19. In 302 ells English, how many yards?

20. In 24 hhd. 18 gal. 2 qt. of molasses, how many gills? 21. In 76 A. 1 R. 8 P., how many square inches?

22. In 445577 feet, how many miles?

23. In 37444325 square inches, how many acres?

24. How many times will a wheel, 16 feet and 6 inches in circumference, turn round in a distance of 84 miles ?*

25. What will 28 rods 129 square feet of land cost, at $15 a square foot?

26. What will be the cost of a pile of wood, 36 feet long, 6 feet high, and 4 feet wide, at 50 cents a cord foot?

27. A man has a journey to perform of 288 miles. He travels the distance in 12 days, traveling 6 hours each day: at what rate does he travel per hour?

28. How many yards of carpeting, 1 yard wide, will carpet a room 18 feet by 20?

29. If the number of inhabitants in the United States were 24 millions, how long will it take a person to count them, counting at the rate of 100 a minute?

30. A merchant wishes to bottle a cask of wine containing 126 gallons, in bottles containing 1 pint each how many bottles are necessary?

31. There is a cube, or square piece of wood, 4 feet each way: how many small cubes, of 1 inch each way, can be sawed from it, allowing no waste in sawing?

32. A merchant wishes to ship 285 bushels of flax-seed in casks containing 7 bushels 2 pecks each: what number of casks are required?

33. The square measure of a floor is 648 sq. ft., and its width is 18 ft.: what is the length?

34. The square measure of a board is 9 sq. feet, and its length is 12 feet: what is its width?

35. A cellar, 34 feet by 25 feet, and 8 feet deep, is to be dug. Supposing that 3 cart-loads of earth make 1 solid yard, how many loads of earth would be carted?

36. How many bricks will pave a sidewalk 25 ft. by 10, the dimensions of a brick being 8 in., 4 in., and 2 in.?

DENOMINATE FRACTIONS.

206. A DENOMINATE FRACTION is one in which the unit of the fraction is a denominate number. Thus, of a yard is a denominate fraction.

207. REDUCTION of denominate fractions is the operation of changing a fraction from one unit to another, without altering its value.

CASE I.

OPERATION.

208. To change from a greater unit to a less. 1. In of a yard, how many inches? ANALYSIS. Since yards are reduced to feet by multiplying by 3, and feet are reduced to inches by multiplying by 12, if be multiplied by 3 and 12, the product will be inches: Hence,

4

× 12 = 20 in.

Rule.-Multiply the fraction by the scales till you reach

the required unit.

Examples.

2. Reduce of a £ to the fraction of a penny.

ΙΣ

3. Reduce 3 of a £ to the fraction of a farthing.

4800
3

4. Reduce of an Ell Eng. to the fraction of a nail.

3

5. Reduce of a hogshead to the fraction of a quart.

648

6. Reduce of a bushel to the fraction of a pint.

8640

7. Reduce 1 of a pound Troy to the fraction of a grain.

8. Reduce 14 of a cwt. to the fraction of an ounce.

25600

9. Reduce .125 cwt. to the decimal of an ounce.

10. Reduce .3125 of a mile to the decimal of an inch.

11. Change .1875 of a day to minutes.

12. Reduce .29763 of a degree to the decimal of a second. 13. Reduce .1723 lb. to the decimal of a grain.

14. Reduce 2.333 £ to pence.

CASE II.

209. To change from a less unit to a greater.

1. In 20 inches, how many yards?

ANALYSIS.-Inches are reduced to yards, by dividing by 12 and 3 in succession. To divide by 12 and 3, is the same as multiplying by ; therefore, 20 × × = yd.

12

and

5

OPERATION.

20

1

12 3

1 = § yd.

Rule.—Divide the fraction by the scales, in succession, till the required unit is reached.

Examples.

2. Reduce of a gallon to the fraction of a hogshead. 3. Reduce of a nail to the fraction of a yard. 4. Reduce of

5. Reduce of 3

of a foot to the fraction of a mile.

pwt. to the fraction of a pound Troy.

6. Reduce .3125 pt. to the decimal of a gallon.

NOTE.-Divide by the units of the scale, and observe the rule for pointing in division of decimals.

7. Change .89725 oz. to the fraction of a cwt.

8. Change .9825 of a penny to the decimal of a pound. 9. Reduce 6.875 seconds to the decimal of a day. 10. What decimal of a yard are 27.9175 nails? 11. To what decimal of a mile are 262.318125 feet equal? 12. Reduce .009375 pt. to the decimal of a bushel? 13. Reduce 7 drams to the decimal of a lb. avoirdupois. 14. Reduce .056 of a pole to the decimal of an acre. 15. Reduce 14 minutes to the decimal of a day. 16. Reduce 21 pints to the decimal of a peck. 17. Reduce 375678 feet to the decimal of a mile. 18. Reduce .5 quarts to the decimal of a barrel.

206. What is a denominate number?

207. What is Reduction of Denominate Fractions? 208. What is Case I.? Give the rule.

209. How do you change from a less unit to a greater?

CASE III.

210. To find the value of a common fraction in integers of lower denominations.

1. What is the value of of a pound Troy?

Multiply the numerator of the fraction by the scale, and divide the product by the denominator; if there is a remainder, treat it in the same way, till the required denomination is reached. The quotients of the several operations will form the answer.

Examples.

0

32

1. What is the value of 2. What is the value of 3. What is the value of 4. What is the value of 5. What is the value of 6. What is the value of 7. What is the value of 8. What is the value of 9. What is the value of 10. What is the value of 11. What is the value of 12. What is the value of 13. What is the value of 15 14. What is the value of 1

7

3

19

of a tun of wine?

of a yard?

of a month?

of a chaldron ?
of a mile?

of a ton?

of 3 days?

of of 63 bushels of grain? hhd.?

of a cwt.?

of a hogshead of wine?

of an acre of land ?

tons?

of a common year?

CASE IV.

211. To find the value of a decimal in integers of lower denominations.

1. Find the value of .890625 of a bushel.

ANALYSIS.-Multiplying the decimal by 4 (since 4 pecks make a bushel), we have 3.5625 pecks. Multiplying the new decimal by 8 (since 8 quarts make a peck), we have 4.5 quarts. Then, multiplying this last decimal by 2 (since 2 pints make a quart), we have 1 pint.

OPERATION.

.890625 bu. 4

3.562500 pk.

8

4.500000 qt.

2

Ans. 3 pk. 4 qt. 1 pt.

1.000000 pt.

Rule.

I. Multiply the decimal by the scale, pointing off as in multiplication of decimal fractions:

II. Multiply the decimal part of the product as before, and continue the operations to the lowest denomination : the integers at the left, form the answer.

Examples.

1. What is the value of .002084 lb. Troy?
2. What is the value of .625 of a cwt.?
3. What is the value of .625 of a gallon?

4. What is the value of £.3375 ?

5. What is the value of .3375 of a ton?

6. What is the value of .05 of an acre?

7. What is the value of .875 of a pipe of wine?
8. What is the value of .125 of a hogshead of beer?
9. What is the value of .375 of a year of 365 days?

210. How do you find the value of a common fraction in integers of lower denominations?

211. How do you find the value of a decimal in integers of lower denominations?