The same rule holds in subtraction. Take for illustration the numbers given in the second example of addition. From 6.8 subtract 5. 23. Now as in the minuend there are no hundredths, we must borrow 10 in this place, and we shall have a remainder of 7 hundredths; adding 1 tenth to the subtrahend, to compensate for the 10 hundredths added to the minuend, we have in the place of tenths a remainder of 5; finally, in the place of units we subtract 5 from 6: the answer is 1. 57. In performing this operation, you may, if you please, call the 8 tenths 80 hundredths; then 23 hundredths from 80 hundredths leaves 57 hundredths. By performing slowly and with care examples of your own selection, you will see the verification of the rule given above, both for addition and subtraction.

Add 2.4 to 3.8. Add .6 to 1.3. Add .4 to .3. Add .37 to .25. Add 3.7 to .24. Add 1.08 to .05.

From 4.6 subtract 2.4.

From 7.1 subtract 6.4. From .18

subtract .13. From 4.5 subtract .6.

In these examples each step should be explained by the pupil as he performs it.

MULTIPLICATION OF DECIMALS.

The rule in multiplication we shall find to be different from the above.

1. First, we will multiply 2.4 by 3. If we regard the multiplicand as a whole number, the answer will be 72. But by regarding the multiplicand as a whole number, -as 24 instead of 2 and 4 tenths, we regarded it ten times greater than it really is; the answer, therefore, is ten times too great. Instead of 72 it must be 7.2.

2. Multiply 6.2 by 3.4. By regarding both as whole numbers we obtain the answer 2108. Now in calling the multiplicand 62 instead of 6.2 we treated it as 10 times greater than it is. The answer must therefore be 10 times too great, even if the multiplier were a whole number. We must therefore divide it by 10, or write 210.8. But the multiplier also is 10 times too great; the answer must therefore be divided again by 10, in order to bring it right. Thus the answer will stand 21.08.

3. Again; multiply .62 by 3.4. Here we obtain the same

figures as before, 2108; but by treating the multiplicand as a whole number, we regarded it as 100 times too great; the answer therefore must be divided by 100, or written 21.08. But the multiplier, calling it a whole number, was taken 10 times greater than it is; the answer must be again divided by 10, and thus it will stand 2.108.

4. Once more; multiply .62 by .34. The figures of the answer are as before, 2108, but by regarding both the factors as whole numbers, we take each 100 times greater than it is; we must therefore divide by 100 to correct the error in the multiplier, and again by 100 to correct the error in the multiplicand. This will remove the point four places to the left, and the true answer will be .2108. By examining these examples you will see that the pointing in each case conforms to the following rule.

Point off as many figures for decimals in the answer as there are decimal places in both the factors taken together. 5. Multiply 2.7 by .3. 6. Multiply .6 by .7. 7. Multiply 6. by .7. 8. Multiply .02 by .3. 9. Multiply .02 by .03. 10. Multiply .01 by .01.

DIVISION OF DECIMALS.

1. Divide 48 by 12. Ans. 4.

2. Divide 4.8 by 12. The figure expressing the answer is 4, as in the first case; but, observe, the dividend is only one tenth as large as before; the quotient, therefore, is only one tenth as large. Instead of 4. it is .4.

3. Divide .48 by 12. The figure of the quotient is still 4, but as the dividend is only one hundredth part as large as in the first example, the quotient will be only one hundredth part of 4, or 4 hundredths, written thus, .04.

4. Again; divide 48 by 1.2. The quotient is still 4, but we must investigate the question to see where this 4 must stand. You observe that the divisor is now only one tenth of 12. Now if the divisor is only one tenth as great as it was before, you must consider how that will affect the quotient. You will perceive on reflection that as you diminish the divisor you increase the quotient. If you make the divisor half as great, the quotient will be twice as great, and so proportionally of other numbers. Now as, in this instance, the divisor is one

tenth as great as before, the quotient must be ten times greater. The figure 4, then, which is the quotient figure, instead of standing in the place of units, as before, must stand in the place of tens; that is, it must be 40, the cipher merely showing that the 4 stands in the place of tens.

5. Once more: divide 48 by .12. Here again you have 4 for the quotient figure, for you can have no other; but on comparing this example with the first, you perceive the divisor is only one hundredth part as great; the quotient must therefore be one hundred times greater, that is, it is 400, the ciphers merely removing the 4 into the place of hundreds.

On examining these examples carefully, you will see that each answer is unquestionably correct. "But by what rule," you ask, “ are these examples wrought?" They are not wrought by rule, but by reasoning on the numbers themselves; and the more you habituate yourself to reason in arithmetic, the less need you will have to depend on rules.

With this suggestion I will now state a rule, which you may at any time follow, when you have not time to look into the reason of the operation.

