26. Multiply $150.182 by 7; that is,

find 7 times $150.183,+ of $150.183.

We first multiply by the 7, as in the preceding exercises. To multiply by the we say, of 15 is 7 with one over; of 10 is 5; of 18 is 9, supplying the vacant place before 9 with

0;

of is 3. We then add together the two products, and make two decimal figures, for the two in the multiplicand.

But it will often be preferable to reduce the vulgar fractions in the multiplicand and multiplier to their equivalent decimals.

In this example we shall have $150.1875×7.5=$1126.40625; which is 1126 dollars, 40 cents, 6 mills and 5 tenths of a mill.

27. A farmer sold 15 acres of land, at 27 dollars 37 cents per acre; required the sum he should receive in payment. Ans. $431.156'.

28. What should be paid for 31⁄2 barrels of flour at 6 dollars 582 cents a barrel, and 11ğ bushels of meal at 43 cents a bushel? Ans. $28.084'.

29. Bought a piece of cloth containing 394 yards, for $238.50; of which 20 yards have been sold at $7.12 per yard. What will be the gain or loss on the whole, if the remainder be sold at $8.06 per yard? Ans. Gain $63.

30. Bought 45 hundred weight of hemp, at $6.25 per hundred weight; which has been made into rope and bagging, at an expense of $130.183. For what sum must the manufactured articles be sold to clear $50? Ans. $464.5625.

31. A trader bought 120 mules at an average price of $39.50; of which he has sold 20 head at $54.624, and 33 head at $59 a head. What will be his entire profit or loss if the rest be sold at $30.50 a head? Ans. Profit $343.

32. A barters to B 35 yards of broadcloth at $7.5 a yard, for 135 yards of silk at $.934 a yard, the difference in value between the two commodities to be paid in money. them must receive money, and how much?

Which of

Ans. B must pay $139.6875.

DIVISION OF DECIMALS.

RULE XXXII.

§ 152. For the division of decimals.

1. Divide as in integers, and in the right of the quotient make as many decimal figures as there are decimal figures in the dividend more than in the divisor; prefixing Os to the quotient, when necessary to make up the number.

2. When the divisor has more decimal figures than the dividend, or is greater than the dividend (regarding both as integers,) annex decimal Os to the dividend, to supply the deficiency.

3. Ciphers may always be annexed to the remainder, and the division continued to any required exactness,-observing that the Os so annexed must be counted as decimal figures belonging to the dividend.

4. Integral Os in the right of the divisor may be omitted, provided the same number of integral figures be made decimals in the dividend,—Os being prefixed to the dividend, when necessary to make up the number. When the divisor is 10, or 100, &c., the quotient is thus immediately obtained.

EXAMPLES.

1. To divide .965 by .5, that is, to find how often .5 is contained in .965.

Dividing as in integers, we find the quotient 193; in which we make two decimal figures, since the dividend has two more decimal figures than the divisor.

2. To divide .375 by 125; that is, to find what part .375 is of 125

125).375 (.003

Having found the quotient 3, we prefix to it two Os and the decimal point, to make three decimal figures; since the dividend has three decimal figures, while the divisor has none.

3. To divide 7 by 1.2.

1.2) 7.0

5.8 3'

Or 5.831.

The dividend having no decimal figure in it, while the divisor has one, we annex a decimal 0 to the dividend.

Having found the quotient figure 5, we annex a 0 to the remainder 10, and say 12 in 100, 8 times and 4 over; annexing another 0 to the 4, we say 12 in 40, 3 times, and 4 over.

Thus the division might be continued. Or, we may form a fraction of the remainder and divisor, and annex it to the quotient.

The three Os annexed make three decimal figures belonging to the dividend; hence we make two decimal figures in the quotient.

4. To divide 8.4 by 300.

300)8.40 (.0 28.

The dividend 8.4 being less than the divisor, we annex a decimal 0 to the dividend, and divide 300 into 8.40.

Or, rejecting the two integral Os in the right of the divisor, and making two integral figures decimals in the dividend,—prefixing a 0 to make up the number,

.084 3.028,

we divide .084 by 3, and find the same quotient as before. In like manner, 345÷100 = 3.45;

345 1000 .345 ;

345 10000.0345 ; and so on:

in which cases the quotient is immediately obtained, by making as many integral figures decimals in the dividend, as there are integral Os in the right of the divisor,-prefixing Os to the dividend, when necessary to make up the number.

The dividend must contain just as many decimal figures as both the divisor and quotient, because the dividend is equal to the product of the divisor and quotient.

Thus in the first example, .965=,5×1.93.

Hence also, the number of decimal places in the dividend cannot be taken less than the number in the divisor; for then the product would have a less number of decimal places than one of its factors,-which is impossible.

We prefix the Os to the quotient, as in the second example, because in no other position of the Os would the divisor, multiplied by the quotient, produce the dividend.

EXERCISES.

