 131 and 133. Thus,

The ratio of 4 to 12 is 3; of 4 to 12×5, is 3×5; and The ratio of 4÷2 to 12, is 6, which is equal to 3×2.

2d. To divide the consequent or multiply the antecedent, divides the ratio. Arts. 132 and 134. Thus,

The ratio of 3 to 24 is 8; of 3 to 24÷÷2, is 4,=8÷2; and The ratio of 3x2 to 24, is 4, which is equal to 8÷2.

3d. To multiply or divide both consequent and antecedent by the same number, does not alter the ratio. Arts. 134, 135. Thus, The ratio of 6 to 18, is 3; of 6x2 to 18X2, is 3; and The ratio of 62 to 182, is 3.

ART. 195. A single ratio, as 2 to 6, is a simple ratio. A compound ratio is the product of two or more simple ratios.

Thus,

In this case, 3 multiplied by 5, is said to have to 10×6, the ratio compounded of the ratios of 3 to 10 and 5 to 6.

ART. 196. Ratios may be compared with each other, by reducing to a common denominator the fractions by which they are expressed: thus,

To find the greater of the two ratios, 2 to 5, and 3 to 8, we have and §, which, reduced to a common denominator, are 15 and 16; and, as 15 is less than 16, the ratio of 2 to 5, is less than the ratio of 3 to 8.

XIV. PROPORTION.

ART. 197. Proportion is an equality of ratios. Four numbers are proportional, when the first has the same ratio to the second that the third has to the fourth.

REVIEW.-194. How is a ratio affected by multiplying the consequent, or dividing the antecedent? By dividing the consequent, or multiplying the antecedent? By multiplying or dividing both consequent and antecedent by the same number? Why? Illustrate each.

Thus, the two ratios, 2 4 and 3: 6, form a proportion, since, each being equal to 2.

ART. 198. PROPORTION IS WRITTEN IN TWO WAYS: 1st. By placing a double colon between the ratios. Thus, 2 : 4 :: 3 : 6.

2d. By placing the sign of equality between them.

The first is read, 2 is to 4 as 3 is to 6; or, 2 has the same ratio to 4, that 3 has to 6. The second is read, the ratio of 2 to 4 equals the ratio of 3 to 5.

REM.-1. The least number of terms that can form a proportion is four, since there are two ratios each containing two terms.

But, one of the terms in each ratio may be the same; thus, 2:4:48. The number repeated is called a MEAN proportional between the other two terms.

2. The terms ratio and proportion are often confounded with each other. Two quantities having the same ratio as 3 to 4, are improperly said to be in the proportion of 3 to 4. A ratio subsists between two quantities; a proportion only between four.

ART. 199. The first and last terms of a proportion are called the extremes; the second and third, the means. Thus, in the proportion 2 3 4 6, 2 and 6 are the extremes, and 3 and 4 the means.

ART. 200. In every proportion, the product of the means is equal to the product of the extremes.

ILLUSTRATIONS.-If we have 3 4

the ratios of each couplet being equal (Art. 206), we must have.

Reducing these fractions to a common denominator (Art. 155), gives

68,

4

8

}

4 X 6

3X8

18

18

REVIEW.-195. What is a simple ratio? A compound ratio? Give examples of each. 196. How compare ratios with each other? 197. What is proportion? When are four numbers proportional? Give examples.

198. How is proportion written? How is the first read? The second? REM. 1. What the least number of terms that can form a proportion? AWnt of the terms ratio and proportion?