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359. PROP. II.The fraction remaining after the division of one integer by another expresses the part the REMAINDER IS of the divisor.

Thus, 42 : 11 = 3%. The divisor 11 is contained 3 times in 42 and 9 left, which is 9 parts or I of the divisor 11. Hence we say that the divisor 11 is contained 3+1 times in 42. We express the I decimally by reducing it according to (338). Hence, 34 = 3.81.

Divide the following and express the remainder decimally, carrying the decimal to four places:

1. 324 : %. 4. 89 = 103. 7 5374 • 183. 2. 473 ; 23. 5. 65 ; 17. 8. 3000 = 547. 3. 783 – 9%. 6. 37 = 43. 9. 1000 = 101.

360. PROP. III.Division is possible only when the dividend and divisor are both of the same denomination (155—I).

For example, i to, or .3 :.07 is impossible until the dividend and divisor are reduced to the same fractional denomination ; thus, .3 .07=.30 = .07 = 4 = 4.285714.

ILLUSTRATION OF PROCESS, 361. Ex. 1. Divide .6 by .64.

(1.) .6 • .643.60 : .64

(2.) 60 = 6 4 = =.9375 EXPLANATION.—1. We reduce, as shown in (1), the dividend and divisor to the same decimal unit or denomination (290).

2. We divide, according to (290), as shown in (2), the numerator 60 by the numerator 64, which gives 4. Reducing &d to a decimal (338), we have .6 5.64 = .9375. Ex. 2. Divide .63 by .0022.

(1.) .63 = .0022 =.6300 •.00 2 2
(2.) 6300 : 22 = 2 8 6 4 = 38 6.3 6

EXPLANATION.-1. We reduce, as shown in (1), the dividend and divisor to the same decimal unit by annexing ciphers to the dividend (350).

2. We divide, according to (290), as shown in (2), the numerator 6300 by the numerator 22, giving as a quotient 286 41.

3. We reduce, according to (338), the 4 in the quotient to a decimal, giving the repetend .36. Hence, .63 = .0022 = 286.36. Ex. 3. Divide 16.821 by 2.7

(1.) 16.821 ; 2.7= 16.821 ; 2.700
(2.) 16.821 : 2.700= 1000 11000

16821 2700
(3.) 27100) 168/21 ( 6.23

162

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EXPLANATION.–1. We reduce, as shown in (1), the dividend and divisor to the same decimal unit by annexing ciphers to the divisor (350).

2. The dividend and divisor each express thousandths as shown in (2). Hence we reject the denominators and divide as in integers (290).

3. Since there are ciphers at the right of the divisor, they may be cut off by cutting off the same number of figures at the right of the dividend (142). Dividing by 27, we find that it is contained 6 times in 168, with 6 remaining.

4. The 6 remaining, with the two figures cut off, make a remainder of 621 or 2017. This is reduced to a decimal by dividing both terms by 27. Hence, as shown in (3), we continue dividing by 27 by taking down the two figures cut off.

The work is abbreviated thus:

We reduce the dividend and divisor to the same decimal unit by cutting off from the right of the dividend the figures that express lower decimal units than the divisor. We then divide as shown in (3), prefixing the remainder to the figures cut off and reducing the result to a decimal.

From these illustrations we obtain the following

362. RULE.Reduce the dividend and divisor to the same decimal unit; divide as in integers and reduce the fractional remainder in the quotient, if any, to a decimal.

EXAMPLES FOR PRACTICE. 363. In the following examples carry the answer in each case to four decimal places:

1. Divide 53.28 by 3.12; by 7.3; by 9.034. 2. Divide 27 by 4.03 ; by.72; by 2.37. 3. Divide $725.42 by $.37; by $3.08; by $.95%. 4. Divide $.93 by $.847; $73.094 by $.754; $.374 by $.74. What is the value of 5. $75.83 : $100.

8. $10000 : $.07. 6. (3 of .73) = .09. 9. 8.345 -- 2.0007. 7. 7347 = 4.51.

10. (84 +12.07) • (15.03—3). 11. ($354.07 — of $10.84) = of $7.08. 12. (= .03 x 64) = (f of f of 123). 13. ($3.05; : ) – (4 of $1.08 ; ). 14. (of $324.18 – $7) $2.0005.

