= 8. = .5; 0.05; 1000 = .005; 10000 Observe that there are as many decimal places in each decimal as there are naughts in the denominator of the equivalent common fraction. Find the product of the following decimals by multiplying the numerators and the denominators, separately, and expressing the result as a decimal; thus, The product of two decimals contains as many decimal places as the sum of the decimal places in both factors. Written Work 1. Multiply .25 by 5. (a) .25 5 1.25 5×5 hundredths =25 hundredths or 2 tenths and 5 hundredths. Write the 5 in hundredths' place, and carry the 2. 5 x 2 tenths = 10 tenths; 10 tenths+2 tenths 12 tenths, or 1 and 2 tenths. Hence, 5 x .25 = 1.25. 2. Multiply .26 by .12. (b) .26 .12 52 26 .0312 1. What is the sum of the decimal places in the two factors? 2. The product, then, must contain now many places? When the product has not enough decimal places, supply the deficiency by prefixing naughts. 54. How much must be paid for 85 acres of land at $45.75 per acre? The 55. Three brothers divided an estate worth $9600. first received .125 of it, the second .375 of it, and the third the remainder. How much did each receive? 56. A bricklayer furnished 2626 thousand bricks at $7.75 a thousand, and worked 65 days at $2.75 a day. What was the amount of his bill? 57. If there are 39.37 inches in a meter, how many inches are there in 12 meters? how many yards? Multiplying by moving the decimal point. 1. Multiply 6.385 by 10; by 100; by 1000. 10 × 6.385 63.85 = 100 × 6.385 = 638.5 1000 × 6.385 = 6385 1. How do you multiply a number by 10? 2. How may you multiply a decimal by 10? by 100? by 1000? 3. How is the value of a number affected by moving the decimal point one place to the right? two places? three places? Moving the decimal point one place to the right multiplies the number by 10; two places by 100; three places by 1000. 2. Multiply 6.1234 by 10; by 100; by 1000. 3. Multiply .0342 by 10; by 100; by 1000. 4. Multiply 1.3412 by 10; by 100; by 1000. DIVISION OF DECIMALS Dividing a decimal or a mixed decimal by an integer. 1. Find of 48 hundredths; of 64 hundredths. 2. Find of.25; of .35; .45; .75. 3. Find of 6 and 36 hundredths; 12 and 24 hundredths. 4. Find of 12.36; 24.42; 48.06; 54.06. Observe that in each problem a decimal or a mixed decimal when divided by an integer is simply separated or partitioned. When the divisor is an integer, division simply separates or partitions the dividend into equal parts. Thus, of 20 and 685 thousandths (20.685) = 1 and 379 thousandths (1.379). A decimal or a mixed decimal is divided by an integer by placing a decimal point above or below the decimal point in the dividend, before beginning to divide, and dividing as in the division of integers. Making the divisor an integer. 1. 6.251.25=5 2. 62.5 12.5 = 5 3. 625.125. = 5 Study of Problems 1. What is the first quotient? the second? the third? 2. What was done to the first problem to make the second? to the second to make the third? 3. How is a decimal affected by moving the decimal point one place to the right? two places? 4. How did moving the decimal point to the right the same number of places in both dividend and divisor of each problem affect the quotient? Multiplying both dividend and divisor by the same number does not change the quotient. Written Work Since multiplying both dividend and divisor by the same number does not change the value of the quotient, make the divisor an integer before beginning to divide. 1. 6.48 .4 64.84. = 4)64.8 16.2 1. Make the divisor an integer by moving the decimal point one place to the right in both dividend and divisor. 2. Show that this does not affect the quotient. 3. Solve, placing the decimal point directly below the point in the dividend, before beginning to divide. Divide as in integers, placing the decimal point directly above or below the decimal point in the dividend, before beginning to divide. HAM. SCH. ARITH. -6 |