11. What is the sum of 2.4999; 47 121212; 0·1, and 411.001 ? Ans. 460 722112.

12. What is the sum of 433-9; 777-5; 67:06, and 35.88? Ans. 1314.34.

SUBTRACTION OF DECIMALS.

37. FROM what has been said under Art. 35, we infer the following

RULE.

Place the smaller number under the larger, so that the decimal point of the one may be directly under that of the other. Then proceed as in subtraction of whole numbers.

7. From 110-001001 subtract 11·010002.

Ans. 98 990999.

MULTIPLICATION OF DECIMALS.

282

38. LET us multiply 0·47 by 06. If we put these decimals in the form of vulgar fractions, they will become, and these, multiplied by Rule under Art. 25, give %=20. Now it is obvious that there will be, in all cases, as many ciphers in the denominator of the product as there are in the denominators of both the factors added together.

Hence, the following

1000

RULE.

Multiply the two factors after the same manner as in whole numbers; then point off, from the right of the product, as many figures for decimals as there are decimal places in both the factors. If there are not so many places of figures, supply the deficiency by prefixing ciphers.

EXAMPLES.

1. Multiply 3.753 by 1.656.

Operation.

3.753

1.656

22518

18765

22518

3753

Ans. 6'214968.

2. What is the product of 0.005 into 0.017?

Ans. 0.000085.

3. What is the product of 0.376 into 0.0076894? Ans. 0.0028912144.

4. What is the product of 0.576 into 0-3854?

Ans. 0.2219904.

5. What is the product of 0.43 into 0.65?

Ans. 0.2795.

6. What is the product of 3.9765 into 4:378?

Ans. 17.409117.

7. What is the product of 415-314 into 7-3004 ?

Ans. 3031-9583256.

8. What is the product of 742 into 11-1415?

Ans. 82-66993.

ABRIDGED MULTIPLICATION OF DECIMALS.

39. ABRIDGED multiplication may be advantageously employed, when one or both of the factors are expressed approximately in decimals.

Suppose we wish the product of and; if we employ the rule under Art. 25, we shall find × =78.

In decimals, we have =0.33333, &c.; =0·16666, &c., and=0·05555, &c.

We will now multiply together the decimal values of , and, employing, in the first operation, 3 decimal places in each factor; 4 places in the second operation, and 5 in the third, as follows:

In the first operation, the result is true to only 3 places of decimals; in the second, it is true to 4, and in the third, to 5; and were we to employ a greater number of decimals in each factor, the product would be found to be accurate to only as many decimal places as there were in each factor. And in all cases, when the factors are approximate decimals, the whole number of decimals. obtained in the product, by the usual method of multiplication, is not accurate.

By the following rule, we may very much abridge the labor of multiplying, and still obtain the product with the same degree of accuracy as by the usual rule.

Our rule for contracting the work of multiplying decimals, is as follows:

RULE.

I. Multiply the multiplicand by the left-hand figure of the multiplier.

II Multiply the multiplicand, deprived of its righthand figure, by the second figure of the multiplier, counting from the left.

III. Multiply the multiplicand, deprived of its two

right-hand figures, by the third figure of the multiplier, counting from the left.

Continue this process until all the figures of the multiplier have been used. Observe to place the successive products so that their right-hand figures shall be directly under each other.

NOTE. In omitting successively the different figures on the right of the multiplicand, we must so far use them as to determine what there would be to carry into the next column.

The student may, perhaps, find some difficulty in fixing the decimal point in the right place. Whenever he is at a loss in this respect, he can multiply a few of the left-hand figures of each factor by the common method, by which means he will be enabled to determine the true place for the decimal point.

Or, which perhaps would be more simple, let the decimal point be fixed in the first partial product, which may be done by the usual rules for decimals.

EXAMPLES.

1. Multiply 0-37894 by 067452.

Operation.

0.37894

0.67452

0.227364

26526

1516

189

7

Ans. 0.255602

Explanation.

First. We multiply the multiplicand 0:37894 by 6,

the left-hand figure of the multiplier, which gives the first partial product, 227364