There must be as many decimals in the quotient as the decimals in the dividend exceed those in the divisor: when there are fewer decimals in the dividend than there are in the divisor, ciphers must be added so as to make the number equal.

We will now review the foregoing examples, and observe their conformity with the above rule. Example 1 has no decimals in the divisor or the dividend, therefore none in the quotient. Ex. 2, the dividend has one decimal, the divisor none; the quotient has therefore one. Ex. 3, the dividend has two decimals, the divisor none; the quotient has two. Ex. 4, the dividend has none, the divisor one; there must then be a cipher added to the dividend, and then the quotient will be in whole numbers. Ex. 5, the dividend has none, the divisor two; there must then be two ciphers added, and then the quotient will be in whole numbers.

6. Divide 45 by 15. 15. Divide 45 by 1.5. 7. Divide 66 by 11.

66 by .11.

Divide 4.5 by 15.
Divide 45 by .15.

Divide .45 by

6.6 by 11. .66 by 11. 66 by 1.1.

In calculations of Federal money, cents and mills are regarded as decimals; the point therefore separating the whole

numbers from the fractions must be placed between the dollars and the cents. Thus 24.00 is 24 dolls.; 2.40 is 2 dolls. 40 cents; 0.24 is 24 cents.

8. A man divided $24.00 among 3 men; how much did each receive?

9. A man divided $2.40 among 3 men; how much did each receive? Divide 2.4 by 3.

10. A man divided $0.24 among 3 men; how much did each receive? Divide $0.24 by 3.

11. A man divided 36 dollars among 4 persons; how much did each receive? Divide 36 by 4.

12. A man divided $3.60 among 4 persons; how much did . each receive? What is one fourth of $3.60?

13. A man divided $0.36 among 4 men; how much did each receive? What is one fourth of ,36?

SECTION XIII.

REDUCTION OF VULGAR FRACTIONS TO DECIMALS.

We have now seen that Decimal Fractions have this great advantage over Vulgar Fractions, that they conform to the same law of notation as whole numbers, and may be added, subtracted, multiplied and divided in the same manner, and with the same ease as whole numbers. It is desirable, therefore, to introduce them in a great many cases instead of Vulgar Fractions. The next question that arises, therefore, is, can a Vulgar Fraction be changed to a decimal, having the same value; and how can it be done? Take the fraction we wish to reduce it to tenths; or in other words to express it in tenths. Now we can change any number to tenths by multiplying it by 10. Thus 3 is 30 tenths, 4 is 40 tenths. We will now take and change the numerator 1 to tenths, and it will stand .10: but the fraction was not one, but one half of one; 10 therefore is twice as great as it should be; we must divide it, therefore, by 2; that is, by the denominator, and it will be .5. To reduce a vulgar fraction, then, to a decimal: add a cipher to the numerator, and divide by the

denominator. If one cipher is not enough to render the division complete, add more.

Reduce to a decimal ; change the numerator to tenths; it will be .10, but the quantity to be reduced to tenths was not one, but one fifth of one; 10, therefore, is 5 times greater than it should be; dividing by 5, the answer is .2.

Reduce to a decimal the fraction, explaining each step in the operation.

Reduce to a decimal the fraction .

Reduce to a decimal the fraction

Reduce to a decimal the fraction .
Reduce to a decimal the fraction .

I will here direct your attention to a fact that it is interesting to notice. If the denominator of the vulgar fraction is one of the factors of 10, that is, if it is either 2 or 5, the decimal figure will be as many times the other factor as there are units in the numerator of the vulgar fraction. This will appear self-evident when we express the numbers by their factors. Thus in obtaining the decimal for we divide 10 by 2; but 10 is 2X5, therefore in dividing by 2, we simply expunge the factor we divide by, and leave the other: 2) 2X5. So in the fraction, we obtain the decimal by dividing 10 by 5, which expunges the factor 5, 5)5×2; in reducing we divide 2×10 by 5, thus: 5) 2×2×5, leaving twice the factor 2; in §, 5) 3×2×5, leaving 3 times the factor 2; in, 5) 2×2×2×5, leaving 4 times the factor 2.

2. We will now take the fraction; proceeding as before we wish to divide 10 by 4, thus, 2×2) 2×5; here we see the division cannot be complete, for the divisor contains the factor 2 twice, while the dividend has it only once. If, however, we had multiplied the original numerator 1 by 100, instead of 10, we should have had 10 twice as a factor in the dividend, and of course each factor of 10 twice; 100 is 10×10, and 10 is 2X5. It would have stood then thus, 2×2)2×5×2×5; the division is now complete, for the dividend contains the factor 2 as many times as the divisor has it. Expunging these we have remaining the factor 5 taken twice, or .25.

This process you may observe conforms to the rule, to add as many ciphers as may be necessary to render the division complete.

3. Reduce the vulgar fraction to a decimal. 30 is composed of the prime factors 3×2×5; it contains 2 only once,