1. What is the difference between 659,722 and 437,591 ? Ans. 222,131. 2. If I pay 67 dollars 47 cents towards a debt of 179 dollars 65 cents, how much of the debt remains unpaid? Ans. $112,18.

3. A handed B a 10 dollar bill to pay a debt of 8 dollars and 9 cents; how much money must B return to A ? Ans. $1,91. 4. If I pay all but $37,093 of a debt amounting to $220,10, how much do I pay? Ans. $183,007. 5. What is the difference between 4 cents, 7 mills, and 10 dollars? Ans. $9,953.

6. If 471,972 feet of boards be sold from a pile consisting of three thousand, how many feet of boards will remain ? Ans. 2528,028.

7. Subtract nine mills from 9 dollars.

Rem. $8,991. 8. What is the difference between six thousand, five hundred and seventy eight ten thousandths, and ninety-three thousandths? Ans. ,5648.

63

9. What is the difference between 1 and 3? Ans. 1,1157+.

Remark. It may be proper to observe, that in this, and the two preceding rules, when vulgar fractions are changed to decimals, and the operations performed with the decimals, some of the decimal figures may be different from what they would be were the division in reducing them continued farther. But if three or four places of decimals are found, the difference is so small a part of an unit, that the error is of no consequence in most kinds of business.

2,2 2 2 2 2+, or,2 2 2 2 2 2+
0 9 0 9 0+, or ,0 9 0 9 0 9+

,1 3 1 3 2

,1 3 1 3 1 3

Explanation.-By carrying the decimal to six

places in the second part of the operation, we have 1 in the fifth place; but in the first part of the operation, in which we carry the decimal to 5 places only, the figure in the fifth place is 2.

11. What is the difference between one million and one millionth ? Ans. 999999,999999.

12. An engineer wishing to ascertain the height of a certain pond above a neighbouring lake, found the summit level of the intervening land to be 24 ft. above the surface of the lake, and 13,3 feet above that of the pond. How much higher was the surface of the pond than that of the lake? Ans. 10,7 ft.

DIVISION OF DECIMAL FRACTIONS.

Illustration.-In dividing decimals, or whole numbers and decimals, the object is to find how many times the divisor can be subtracted from the dividend, or to divide the dividend into several equal parts.

A decimal may be divided by a decimal, and yet the quotient be a whole number.

,5),5 ( 1

5

,2),8 (4

8

2,3) 4,6(2

4 6

If we divide 5 tenths by 5 tenths, the quotient is an unit, since 5 tenths can be taken from 5 tenths once.

So if we divide 8 tenths by 2 tenths, 4, the quotient, is a whole number, for 2 tenths can be subtracted 4 times from 8 tenths.

Also, the quotint of 4,6, divided by 2,3, is 2 units, since 3 tenths can be taken 2 times from 6 tenths, and 2 units 2 times from 4 units.

2) 4, 8 (2, 4

4

88

But the quotient of 4,8 divided by 2, is a mixed number. The first figure in the quotient is a whole number found by dividing the 4 units in the dividend by 2. The next part of the dividend to be divided is only 8 tenths, and the divisor being a whole number, is equal to 20 tenths; therefore the divisor cannot be taken once from the remaining part of the dividend. If we were to set 20 tenths under the remainder, 8 tenths, the fraction would express the remaining part of the quotient; but equal, the same number of tenths which we find by dividing in the common way. The product of the divisor multiplied by the last quotient figure, must never be greater than that part of the dividend which we are dividing. The product of 2 multiplied by 4 tenths, is 8 tenths, which can be taken from the remaining part of the dividend, consequently the right hand figure in the quotient is a fraction. The 4 tenths show what part of the divisor can be taken from 8 tenths.

At $2,5 per yard, how many yards of cloth can be bought for $6,75?

2,5) 6,7 5 ( 2,7

50

175

1 7 5

There will be as many yards as 2,5 can be taken times from 6,75. The left hand figure of the quotient is a whole number. The second dividend is 1 and 75 hundredths, a less number than the divisor, and only a part of the divisor can be taken from it, which shows that the next figure in the quotient will be a fraction. Multiplying the divisor by 7 tenths, produces a product of 1 and 75 hundredths, a number just equal to that part of the dividend which is to be divided the second time; 7 tenths must therefore

be the last quotient figure. Had there been another decimal place in the dividend, by bringing it to the right of hundredths, the remaining part of the dividend must have been less than that last divided, and the next quotient figure must have been less than tenths. By a similar method of reasoning, it might be shown, that for every additional place in the dividend, there would be an additional place in the quotient.

In the three first examples, the decimal places in the divisor and dividend are equal, and the quotients whole numbers; in the two last, there is one decimal place more in each dividend than in the divisor, and one decimal place in the quotient.

From what has been shown, it appears, that when the decimal places in the dividend and divisor are equal, the quotient may be a whole number, but when the decimal places in the dividend are the greatest, there will be as many decimal places in the quotient as the decimal places in the dividend are more than those in the divisor.

RULE.

Divide as in whole numbers, and point off as many places for decimals in the quotient, as the decimal places in the dividend are more than those in the divisor.

1. How many times can 6,12 be taken from 104,958 ?

Operation.

6,1 2 ) 1 0 4,9 5 8 (17,1 5 Ans.

6 1 2

4 3 7 5

4 2.8 4

918
6 12

3060

Explanation. The cipher annexed to the last remainder, does not increase the value of the remainder, but only represents another decimal place in the dividend. Two decimal places are pointed off in the quotient, because the three given decimals, and the cipher annexed to the remainder, make 4 places, two more than the number of decimal places in the divisor.

2. If a yard of cassimere cost $2,31, how many yards can be bought for $50,623?

Ans. 21,91+yards. 3. If I sell 169,8 pounds of butter for $23,26, what do I receive per pound? Ans. $0,136+. 4. A man sells cider to the amount of $65, at $1,95 per barrel; how many barrels does he sell? Ans. 33,33+. 5. How much sugar, at 12 cents 5 mills per pound, can be bought for $15,50?

they cost a piece?

Ans. 124. 6. If a dozen of buttons cost 75 cents, what do Ans. 6 cents and 21 mills. 7. What is the cost of one bushel of potatoes, if 35 bushels cost $12? Ans. $0,342+. 8. If a labourer receive $4,25 for 6 days' work, what is that per day? Ans. $0,708+. 9. F owes G $15,58, and is to pay him in rye at 67 cents per bushel; how much rye will be required to discharge the debt? Ans. 23,25+bush. 10. When buttons are sold at 9 cents a dozen, what are they a piece?

1 2 ),0 9 0 (,0 0 7 5

8 4

60 60%

One button cost 75 ten thousandths of a dollar, or 7 mills.

Explanation. The number of cents the 12 buttons cost being less than 12, the price of one button