LESSON LIX.

CASE II.-THE DIVISOR BEING A DECIMAL NUMBER.

1. PROPOSITION. If you increase the same number of times both the dividend and the divisor, will it change the quotient?

Ans. If the dividend and divisor be both multiplied or divided by the same number, the quotient remains the

same.

2. Can you explain why it is so?

Ans. If one dividend contains one divisor a number of times, several dividends will contain as many divisors the same number of times.

Let us suppose, as an illustration, that there are a number of baskets, say 5, with 12 oranges in each, and 4 boys to divide each basket; each boy will get 3 oranges for his share; 12 oranges will be one dividend, 4 one divisor, and 3 the quotient. Evidently, it matters not whether 4 boys help themselves out of one particular basket, or all the oranges out of the 5 baskets are put together, and the 5 sets of boys help themselves out of the pile. In either case, each will get 3 oranges, and none will be left. The 5 baskets will be 5 dividends; the 5 sets of boys 5 divisors; and 3 oranges will still be the quotient.

3. How is one decimal number divided by another? The only difficulty of this division is the fixing of the position of the units' point; otherwise it is like simple division.

4. What is the simplest way to perform this operation without mistake?

Ans. The following is the safest practical rule; it reduces the operation to the preceding.

RULE.*

Change the divisor into a whole number, by striking off the decimal point; remove the point of the dividend as many places to the right as there were decimal figures in the divisor, annexing ciphers, if necessary; and then divide by the altered divisor, according to the preceding rule.

FIRST EXAMPLE-MORE DECIMALS IN THE DIVIDEND.

5. Divide 36.864 by 4.8.

The divisor having only one decimal place, the point in the dividend is moved only one place to the right; and we now divide 368.64 by the integer 48, instead of the original numbers, which will give the same quotient.

6. How do you understand that you thus get the same quotient?

Ans. By removing the units' point one place to the right, we increase each number ten times (LVI. 3); both the dividend and divisor are thus equally multiplied and the quotient is not changed (1).

* I have tried with pupils every other mode of dividing decimals, and found this way to be the only one they could understand and practise without mistakes, especially when the quotient is extended.

SECOND EXAMPLE-EQUAL NUMBER OF DECIMALS.

7. Divide 65.74 by 3.46.

In this case, the decimal point need not be considered; we divide as if both were whole numbers; since not noticing the units' point amounts to multiplying both equally; which does not change the quotient (1).

It is evident that when the dividend and divisor have the same number of decimals, the quotient is a whole number.

EXERCISES.

1. Divide: 1201.725 by 3.675.

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Ans. 327. Ans. 4086.

Ans. 45.

Ans.

THIRD EXAMPLE-MORE DECIMALS IN THE DIVISOR.

8. Let it be proposed to divide 657.4 by 0.346.

When there are more decimal figures in the divisor than in the dividend, the removal of the units point requires the addition of Os to the dividend, as may be seen in the annexed operation, and the quotient is a whole number.

In this case, as in the the third example, zeros must be added to the dividend, and the quotient is a whole

number.

11. You observe that the quotient is a whole number; how do you understand that it can be so large a number, when the dividend is so small?

Ans. However small a quantity may be, we may always conceive of a much smaller one, which it will contain a great many number of times.

DECIMAL REMAINDERS.

12. What is done if the division of decimals leaves a remainder?

Ans. If a decimal dividend leaves a rèmainder, it is either neglected as too small, or it may be placed over the altered divisor, and thus annexed to the quotient.

Suppose, for instance, that we divide 12.791 by 1.6, the quotient is 7.99 with a remainder 7 hundredths, which we place, not over the original divisor 1.6, but over the altered divisor 16: it means that the remaining 7 hundredths are yet to be divided by 16 in some way. More generally, however, it is omitted, and sometimes the sign is placed after the quotient, to indicate that it is not an exact quotient.

PROOF OF DECIMAL MULTIPLICATION AND DIVISION.

13. How are decimal multiplication and division proved?

Multiplication is proved by division, and division by multiplication, exactly as in simple numbers; and no farther explanation is needed here, as we know how to perform both operations.

EXERCISES WITH PROOFS.

Several decimal points are omitted purposely, as well as additional zeros.

Answers.

10.59.

*849.9.