TO REDUCE A Whole or MIXED NUMBER TO AN Reduce 2 to an improper fraction. This is the reverse of the previous case. We have given the divisor 2, the quotient 2, and the remainder 1. The dividend 2× 2+1=5. Dividend 5 and divisor 2 are thus expressed, §. Reduce 2 to an improper fraction having 2 for its denominator. The divisor being 2, and the quotient 2, with no remainder, the dividend is 2 x 2 = 4. Or, since 1=2, 2=2x=.. Ans. . So, to divide by a fraction, we invert the fraction, taking the denominator for a numerator, the numerator for a denominator, and multiply. CANCELLATION. What is the product of 16, 39, 27, 11, and 18? Instead of performing the several operations as we go along, we will merely write the various numbers as factors, and perform the operations afterwards; thus, Since dividing both numerator and denominator by the same number does not alter the value of the fraction, we may strike out all factors that are common to both. 4 of the denominator goes in 16 of the numerator 4 times, so we cancel the factor 4 by drawing a line through the 4, another line through the 16, and writing 4 above the 16. 13 in 39, 3 times; 9 in 27, 3 times; in 36, 4 times. cancels the 4 left of the 16. The 4 which is left of the 36 The factors that remain are 3, 3, 11, and 13 in the numerator, and 7 in the denominator. Cancellation is a convenient method of 107 321 1284 REDUCING A FRACTION TO ITS LOWEST TERMS. 107 17148 1429 = We see by inspection that 4 is a factor of both terms, so we cancel the factor 4. 321 and 4287 are both divisible by 3. We cancel the factor 3. 107 cannot be divided by 2, 3, 4, 5, 6, 7, 8, 9, 10, or 11. 11 x 11 121, a number greater than 107. If 107 had a factor greater than 11, its corresponding factor would have to be less than 11. 107 has no factor less than 11. Hence, 107 is a prime number, and we have cancelled all the common factors, 107 not being a factor of 1429. If the factors of the numerator and the denominator cannot be readily found by inspection, find the greatest common divisor of the numerator and the denominator, and divide both by this number. 4287 1429 To divide one number by another is to find how many times the second is contained in the first. The quotient indicates the comparative value of the numbers. 3÷ 5 = §, and so 3 is of 5. To find what part one number is of another, we divide the number which is the part by the one of which it is the part. 3 times a certain number is 12. What is the number? We have given one factor and the product, and are required to find the other factor. This we do by division. 16 is of what number? Ans. 12÷ 3 = 4. The product being 16, and one factor, the other factor is of of 15 of a certain number is 18. What is the Decimals may be added and subtracted the same as whole numbers, merely observing the position of the decimal point. In a similar manner, we may multiply a decimal and a whole number, or divide a decimal by a whole number. We 24 515 thousandths. may continue the division to as many decimal places as we wish, thus : 256) 5172.14 (20.20367 512 521 512 940 768 1720 1536 1840 1792 48 Divide 12.03 by 42307. 42307) 12.0300 (.0002 84614 35686 We annex ciphers to the dividend until we get a number large enough. to contain the divisor at least once, obtain the first significant figure of the divisor, point off as many decimal places as there are in the dividend, and continue the division as far as desired. Multiply 32.142 by 1.23. 32.142 1.23 96426 64284 32142 39.53466 32.14232142. 1.23-188. 32142 and 123 being the numerators, we multiply them together, obtaining 3953466 for the numerator of the product. Thus, in multiplication, we shall always have to point off as many places in the product as there are decimal places in both of the factors. Divide 36.4728 by 12.23. The divisor being a decimal, we may change it into a whole number by moving the point the necessary number of places to the right. We must also move the point of the dividend the same number of places, thus multiplying both divisor and dividend by the same number. As has been shown under the head of fractions, this does not affect the quotient. 1223.) 3647.28 (2.98. 2446 We may put the point into the quotient when we come to it, and then continue the division to as many decimal places as is desired, or, instead of shifting the point in divisor and dividend, as has been done in this example, we may divide until the last digit of the dividend has been brought down and used, and then point off as many places in the quotient as there are decimal places in the dividend less those in the divisor. 12012 11007 10058 9784 Divide 6 by .03275. There being four decimal places less in the dividend than there are in the divisor, we annex four ciphers to the dividend, and then divide as though dividend and divisor were both whole numbers. |