9. Find the sum of 234 12, 23-412, 2.3412, 0-23412 Ans. 260.10732. 10. What is the sum of 111-111, 12-1212, 13-1313, 14.14142 Ans. 150-5049 SUBTRACTION OF DECIMAL FRACTIONS. 51. There is no difference between the subtraction of decimals and that of whole numbers, provided we are careful to keep the decimal points directly under each other, so that like orders may stand under each other. Hence this RULE. Place the less number under the greater, so that the decimal points shall be directly under each other; subtract, as in whole numbers. In the difference place the point under the points of the numbers above. How do you place the numbers in subtraction? Then how do you proceed? MULTIPLICATION OF DECIMAL FRACTIONS. 52. A tenth taken once, must give 1 tenth for a pro duct; if taken only one-tenth of a time, the product will be one-tenth of a tenth, or one hundredth; that is, x , or decimally expressed 0.1 x 0.1 0.01. This is evidently true, since if the tenth-part of any thing be divided into 10 equal parts, each subdivision will be a hundredth-part of the whole. So of Too, and so on. Multiply 0-136 by 0.78. If we supply the denominators of these decimal fractions, which denominators are always understood, we shall have Hence, multiplying by 7%, (ART. 45,) we find 136 1000 13878=136 X 73=10608=0·10608. 00 100000 From which we see that the number of decimal places in the product, always denoted by the number of zeros in the denominator, which is understood, is equal to the number of decimal places in both factors. Hence we have this RULE. Multiply as in whole numbers, and give as many decimal places in the product as there are in both the factors. When there are not as many places in the product, prefix ciphers. How do you multiply decimals? How many decimal places must there be in the product? When the whole number of figures in the product is not as great, how do you proceed? In this example, the multiplicand has 3 decimal places, and the multiplier has 2; therefore, by the rule, the product must have 5 places, and since the product consists of but 4 figures, we prefix one cipher before making the decimal point. 2. Multiply 0-561 by 0.786. Ans. 0.440946. Ans. 12-129324. Ans. 2221.7778. 53. A decimal number may be multiplied by 10, 100, 1000, &c., by removing the decimal point as many places to the right as there are ciphers in the multiplier; and if there are not so many figures, make up the deficiency by annexing ciphers. How may a decimal number be multiplied by 10, 100, 1000, &c.? When there are not as many decimal figures in the multiplicand as there are ciphers in the multiplier, how do you proceed? DIVISION OF DECIMAL FRACTIONS. 54. In multiplication of decimals, we know that the number of decimal places in the product is equal to the sum of those in both the factors. Now, since the product divided by one of the factors must produce the other fac tor or quotient, it follows, that in division the decimal places of the dividend must be equal to the number of places in the divisor and quotient taken together. Hence, the number of decimal places in the quotient must equal the excess of those in the dividend above those in the divisor. Divide 5.81224 by 5.432. Dividing 581224 by 5432 we find 107 for the quotient Since 5 figures of the dividend are decimals, and only 3 figures of the divisor are decimals, it follows that two figures of the quotient 107 must be decimals, so that 1-07 is the quotient sought. Hence the following RULE. Divide as in whole numbers; give as many decimal places in the quotient as those in the dividend exceed those in the divisor; if there are not as many, supply the deficiency by prefixing ciphers. How do you divide one decimal by another? How many decimal places must the quotient have? If the whole number of figures in the quotient is not as great as the number of decimals required, how do you proceed? In this example, the dividend contains 6 decimal places, and the divisor but 1; therefore, by the rule, the quotient ought to contain 5; but as there are but 4 figures in the quotient, we make up the deficiency by prefixing a cipher before making the decimal point. 2. Divide 3.810688 by 1.12. 3. Divide 0-109896 by 0.241. 4. Divide 1.12264556 by 1·0012. 5. Divide 0.01764144 by 0·0018. Ans. 3.4024. Ans. 0.456. Ans. 1.1213. Ans. 9.8008. 55. When there are not as many decimal places in the dividend as in the divisor, we may, (by ARt. 49,) annex as many ciphers to the dividend as we please, if we do not change the place of the decimal point. When the number of decimal places is the same in both dividend and divisor, the quotient will be a whole number. As for example, divided by 3, which is a whole number; that is, 0-6 divided by 0·2=3, a whole number. When there are not as many decimal places in the dividend as in the divisor, how do you proceed? When the number of decimal places in the dividend is the same as in the divisor, what will the quotient be? 8. Divide 0-7 by 0·07. 9. Divide 0-25 by 0·0005. 10. Divide 0.125 by 0·000005. Ans. 10. Ans. 500. Ans. 25000. |