8. Reduce 16 to an improper fraction. 9. Reduce 143 to an improper fraction. 10. Reduce 1261 to an improper fraction. 11. Reduce 14911 to an improper fraction. 12. Reduce 161 to an improper fraction. 13. Reduce 171 to an improper fraction. 14. Reduce 983 to an improper fraction. 15. Reduce 116 to an improper fraction. 16. Reduce 718 to an improper fraction. 17. Reduce 100109 to an improper fraction. 18. Reduce 478 to an improper fraction. 19. Reduce 871 to an improper fraction. 20. Reduce 16719 to an improper fraction. 21. Reduce 61310 to an improper fraction. 22. Reduce 159100 to an improper fraction. 23. Reduce 999,99 to an improper fraction. 116 CASE IV. To reduce improper fractions to integers or mixed numbers. 1. How many dollars in 2 half dollars? In 4? In 5? 2. How many dollars in 5 quarters? In 6? In 7? In 8? 3. How many dollars in 16 eighths? In 24? 4. Reduce to a mixed fraction. OPERATION. 19)117(6 Ans. 114 3 In 30? To apply this question, we may suppose a certain number of dollars to have been cut into 19 equal parts each, and we wish to know how many dollars 117 of these parts contain. To effect this, we must divide 117 by 19. Hence the following RULE. Divide the numerator by the denominator, and if there be a remainder, place it over the denominator at the right hand of the integer. 5. Reduce 167 to a mixed number. 15 6. Reduce 1631 to a mixed number. 116 7. Reduce 131 to a mixed number. 8. Reduce to a mixed number. Ans. 11. Ans. 14. Ans. 71. Ans. 3. CASE V. To reduce compound fractions to simple fractions. 1. What is of } ? OPERATION. This question may be analyzed by saying, x= Ans. If of an orange be divided into 4 equal parts, one of these parts is of the orange; and, if of be, it is evident, that of will be seven times 104 as much. And 7 times is If, therefore, of be, 2 of } will be 3 times as much; and 3 times is. We therefore induce the following RULE. Change mixed numbers and whole numbers, if there be any, to improper fractions; then multiply all the numerators together for a new numerator, and all the denominators together for a new denominator; the fraction should then be reduced to its lowest terms. But, if there be numbers in the numerator similar to those in the denominator, they may be cancelled in the operation. And, if there be any two numbers, one of which is a numerator and the other a denominator, which may be divided by a number without a remainder, the quotients, arising from such division, may be used in the operation of the question instead of the original numbers. To find the least common multiple of two or more numbers; that is, the least number, that may be divided by them without a remainder. 1. What is the least common multiple of 4, 6, 8, 16, 20? OPERATION. 4)4 6 8 16 20 2) 1 6 2 4 5 1 3 1 2 5 We examine these numbers, and find that 4 will divide more of them than any other. Having divided them and written their quotients and undivided numbers in a line beneath, we again examine them as before, and find that 2 will divide more of the remaining numbers and quotients than any other number. After this division, we perceive, that no number but a unit will divide the quotients. We then multiply all the divisors and quotients together, and find the product to be 240, which is the least common multiple, or the least number, that can be divided by 4, 6, 8, 16, and 20, without a remainder. Hence the following 4× 2 × 3 × 2 × 5=240 Ans. RULE. Divide by such a number as will divide most of the given numbers without a remainder, and set the several quotients, with the several undivided numbers in a line beneath, and so continue to divide, until no number greater than unity will divide two or more of them. Then multiply all the divisors, quotients, and undivided numbers together, and the product is the least common multiple 2 What is the least common multiple of 6, 8, 10, 18, 20 and 24? Ans. 360. 3. What is the least common multiple of 14, 19, 38 and Ans. 798. 57? 4. What is the least common multiple of 20, 36, 48 and 50? Ans. 3600. 5. What is the least common multiple of 15, 25, 35, 45 and 100? Ans. 6300. 6. What is the least common multiple of 100, 200, 300, 400 and 575? Ans. 27600. 7. I have four different measures; the first contains 4 quarts, the second 6 quarts, the third 10 quarts, and the fourth 12 quarts. How large is a vessel, that may be filled by each one of these, taken any number of times full? Ans. 60 quarts. CASE VII. To reduce fractions to a common denominator. 1. Reduce, and to other fractions of equal value, having the same, or a common denominator. OPERATION. First Method. Ans. 1,28,48 33 4)8 12 16 4×2×3×2=| 48 common denominator. Having found the common denominator, 48, by the last case, we divide it by the denominators 8, 12, and 16; and the quotients 6, 4, and 3, we multiply by the numerators 7, 5, and 11, and the products 42, 20, and 33 are numerators to be written over the common denominator; thus, 28, 33. OPERATION. Second Method. 7X12X16=1344 numerator for 7. 5X 8X16. 640 numerator for . 11X 8X12=1056 numerator for . 8X12X16=1536 common denominator. The numerators are produced by multiplying the numerators of the given fraction by each of the other denominators, and the common denominator is obtained by multiplying all the denominators. By this process, we obtain the following Ans. 1344 640 1536, 1536, 1536 1056 The pupil will perceive, that this method does not express the fractions in so low terms as the other; although they both have the same value. From the above illustrations, we deduce the following RULE. Find the least common multiple of all the denominators by Case VI., and it will be the denominator required. Divide the common multiple by each of the denominators, and multiply the quotients by the respective numerators of the fractions and their products will be the numerators required. Or, multiply each numerator into all the denominators except its own for a new numerator; and all the denominators into each other for a common denominator. 14, 21 2. Reduce, §, 1 and . 3. Reduce, f, 1 and 2. 4. Reduce, 14, 8 and 28. 5. Reduce, 1, 14 and 6. Reduce, }, ‡ and 2. 7. Reduce, 1, 2 and }. 8. Reduce, }, † and 7. 9. Reduce, Į, Į and 3 }. 10. Reduce, }, † and 4 1. 11. Reduce, 14, 11, and 1. 12. Reduce, 7, 17 and 4. 13. Reduce,, and . 14. Reduce, 1, 7 and 7. 15. Reduce, 7, 8, and 5). Let the following questions be performed by the second method. 16. Reduce, and to fractions having a common denominator. 96 105 Ans. 80 630 630 630. Ans. 546 572 616 1001 1001, 1001. 4836 624 624 624. Ans. 364 192 612 Ans. 1485 1020 2295, 2295, 2295. 540 26316 2295, 2295, 2295 • 336 Ans.,,, 42 2013 Ans. 1, 3, 396 Ans. 1,,, W. 36 816 |