1. Divide 22.36 by 4.3 and 3.25 by 1.3.
2. Divide 12.25 by 3.5 and 87.9 by .3.
3. Divide .0425 by .05 and 5 by .0625.
4. Divide 16.776 by 7.2 and .816 by .04.
5. Divide 1.5 by .375 and 75 by 12.
6. Divide 4.9 by 70 and 7.02 by 3.
7. Divide 5.85 by .65 and 5.92 by .08.
8. Divide 15.57 by .45 and .001638 by .07.

9. Divide 8 by 3.2 and 234.375 by 25. 10. Divide .0276 by 23 and .08 by 32.

Ans. 5.2 and 2.5. Ans. 3.5 and 293. Ans. .85 and 80. Ans. 2.33 and 20.4.

Ans. 4 and 6.25. Ans. .07 and 2.34. Ans. 9 and 74.

Ans. 34.6 and .0234.

Ans. 2.5 and 9.375. Ans. .0012 and .0025.

In the next exercises, let the quotient be continued to thousandths, and be expressed by an approximate decimal.

11. Divide 13.29 by 2.8 and .278 by .07.

Ans. 4.746' and 3.971'.
Ans. .025' and .012'.
Ans. .082' and .010'..
Ans. .664' and .612'.
Ans. .766' and 175'.

12. Divide 2.37 by 93 and .0011 by .09. 13. Divide .737 by 8.9 and .09 by 8.3. 14. Divide 8.641 by 13 and .643 by 1.05. 15. Divide .023 by .03 and .013 by .074.

Another Method of Reducing a Vulgar Fraction to a Decimal.

§ 153. The quotient of a less integer divided by a greater may be represented by a proper vulgar fraction; thus 3÷4 is 3.

If the less integer be then divided by the greater, decimally, the vulgar fraction will be reduced to a decimal.

3÷4=3.00÷4=.75; or 3.75.

The application of Rule XXXII to this case, is substantially the same with that of Rule XXVIII.

16. Divide 4 by

15

34

to a decimal. Ans. .264'.

to a decimal.

Ans. .108'.

to a decimal.

Ans. .085.

25

to a decimal.

Ans. .073'.

to a decimal.

Ans. .538'.

18 to a decimal.

Ans. .526'.

15, or reduce to a decimal. Ans. .266', 17. Divide 9 by 34, or reduce 18. Divide 13 by 120, or reduce 19. Divide 17 by 200, or reduce 20. Divide 25 by 339, or reduce 21. Divide 7 by 13, or reduce 22. Divide 10 by 19, or reduce 23. Divide 21 by 121, or reduce 24. Divide 73 by 300, or reduce 25. Divide 99 by 500, or reduce to a decimal. Ans. .198.

73 300

26. How many gallons of wine, at 1 dollar 371⁄2 cents per gallon, may be bought for 25 dollars and 50 cents?

Expressing the cents in decimals of a $, the number of gallons is the number of times $1.37 cents is contained in $25.5; that is, $25.5 $1.37,-$25.5-$275.

But in the Division of Federal Money, instead of fractions of a cent, it will generally facilitate the operation to take their equivalent decimals. We have then

$25.5÷$1.375. Ans. 18.545' gallons.

27. What quantity of coal, at 18 dollars and 75 cents per ton, may be purchased for 13 dollars?

$13 will buy the same part of a ton that $13 is of $18.75. Ans. .693' of a ton.

28. How many hundred weight of flour, at 2 dollars 18 cents per hundred, may be purchased for 25 dollars? Ans. 11.428' hundred weight. 29. What quantity of land, at the rate of 25 dollars per acre, may be bought for 9 dollars 62 cents?

Ans. .3849 of an acre. 30. How many bushels of clover seed, at 5 dollars 183 cents per bushel, may be bought for 30 dollars?

Ans. 5.783' bushels. 31. How many yards of cloth, at 4 dollars and 50 cents a yard, may be purchased for 19 dollars 75 cents?

Ans. 4.388' dollars. 32. How many barrels of corn, at 3 dollars and 85 cents per barrel, may be bought for 100 dollars? Ans. 25.974' barrels. 33. What quantity of bacon, at 8 dollars 314 cents per hundred weight, may be purchased for 5 dollars 6 cents?

Ans. .609' of a hundred weight. 34. What quantity of iron, at 45 dollars 50 cents per ton, may be bought for the sum of $7.064+$13.5 ?

Ans. .451' of a ton.

35. How many barrels of apples, at 2 dollars 12 barrel, may be bought for $75.83-$20.56?

cents per

Ans. 26.011' barrels.

36. What should be paid for a ton of hay, when .7 of a ton sells for 13 dollars 12 cents?

of

If 7 tenths of a ton cost $13.12, 1 tenth would be worth $13.12; and a WHOLE TON, or 10 tenths, would be worth 10 of $13.12; that is, such a part of this sum as is expressed by the reciprocal of 7; or $13.12÷÷.7.