53 5 × 12 (1.) = 69 = 32' according to (300). (2.) 53÷2=23÷, and 23 ÷ } = {} ÷ = 4, the same result as obtained by the method of multiplying by the least common multiple of the denominators of the partial fractions. EXAMPLES FOR PRACTICE. 303. Reduce to simple fractions, and explain as above: When the numerator or denominator contains two or more terms connected by a sign, perform the operation indicated by the sign first, then reduce to a simple fraction. Reduce the following to simple fractions: 304. PROB. II.-To reduce a fraction to any given denominator. 1. Examples where the denominator of the required fraction is a factor of the denominator of the given fraction. Reduce to a fraction whose denominator is 8. 17 17÷3 = 24 24 ÷ 3 8 EXPLANATION.-We observe that 8, the denominator of the required fraction, is a factor of 24, the denominator of the given fraction. Hence, dividing both terms of by 3, the other factor of 24, the fraction is reduced (235-III) to a fraction whose denominator is 8. 2. Examples where the denominator of the required fraction is not a factor of the denominator of the given fraction. Reduce to a fraction whose denominator is 10. 2. The denominator 130 now contains the factors 13 and 10. Hence, dividing both terms of the fraction by 13 (235-III), as shown in (2), 80 130 a fraction whose denominator is 10. From these examples we obtain the following: 305. RULE.-Multiply both terms of the fraction by the given denominator, and then divide them by the denominator of the fraction. Observe that when the given denominator is a factor or multiple of the denominator of the fraction, it is not necessary to multiply by it, as will be seen in the first example above. 306. 1. Reduce &,, 13, 14, 3, and 48 each to thirds. 2. How many sevenths in ? In ? In ? 3. Reduce,,,,, and each to sevenths. 4. In 4 how many twentieths? 5. Reduce,,, 4, and How many ninths? to hundredths. 6. How many tenths in ? In 3? In? In }} ? 7. Express as hundredths 4, 8, 7, 13, and 123. 8. How many thousandths in ? In 18? In? In ? 9. Reduce to hundredths ; 10. How many hundredths in 1? In 17? In ? In? In 2? In 44? In 7? In 94? REVIEW EXAMPLES. 30%. 1. How many thirtieths in , and why? In ? 3 2. Reduce, f, and each to twenty-eighths. 5. Reduce to a fraction whose numerator is 12; is 20; is 2; is 3; is 7 (235-III). 6. Reduce to a common numerator and 4 (252). 8. Find the value of (4 of † — 11) ÷ (†1 + 2 31 9. If of an estate is worth $3460, what is 4 of it worth? 10. $4 is what part of $8? Of $12? Of $32? Of $48? Write the solution of this example, with reason for each step. 11. If a man can do a piece of work in 150 days, what part of it can he do in 5 days? In 15 days? In 25 days? In 71 days? In 3 days? In 12 days? 12. A's farm contains 120 acres and B's 280; what part of B's farm is A's? 13. 42 is § of what number? Ans. . Write the solution of this example, with reason for each step. 14. $897 is of how many dollars? 15. of 76 tons of coal is of how many tons? 16. A piece of cloth containing 73 yards is of another piece. How many yards in the latter? 17. Bought a horse for $286, and sold him for of what he cost; how much did I lose? 18. 84 is of 8 times what number? Write the solution of this example, with reason for each step. 19. A has $694 in a bank, which is of 3 times the amount B has in the same bank; what is B's money? 20. Two men are 863 miles apart; when they meet, one has traveled 8 miles more than the other; how far has each traveled? 21. If of a farm is valued at $4732§, what is the value of the whole farm? 22. The less of two numbers is 4325, and their difference 123 Find the greater number. 23. A man owning of a saw-mill, sold of his share for $2800; what was the value of the mill? 24. What number diminished by and of itself leaves a remainder of 32 ? 25. I put of my money in the bank and gave had left to a friend, and had still remaining $400. had I at first? Ans. 504. of what I How much Ans. $1800. 26. Sold 342 bushels of wheat at $15 a bushel, and expended the amount received in buying wood at $4 a cord. How many cords of wood did I purchase? 27. If of 4 pounds of tea cost $21, tea can be bought for $74? For $128? 28. If 5 be added to both terms of the will its value be changed, and why? Ans. 123 cords. how many pounds of For $? fraction 4, how much 29. I exchanged 47% bushels of corn, at $ per bushel, for 24 bushels of wheat; how much did the wheat cost a bushel? 30. A can do a piece of work in 5 days, B can do the same work in 7 days; in what time can both together do it? 31. Bought of 844 acres of land for of $35847; what was the price per acre? 32. A boy while fishing lost of his line; he then added 8 feet, which was of what he lost; what was the length of the line at first? Ans. 15 feet. 33. A merchant bought a quantity of cloth for $28494, and sold it for of what it cost him, thereby losing $4 a yard. How many yards did he purchase, and at what price per yard? 34. A tailor having 276 yards of cloth, sold of it at one time and at another; what is the value of the remainder at $3 a yard? 35. A man sold of his farm at one time, and the remainder for $180 at $45 an acre; were there in the farm? at another, how many acres 36. A merchant owning of a ship, sells of his share to B, and of the remainder to C for $6004; what is the value of the ship? REVIEW AND TEST QUESTIONS. 308. 1. Define Fractional Unit, Numerator, Denominator, Improper Fraction, Reduction, Lowest Terms, Simple Fraction, Common Denominator, and Complex Fraction. 2. What is meant by the unit of a fraction? Illustrate by an example. 3. When may be greater than ? than ? 4. State the three principles of Reduction of Fractions, and illustrate each by lines. 5. Illustrate with lines or objects each of the following propositions: I. To diminish the numerator, the denominator remaining the same, diminishes the value of the fraction. II. To increase the denominator, the numerator remain ing the same, diminishes the value of the fraction. III. To increase the numerator, the denominator remaining the same, increases the value of the fraction. IV. To diminish the denominator, the numerator remaining the same, increases the value of the fraction. 6. What is meant by the Least Common Denominator? 7. When the denominators of the given fractions are prime to each other, how is the Least Common Denominator found, and why? |