26. John spent 85 cents for a knife, one dollar and a half for a sled, and a quarter of a dollar for a ball. What change should he have received from a $5 bill?

161. 1. 5 times .1 are how many tenths?

2. 3 times .3 are how many tenths?

3. 4 times .6 are how many tenths?

SOLUTION. 4 times .6 are 24 tenths, equal to 2 units and 4 tenths, or 2.4.

4. Multiply .8 by 6; by 7; by 8.

5. Multiply .09 by 5; by 6; by 7.

NOTE.-In decimals, as in integers, the product is the same denomination, or kind, as the multiplicand.

6. Multiply 1.2 by 3; by 4; by 5.

7. Multiply .1 by .1.

SOLUTION..1X.1=0x00, that is, .01.

8. Multiply .3 by .03.

SOLUTION.

.3X.03=1×180=1000, or .009.

NOTE I. In the multiplication of one decimal fraction by another decimal fraction, there is really the same process as in the multiplication of any fraction by another fraction; that is, the numerator of the product is the product of the numerators of the factors; and the denominator of the product is the product of the denominators of the factors. Since the denominator of every decimal fraction is a power of 10, the product of the denominators is that power of 10 which contains as many O's as both, or all, of the denominators of the factors. Thus,

1×1=100; 10×100=1000; 100×100=10000; etc. In multi

plying one decimal fraction by another, therefore, it is necessary to find the product of the numerators only, and make the number of decimal orders in the product of the numerators equal to the number in both of the factors. Thus, .3x.5=.15.

NOTE II. It is frequently necessary to prefix 0's to the product of the numerators, in order to secure the requisite number of decimal orders. Thus, .1×.1=.01; .01.01 .0001; etc.

PRINCIPLE.

3. The number of decimal orders in the product of two or more decimal factors is always equal to the number of decimal orders in all the factors.

9. Multiply .325 by 5.

WRITTEN.

=

SOLUTION. 5 times .325-5 times 325=1888=

PROCESS.

.325

5

1.625.

1.625

10. Multiply .325 by .05.

PROCESS.

.325

.05

.01625.

325

1625 SOLUTION..325 X.05 = X +80=1888 =

10000

.01625

Rule.-Multiply as in integers, and make the number of decimal orders in the product equal to the number of decimal orders in both factors. If necessary, prefix 0's.

21. Find cost of 25 yards of cloth, worth $4.25 per yard. 22. Find value of 3625 bushels of wheat, worth $1.12 per bushel.

23. Find cost of 2450 acres of land, worth $28.50 per

acre.

24. Multiply 1.8456 by 100.

SUGGESTION. Simply remove the decimal point two places to the

right.

25. Multiply 93.426 by 1000.

26. Find product of .18433 by 100.

27. Multiply .4 by 100.

SUGGESTION.-Removing the decimal point one place to the right

1

multiplies by 10; erasing this point and annexing 0 multiplies again

by 10.

28. Multiply 3.5 by 100.

29. Multiply 54.69 by 1000.

DIVISION OF DECIMAL FRACTIONS.

ORAL.

162. 1. Divide .6 by 2.

SOLUTION.-Dividing by 2 is finding one half. One half of .6 is .3.

2. Divide .8 by 2; by 4.

3. Divide .12 by 2; by 3; by 4.

4. Divide 3.6 by 9.

SOLUTION.- 3.6=36 tenths. of 36 tenths is 4 tenths, or .4.

163. The difficulty in division of decimals is the placing of the decimal point in the quotient. But it should be borne in mind that division is the converse of multiplication; that the divisor and quotient are factors of the dividend. Hence, the number of decimal orders in the divisor and quotient must be equal to the number of decimal orders in the dividend. Therefore, the decimal orders in the quotient must equal the excess of the decimal orders in the dividend over those in the divisor.

That is, if there are 5 decimal places in the dividend, and 2 decimal places in the divisor, there must be 3 decimal places in the quotient.

In each of the three preceding examples, there is one decimal place in the dividend, none in the divisor; therefore, there must be one decimal place in the quotient

NOTE. The correctness of the pointing in the quotient may always be tested by proving the work; since the dividend must be the product of divisor and quotient.

PRINCIPLE.

4. The decimal orders in the divisor and quotient are equal in number to the decimal orders in the dividend.

164. The division of a decimal fraction or mixed number by a decimal fraction may be considered in three cases. 1. When the decimal orders in the dividend and divisor are equal in number.

2. When the decimal orders in the dividend are less in number than those in the divisor.

3. When the decimal orders in the dividend exceed in number those in the divisor.

CASE I.

165. 1. Divide .9 by .3.

SOLUTION..9÷.3=1%÷19÷3=3.

2. Divide .7 by .3.

SOLUTION.— .7÷.3=16÷13=7÷3=2. The common fraction may be changed to a decimal (Art. 157).

NOTE.-When dividend and divisor are fractions having a common denominator, their quotient is the quotient of the numerator of the dividend divided by the numerator of the divisor (Art. 148). When dividend and divisor are decimal fractions having the same number of decimal orders, they have a common denominator; and their quotient is the quotient of their numerators, and is an integer or a mixed number. When the dividend is a multiple of the divisor, the quotient is an integer. Thus, 2.4+.8-3. When the dividend is not a multiple of the divisor, the quotient is a mixed number. Thus, 2.5+.8=3, or 3.125.

SOLUTION..21÷.04-100÷10=21÷4=54 or 5.25.

12. Divide .45 by .06; by .07; by .08; by .09.

CASE II.

166. 1. Divide .8 by .04.

SOLUTION.-Annex a sufficient number of 0's to the dividend to make the decimal places in the dividend equal to those in the divisor. Then proceed as in Case I. .8=.80. .80÷.04=10%÷10=80÷4=20. 2. Divide .9 by .004.

SOLUTION.-.9.900; .900÷.004-1000-1000-900÷4=225.

167. 1. Divide .24 by .3.

SOLUTION..24+3=100÷13b=106×1o==.8.

NOTE.-The denominator of the quotient is the quotient of 100+10=10; that is, the denominator of the quotient contains as many O's as the O's in the denominator of the dividend exceed the O's in the denominator of the divisor. Or, when the fractions are written decimally, the quotient contains as many decimal places as the decimal places in the dividend exceed those in the divisor.

ANALYSIS.-Regard both numbers as integers; then 512÷8-64. Dividing the divisor, 8, by 10, multiplies the quotient by 10; that is, 512÷.8=640.

Dividing the dividend, 512, by 100, divides the quotient by 100; that is, 5.12.8-6.40, or 6.4.

Observe that the number of decimal orders in the divisor and quotient equals the number of decimal orders in the dividend.