find the result to be, and that is equal to is evident from the fact, that the ratio of 5 to 15 is equal to the ratio of 1 to 3. And, as the value of a fraction depends on the ratio, which the numerator bears to the denominator, if their ratios are equal, the fractions are also equal. Q. e. d. Hence the following RULE. Divide the numerator and denominator by any number that will divide them both without a remainder; and so continue until no number will divide them but unity. Or, divide the numerator and denominator by the greatest common Ans. 567 9. Reduce it to its lowest terms. 10. What is the lowest expression of ? Ans. 13. III. To reduce mixed numbers to improper fractions. MENTAL OPERATIONS. 1. In 3 dollars how many halves? How many thirds? 2. In 7 apples how many tenths? How many twelfths? 3. In 83 dollars how many sevenths? 4. In 3 5. In 9 6. In 7 oranges how many fourths? gallons how many elevenths? quarts how many fifths of quarts? OPERATION. 7 5 35 3 38 5 We analyze this question by saying, as there are 5 fifths in one quart, there will be 5 times as many fifths as quarts; therefore, in seven quarts and three fifths, there will be 38 fifths, which should be expressed thus, 38. And this fraction, by definition. 2d, on page 76, is an improper fraction. Hence the following RULE. Multiply the whole number by the denominator of the fraction, and to the product add the numerator, and place their sum over the denominator of the fraction. 7. Reduce 8 to an improper fraction. 8. Reduce 15 to an improper fraction. 9. In 187 how many ninths? 10. In 161 Ans. Ans. 17. Ans. 169. Ans. 18848. how many one hundred and seventeenths? 117 11. Change 4311 to an improper fraction Ans. 5142 12. What improper fraction will express 27? 13. Change 111 to an improper fraction? Ans. 360. Ans. 12322. IV. To change improper fractions to integers or whole numbers. MENTAL OPERATIONS. 1. How many dollars in 4 halves? In 5 halves? In 6 halves? In 7 halves? In 12 halves? In 19 halves? 2. How many dollars in 5 quarters? In 9 quarters ? 3. How many dollars in 10 eighths? In 20 eighths? FOR THE SLATE. 4. How many dollars in 7 dollars? OPERATION. 16) 37 (21% 32 Ans. 21. This question may be analyzed by saying, as 16 sixteenths make one dollar, there will be as many dollars in 37 sixteenths as 37 contains 16, which is 2 times, $25. This answer is called a mixed number by definition 5th, page 77. Hence we see the propriety of the following 5 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. Change 178 to a mixed number. Ans. 10. 876 6. Change to a mixed number. 13. Change 125 to an improper fraction. Ans. 10111 Ans. 188. Ans. 1. Ans. 567. Ans. Ans. Ans. 125. V. To change or reduce compound fractions to simple fractions. MENTAL OPERATIONS. 1. What part of an orange is a 2. What part of an apple is a 3. What part of a bushel is a 4. What part of a quart is a 5. What is of ? OPERATION. Ans. one of these parts is of a half? FOR THE SLATE. Ans.. This question may be analyzed by saying, if of an apple be divided into 5 equal parts, that of an apple; and, if of be of will be 7 times as much. 7 of be, of will be 4 is 28. We therefore induce the following , it is evident, that times is; and, if times as much. 4 times RULE. Change mixed numbers and whole numbers, if there be any, to improper fractions; then multiply all the numera tors together for a new numerator, and all the denominators together for a new denominator; the fraction should then be reduced to its lowest terms. 6. What is of 4 of 4? OPERATION. 48 XtX == Ans. 105 8. What is of of 3 of 4? 9. Change of of Ans. 756 1928=176. of 2% of 7 to a simple fraction. Ans. 2. NOTE 1. If there be numbers in the numerators and denominators, that be alike, an equal number of the same value may be cancelled. 10. Reduce of 4 of 4 of 7 to a simple fraction. In performing this question, we perceive that there is a 4 and 5 and 7 among the numerators, and also the same numbers among the denominators; these we cancel before we commence the operation. 11. Required the value of of of 1 of 1 of 5%. = Ans. 5 X 9 X 11 X8X7 717 5 X 9 X 11 X 8 X 7 13. Reduce of off of of 44 to a simple fraction. Ans. . NOTE 2. When there are any two numbers, one in the numerators and the other in the denominators, 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. 14. Reduce 1 of 8 of 7 to a simple fraction. In performing this question, we find that the 15 among the numerators and the 9 among the denominators may be divided by 3, and that the quotients will be 5 and 3. We write the 5 above the 15, and the 3 below the 9. We also find an 8 among the numerators, and a 16 among the denominators, which may be divided by 8, and that the quotients will be 1 and 2. We write the 1 over the 8, and the 2 under the 16. We then multiply the 5, and 1, and 7 together for a new numerator, and the 2, and 3, and 11 together for a new denominator. That the result will be the same by this process as by the other, is evident from the fact, that the multiples of any number have the same ratio to each other, as the numbers themselves. This cancelling principle, when well understood, will often facilitate the operations of many questions, when the divisors and dividends have a common denominator. 15. Reduce of 3 of 11⁄2 of 9§ to a whole number. 16. Divide the continued product of 18, 24, 27, and 30, by the continued product of 20, 21, 9, and 10. 17. Divide the continued product of 20, 19, 18, 17, 16, 15, 14, 13, 12, and 11, by the continued product of 10, 9, 8, 7, 6, 5, 4, 3, 2, and 1. |