NOTE. When a decimal contains several places of figures, their value can be found nearly by inspection. To do this, take the first or the first two figures, as the numerator, of which 10 or 100 is the denominator, then reduce it to its lowest terms:

Thus .332125 is nearly 1 third; .258321 is nearly 1 fourth.

CASE III.

ART. 187. To reduce a decimal of one denomination to an equivalent decimal of another denomination.

1. Reduce .25 qt. to the fraction of a pt. SOLUTION. To reduce quarts to pints, we multiply by 2, the number of pints in a quart.

Therefore, multiply the fraction of a quart by 2, to reduce it to the fraction of a pint. Art. 81.

2. Reduce .5pt to the fraction of a qt. SOLUTION.-To reduce pints to quarts, we divide by 2, the number of pints in a quart.

Therefore, divide the fraction of a pint by 2, to reduce it to the fraction of a quart. Art. 81.

OPERATION.

qt.

.25

2

.50 Ans. .5 pt.

OPERATION.

pt.

2).5

Ans. .25 qt.

*3. Reduce .125 bu. to the fraction of a pk. Ans. .5 *4. Reduce .7 pk. to the fraction of a bu. Ans. .175

Rule for Case III.-Multiply or divide as in Reduction of Whole Numbers, Art. 81, according to the rules for the Multiplication and Division of Decimals. Arts. 181 and 183.

5. Reduce .0625 lb. Troy to the frac. of an oz. Ans. .75 6. .05 of a yd. to the fraction of a na. 7. .00546875 A. to the fraction of a P.

Ans. .8 Ans. .875

REVIEW.-185. What is Case 1? Rule for Case 1? NOTE. When com. fractions can not be exactly expressed in decimals, what is to be done? 186. What is Case 2? Rule for Case 2? NOTE. How find the value nearly of a decimal? 187. How reduce a decimal from a higher to a lower denomination? From a lower to a higher ?

8. Reduce .0004375 mi. to the frac. of a rd. Ans. .14

9. .25 pt. to the fraction of a gal. 10. .6pt. to the fraction of a bu. .3min. to the fraction of a da. 12. .7 rd. to the fraction of a mi.

11.

Ans. .03125

Ans. .009375 Ans. .0002083+

Ans. .0021875

CASE IV.

ART. 188. To find the value of a decimal in integers of

a lower denomination.

1. Find the value of .3125 of a bu.

SOLUTION. First multiply by 4, as in reducing bushels to pecks. Art 81. This gives 1 peck and .25 of a peck; then find the value of .25 pk., by multiplying it by 8, the number of quarts in a peck. This gives 2 quarts; hence, .3125 of a bushel equals 1 pk. 2 qt. Ans.

*2. Find the value of .875 of a yd.

OPERATION. bu. .3125 4

pk. 1.2500 8

qt. 2.00 Ans. 3 qr. 2na.

Rule for Case IV.-Multiply the given decimal by that number which will reduce it to the next lower denomination (Art. 81), and point the product, as in Multiplication (Art. 181). Reduce this decimal in like manner, and so on; the several integers on the left will be the required answer.

6.

.04318 of a mi. Ans. 13rd. 4 yd. 1 ft. 5.8848 in.

7.

.33625 of a cwt.

Ans. 1qr. 8lb. 10 oz.

CASE V.

ART. 189. To reduce a quantity composed of one or more denominations, to the decimal of another quantity of one or more denominations.

OPERATION.

1. Reduce 2ft. 3 in. to the decimal of a yard. SOLUTION.-By Art. 163, the result expressed as a common fraction, is . This reduced to a decimal, becomes .75, the required answer.

2ft. 3 in.=27 in. 1yd. =36 in. 37.75 Ans.

*2. Reduce 2 pk. 4qt. to the dec. of a bu. Ans. .625

Rule for Case V.-Reduce the first quantity and that of which it is to be made a part, both to the same denomination; the less will be the numerator, and the greater the denominator of a common fraction, which reduce to a decimal.

3. Red. 13 hr. 30 min. to the dec. of a da. Ans. .5625 4. 9 dr. to the dec. of a lb. Av. Ans. .03515625 5. .028 of a P. to the dec. of an A. Ans. .000175 6. 7 min. to the dec. of a da. Ans. .0048611+ 7. 4gal. 1qt. 1.28 pt. to the dec. of a hhd, Ans. .07 8. What part is 3 pk. 7qt. 1 pt. of 2bu. 2 pk. 4qt., expressed decimally? Ans. .375 9. What part is 99 pages, of a book of 512 pages, expressed decimally. Ans. .193359375 10. What decimal will express the part that 55 A. 2R. 17 P. is of 229 A. 2 R. 16 P.? Ans. .2421875

ART. 190. PROMISCUOUS EXAMPLES.

1. What cost 9 yards of muslin, at $0.4 per yd., and 12 yards at $0.1875 per yd.?

Ans. $5.85 2. What cost 2.3 yards of ribbon, at $0.45 per yd., and 1.5 yards at $0.375 per yd.? Ans. $1.5975

3. At $2.6875 per yd., what cost 16 yd. of cloth? What 161 yd.? Ans. to last, $43.671875

4. At $0.75 per bushel, how much wheat can be bought for $35.25? Ans. 47 bu.

REVIEW.-188. What is Case 4? What the Rule for Case 4? 189. What is Case 5? What the Rule for Case 5?

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

6. What cost 6 cwt. 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 $19 Ans. $360.8888+ per hhd.?

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. .4 na. 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. .05 of a year, (365.25 days,) take .5 of an hour. Ans. 18 da. 5 hr. 48 min.

16.

17.

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

Ans. 9hr. 40 min. 48 sec.

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

19. Find the value of .3 of a year, integers.

Ans. 109 da.

20. In .005 of a year how many sec.?

21. What decimal of a C. is 1 cu. in?

(365.25 days,) in 13 hr. 48 min.

Ans. 157788. Ans. .000004+

22. At $690.35 per mile, what cost a road 17 mi. 3 fur. 15 rd. 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 10yds., 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 quantities of the same kind, but of different denominations, what is required?