In looking over the above process we find, that the two numbers are multiplied together in the same manner as whole numbers, and as many places are pointed off for decimals in the product, as there are in the multiplicand and multiplier counted together.

It is plain that this must always be the case, for tenths multiplied by tenths must produce tenths of tenths, that is hundredths, which is two places; tenths multiplied by hundredths must produce tenths of hundredths, or thousandths, which is three places; hundredths multiplied by hundredths must produce hundredths of hundredths, that is ten-thousandths, which is four places, &c.

What cost 5 tons of hay, at $27.38 per ton? 5=

5.375.

27.38

5.375

13690

19166

8214

13690

$147.16750 Ans.

In this example there are hundredths in the multiplicand, and thousandths in the multiplic:. Now hundredths multiplied by thousandths must produce hundredths of thousandths, which is five decimal places, the number found by counting the places in the multiplicand and multiplier to.. gether. The answer is 147. dollars, 16 cents, 7 mills, and

I of a mill.

A man owned .03 of the stock in a bank, and sold 2 of his share. What part of the whole stock did he sell?

It is evident that the answer to this question must be expressed in thousandths, for hundredths multiplied by tenths must produce thousandths. of are. But if we multiply them in the form of decimals, we obtain only one figure, viz. 6. In order to make it express T it will be necessary to write two zeros before it, thus, .006.

.03
.2

Ans. .006 of the whole stock.

This result is agreeable to the above rule.

The following is the general rule for multiplication, when there are decimals in either or both the numbers: Multiply as in whole numbers, and point off as many places from the right of the product for decimals, as there are decimal places in the multiplicand and multiplier counted together. If the product does not contain so many places, as many zeros must be written at the left, as are necessary to make the number.

up

Division of Decimals.

XXVIII. A man bought 8 yards of broadcloth for $75.376; how much was it per yard?

$75.376

mills. 75376 (8

72

9422 mills.

33

32 $9.422 Ans.

17

16

16

16

In this example there are decimals in the dividend only. I consider $75.376 as 75376 mills. Then dividing by 8, either by long or short division, I obtain 9422 mills per yard, which is $9.422. The answer has the same number of decimal places as the dividend.

1

Divide 117.54 bushels of corn equally among 18 men. How much will each have?

5 4 11754; this divided by 18 gives

117.54 = 117%

$536% 6.53.

100

100

117.54 (18

108

6.53

95

90

54

54

Or we may reason as follows. I divide 117 by 18, which gives 6, and 9 remainder. 9 whole ones are 90 tenths, and 5 are 95 tenths; this divided by 18 gives 5, which must be tenths, and 5 remainder. 5 tenths are 50 hundredths, and 4 are 54 hundredths; this divided by 18 gives 3, which must be 3 hundredths. The answer is 6.53 each, as before.

If you divide 7.75 barrels of flour equally among 13 men, how much will you give each of them?

1000

=

78

2

596

It is evident that they cannot have so much as a barrel each. 7.757387788. Dividing this by 13, I obtain 596 and a small remainder, which is not worth noticing, since it is only a part of a thousandth of a barrel. 1030 .596. Or we may reason thus: 7 whole ones are 70 tenths, and 7 are 77 tenths. This divided by 13 gives 5, which must be tenths, and 12 remainder. 12 tenths are 120 hundredths, and 5 are 125 hundredths. This divided by 13 gives 9, which must be hundredths, and 8 remainder. We may now reduce this to thousandths, by annexing a zero. 8 hundredths are 80 thousandths. This divided by 13 gives 6, which must be thousandths, and 2 remander. Thousandths will be sufficiently exact in this instance, we may therefore

omit the remainder. The answer is .596 + of a barrel each.

From the above examples it appears, that when only the dividend contains decimals, division is performed as in whole numbers, and in the result as many decimal places must be pointed off from the right, as there are in the dividend.

Note. If there be a remainder after all the figures have been brought down, the division may be carried further, by annexing zeros. In estimating the decimal places in the quotient, the zeros must be counted with the decimal places of the dividend.

At $6.75 a cord, how many cords of wood may be bought for $38?

67

In this example there are decimals in the divisor only $6.75 is 675 cents or $75 of a dollar. The 38 dollars must also be reduced to cents or hundredths. This is done by annexing two zeros. Then as many times as 675 hundredths are contained in 3800 hundredths, so many cords may be bought.

The answer is 54 cords, or reducing the fraction to a decimal, by annexing zeros and continuing the division, 5.62+ cords.

If 3.423 yards of cloth cost $25, what is that per yard?

The question is, if 433 of a yard cost $25, what is that a yard?

According to Art. XXIV., we must multiply 25 by 1000, that is, annex three zeros, and divide by 3423.

3423

The answer is $71933, or reducing the fraction to cents. $7.30 per yard.

If 1.875 yard of cloth is sufficient to make a coat; how many coats may be made of 47.5 yards?

In this example the divisor is thousandths, and the dividend tenths. If two zeros be annexed to the dividend it will be reduced to thousandths.

1875 thousandths are contained in 47500 thousandths times, or reducing the fraction to decimals, 25.33 + times, consequently, 25 coats, and of another coat may be made from it.

From the three last examples we derive the following rule: When the divisor only contains decimals, or when there are more decimal places in the divisor than in the dividend, annex as many zeros to the dividend as the places in the divisor exceed those in the dividend, and then proceed as in whole numbers. The answer will be whole numbers.

At $2.25 per gallon, how many gallons of wine may be bought for $15.375 ?