15. At $2.32, how many yards of cloth can be bought for $373.84 ? · 16. The product of two numbers is 375.04 and one of them is 73.009; what is the other?

Ans. 5.1369 +. 17. How much tea can be bought for $134.84, if 23} pounds cost $17.70 ?

Ans. 179.7866 + pounds. 18. A merchant received $173.25, $32.19, and $89.13. He expended the whole in buying silk $1.371. How many yards of silk did he buy?

19. A farmer sold 1324 bushels wheat at $1.35 per bushel, and 184 bushels corn at $.731 per bushel. He bought coal with the amount received, at $9.54 a ton. How many tons did he buy?

20. What decimal part of a farm worth $3965 can be bought for $1498.77?

Ans. .378. 21. What is the value of 275 acres of land when .57 of an acre is worth $48 ?

22. A merchant lost .47 of his capital, and had to use .13 more for family expenses, and had still remaining $5380. What was his original capital ?

Ans. $13450.

REVIEW EXAMPLES.

364. Answers involving decimals, unless otherwise stated, are carried to four decimal places. What is the cost

1. Of.71 of a pound of tea, if y pounds cost $6.95.
2. Of 4.5 acres of land, if 100 acres cost $7385.
3. Of 93 cords of wood, at $12.60 for 2.8 cords.
4. Of 5384 feet lumber, at $5.75 per 100 feet.
5. Of 13.25 yards of cloth, if 3.75 yards cost $9.933.
6. Of 31460 bricks, at $8.95 per 1000 bricks.
1. Of 1581 pounds butter, if 9.54 pounds cost $3.239.
Reduce each of the following examples to decimals: *
8.. # 12. of 14.

16. (13+3) *. 9. $. 13. 54 - 5%. 17. f of 4 of 14 10. 24. 14. (38+3). 18. (8-157x 24.

8

of .3 83 – 4.3

7.51 20. Four loads of hay weighed respectively 2583.07, 30071, 2567), and 30741 pounds; what was the total weight?

21. Seven car-loads of coal, each containing 13.75 tons, were sold at $8.53 per ton. How much was received for the whole ?

22. At $1.75 per 100, what is the cost of 5384 oranges ?

25

11.

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23. What is the cost of carrying 893850 pounds of corn from Chicago to New York, at $.35per 100 pounds ?

24. If freight from St. Louis to New York is $.394 per 100 pounds, what is the cost of transporting 3 boxes of goods, weighing respectively 783, 325%, and 2867 pounds ?

25. A piece of broadcloth cost $195.381, at $3.27 per yard. How many yards does it contain ?

26. A person having $1142.49% wishes to buy an equal number of bushels of wheat, corn, and oats; the wheat at $1.37, the corn at $.874, and the oats at $.354. How many bushels of each can he buy?

27. Expended $460.80 in purchasing silk, .3 of it at $2.25 per yard, f of it at $1.86 per yard, and the balance at $3.45 per yard. How many yards did I buy of each quality of silk ? 28. What is the value of (* of ? — 2). •36

of | 14 + 8) = .48 29. A produce dealer exchanged 48% bushels oats at 39 cts. per bushel, and 134 barrels of apples at $3.85 a barrel, for butter at 371 cts. a pound; how many pounds of butter did he receive ?

30. A grain merchant bought 1830 bushels of wheat at $1.25 a bushel, 570 bushels corn at 734 cts. a bushel, and 468 bushels oats at 354 cts. a bushel. He sold the wheat at an advance of 174 cts. a bushel, the corn at an advance of 9cts. a bushel, and the oats at a loss of 3 cts. a bushel. How much did he pay for the entire quantity, and what was his gain on the transaction ?

31. A fruit merchant expended $523.60 in purchasing apples at $3.85 a barrel, which he afterwards sold at an advance of $1.07 per barrel ; what was his gain on the sale ?

32. The cost of constructing a certain road was $5050.50. There were 35 men employed upon it 78 days, and each man received the same amount per day; how much was the daily wages ?

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