MULTIPLICATION OF DECIMAL FRACTIONS. 149.-1. Multiply .37 by .8. We e may first write .37 100 1000 37 100' If, now, we multiply the fraction 37 by, we find the product to be 296 the number of ciphers in the denominator of this product is equal to the number of decimal places in the two factors, and the same will be true for any two factors whatever. 2. Multiply .3 by .02. Now, to express the 6 thousandths decimally, we have to prefix two ciphers to the 6, and this makes as many decimal places in the product as there are in both multiplicand and multiplier. Therefore, to multiply one decimal by another, Multiply as in simple numbers, and point off in the product, from the right hand, as many figures for decimals as are equal to the number of decimal places in the multiplicand and multiplier; and if there be not so many in the product, supply the deficiency by prefixing ciphers. QUEST.-149. After multiplying, how many decimal places will you point off in the product? When there are not so many in the product, what do you do? Give the rule for the multiplication of decimals. 4. Multiply one and one millionth by one thousandth. 13. Multiply two hundred and ninety-four millionths, by one millionth. 14. Multiply three hundred, and twenty-seven hundredths by 62. 15. Multiply 93.01401 by 10.03962. Ans. Ans. 16. What is the product of five-tenths by five-tenths? 17. What is the product of five-tenths by five thousandths? CONTRACTION IN MULTIPLICATION. 150. CONTRACTION in the multiplication of decimals is a short method of finding the product of two decimal numbers in such a manner, that it shall contain but a given number of decimal places. 1. Let it be required to find the product of 2.38645 multiplied by 38.2175, in such a manner that it shall contain but four decimal places. In this example it is proposed to take the multiplicand 2.38645, 38 times, then 2 tenths times, then 1 hundredth times, theń 7 thousandth times, then 5 ten-thousandth times, QUEST.-150. What is contraction in the multiplication of decimals? What is proposed in the example? How are the numbers written down for multiplication? and the sum of these several products will be the product sought. Write the unit figure of the multiplier directly under that place of the multiplicand which is to be retained in the product, and the remaining places of integer numbers, if any, to the right, and then write the decimal places to the left in their order, tenths, hundredths, &c. OPERATION. 2.38645 5712.83 715935 4773 239 167 12 91.2042 When the numbers are so written, the product of any figure in the multiplier by the figure of the multiplicand directly over it, will be of the same order of value as the last figure to be retained in the product. Therefore, the first figure of each product is always to be arranged directly under the last retained figure of the multiplicand. But it is the whole of the multiplicand which should be multiplied by each figure of the multiplier, and not a part of it only. Hence, to compensate for the part omitted, we begin with the figure to the right of the one directly over any multiplier, and carry one when the product is greater than 5 and less than 15, 2 when it falls between 15 and 25, 3 when it falls between 25 and 35, and so on for the higher numbers. For example, when we multiply by the 8, instead of saying 8 times 4 are 32, and writing down the 2, we say first, 8 times 5 are 40, and then carry 4 to the product 32, which gives 36. So, when we multiply by the last figure 5, we first say, 5 times 3 are 15, then 5 times 2 are 10 and 2 to carry make 12, which is written down. EXAMPLES. 1. Multiply 36.74637 by 127.0463, retaining three decimal places in the product. QUEST.-When the numbers are so written, what will be the order of value of the product of any figure of the multiplier by the figure directly over it? Where then is the first figure by each product to be written? How do you compensate for the part omitted? 2. Multiply 54.7494367 by 4.714753, reserving five places of decimals in the product. 3. Multiply 475.710564 by .3416494, retaining three decimal places in the product. 4. Multiply 3754.4078 by .734576, retaining five decimal places in the product. 5. Multiply 4745.679 by 751.4549, and reserve only whole numbers in the product. 151. NOTE.-When a decimal number is to be multiplied by 10, 100, 1000, &c., the multiplication may be made by removing the decimal point as many places to the right hand as there are ciphers in the multiplier; and if there be not so many figures on the right of the decimal point, supply the deficiency by annexing ciphers. QUEST.-151. How do you multiply a decimal number by 10, 100, 1000, &c.? If there are not as many decimal figures as there are ciphers in the multiplier, what do you do? DIVISION OF DECIMAL FRACTIONS. 152. Division of Decimal Fractions is similar to that of simple numbers. We have just seen that, if one decimal fraction be multiplied by another, the product will contain as many places of decimals as there were in both the factors. Now, if this product be divided by one of the factors, the quotient will be the other factor (Art. 79). Hence, in division, the dividend must contain just as many decimal places as the divisor and quotient together. The quotient, therefore, will contain as many places as the dividend, less the number in the divisor. Divide as in simple numbers, and point off in the quotient, from the right hand, so many places for decimals as the decimal places in the dividend exceed those in the divisor; and if there are not so many, supply the deficiency by prefixing ciphers. 2. Divide 4.6842 by 2.11. Ans. Ans. Ans. QUEST.-152. If one decimal fraction be multiplied by another, how many decimal places will there be in the product? How does the number of decimal places in the dividend compare with those in the divisor and quotient? How do you determine the number of decimal places in the quotient? If the divisor contains four places and the dividend six, how many in the quotient? If the divisor contains three places and the dividend five, how many in the quotient? Give the rule for the division of decimals. |