112. A common denominator is a denominator common to two or more fractions. A common denominator of two or more fractions having different denominators must be a common multiple of the given denominators. 113. The least common denominator of two or more fractions is the L. C. M. of their denominators. When the fractions are in their lowest terms, and the denominators are prime to each other, their continued product is the L.C.D. WRITTEN EXERCISES. 114. 1. Change, , and to equivalent fractions having a common denominator. EXPLANATION.-Since the denominators are prime to each other, and in their lowest terms, their product 30 is their L.C.M., and hence the L. C. D. of the fractions,, and . To find the numerators, take such part of the common denominator as the given fraction is of 1. Thus, of 30 = 15; of 30 20; and Change to fractions having a common denominator, 2. and §. 4. 4, §, and . 3. and . 5.,, and . 8. Change, and to equivalent least common denominator. EXPLANATION.-First find the L.C.M. of the given denominators, which is 42. This must be the least common denominator of the 2x3x5 = 30 1 of 38 = 18 3 of 38 = 18 # of 38 = it 7)3 of 30 = 24. Hence, etc. 6.,, and . fractions having the ку 3 1 2×3×7=42 14 given fractions. (113.) Reduce mixed numbers to improper fractions and fractions to their lowest terms, before finding the L. C. M. Change to fractions having the least common denominator: 12., 24, 3, and 1. 9.,, and 17. 10., 1, and. 11. ff, 7%, and §. 13. 61,, 7, and 14. 25 ADDITION AND SUBTRACTION. INDUCTIVE EXERCISES. } 115. 1. What is the sum of and? Of and ? ? and ? less? 5. A gentleman who owned a ɛail-boat sold of it. What part did he still own? 6. How many times 1 are ++3? $+8+8? 7. A boy paid $1 for a slate, $% for a reader, and $ for an arithmetic. How much did he pay for all? How are fractions having a common denominator added or subtracted? 8. What is the fractional unit of ? Of ? Can and in their present form be added? Why not? Can one be subtracted from the other? Why not? 9. What change must be made in and before their sum or their difference can be found? 10. Mary paid $2 for some ribbon, and $ for a pair of gloves. How much did she pay for both? ANALYSIS. She paid the sum of $2 and $5. is equal to, and 18 equal to 19; 1 and 19 are 12, or 12. Hence, she paid $1. 11. A man having of a ton of coal, bought of a ton more. How much had he then? 12. A boy having $4, gave $1 for a necktie. What had he left? ANALYSIS.-He had left the difference between $2 and $1. & is equal to, and equals less are. Hence, he had $ left. 12 13. A man owning of an acre of ground, sold † of an What part remained ? acre. How are fractions having different denominators added or subtracted? 14. A farmer sold 31 tons of hay to one man, and 5% to another. How much did he sell to both? ANALYSIS.—The sum of 31 tons and 53 tons. 5 and 3 are 8; and 1 and are ğ, which added to 8 makes 89 tons. 15. A man bought 5 cords of wood at one time, and 7 at another. How much did he buy in all? 16. From a piece of cloth containing 12 yards, 5 yards were cut. How many yards remained? ANALYSIS.-The difference between 12 yards and 5 yards. from leaves, and 5 from 12 leaves 7. Hence, 7 yards remained. 17. If a ton of coal costs $74, and a cord of wood $43, what is the difference in their cost? How are mixed numbers added or subtracted? Of, 1, and †? 18. What is the sum of 1, 3, and 1⁄2? 19. Find the sum of §,, and . 20. Subtract from ; from ; & from 3; & from §. 21. If a grocer buys tea at $ a pound, and sells it at $1, does he gain or lose, and how much? 22. How much greater than 2 is +1+11? 23. What is the difference between 4 and 2? 54 and 74? 24. Write on the slate or board, 1. Give sum of each fraction and the one at its right. 2. Give the difference. 3. Give the sum of each three fractions successively. 4. Give the difference between the sum of two fractions and the one at the right. PRINCIPLE.—The sum, or difference of two or more fractions can be found only when they have a common denominator, and when they express fractional units of the same kind. WRITTEN EXERCISES. 116. 1. From the sum of and take the difference between 1 and 2. 2. A man owning of a cotton-mill, sold of the whole to one man and to another. What part had he left? 3. From the sum of 14 and 9 take the difference between 25 and 16. 4. A clerk earned $75 and other expenses $27. Find the value of 5.4+2+8-14. 6.48+28-51. one month, and paid for board $283, 7. 18-51+31-1. 9. 11+11+51-255. 10. 281+365-(44—13}). Find the difference between 19. 1 and §. 13. A lady having $50, paid $12 for a bonnet, $17§ for a shawl, and $31% for a veil. How much money had she left? Find the sum 14. Of, 18, and . 15. Of,, and . 16. Of 42, 31, and 9. 20. 4 and 24. 22. 633 and 714. 23. 106 and 951. 24. Bought a quantity of coal for $136, and of lumber for $350. I sold the coal for $184, and the lumber for $4164. What was my whole gain? 25. A merchant sold 464 yards of cloth for $127, 64 yards for $2265, and 761 yards for $3123. How many yards of cloth did he sell, and how much did he receive for the whole? 26. A man having $253, paid $6 for coal, $2 for dry goods, and $2 for a pound of tea. How much had he left? 27. A man bought a ton of hay for $15%, a barrel of flour for $91, and a barrel of apples for $3. What change should be given to him for 3 ten-dollar bills? Complete the following equations: 28. 1+-+ = ? 29. 8+23-541 = ? 30. 48-(164-31) = ? MULTIPLICATION. 31. 413+56-243-414 = ? 32. 120-5189014-} = ? 33. 342-(21+1-9) = ? 11. In all examples of multiplication of fractions, either one or both of the factors will be fractional. INDUCTIVE EXERCISES. 1 118. 1. What part of a bushel is 3 times of a bushel? 2. What part of a cord is 5 times of a cord? 3. At a pound, what will 4 pounds of sugar cost? = 8 ANALYSIS.-4 pounds will cost 4 times $3, or $3×4 = $12 - $11; or $+$} = $14. (102, I.) = +4 4. At $ a bushel, what is the cost of 3 bushels of oats? 5. Of 5 bushels? Of 6 bushels? Of 8 bushels? 6. If a horse eats § of a bushel of oats in a day, how much will 2 horses eat? 4 horses? 6 horses? 8 horses? How many ways to multiply a fraction by an integer, and what are they? Show that multiplying the numerator of nominator by 4, multiplies the fraction by 4. by 4, or dividing the de(102, I.) 4 18 by 5; 7. Multiply by 7; 1 by 6; by 8; § by 9. A fraction is multiplied by a number equal to its denominator by cancelling the denominator. Thus, 7 × 8 = 7. Cancelling a factor of the denominator multiplies the fraction by that factor. Thus, 4 = §. 8. Multiply by 9; by 5; by 7; by 8, 1 by 12. 9 |