1. What is the unit of 2 fourths? Of 3 fourths? What is the sum of 2 fourths and 3 fourths?

How many are

and ?

2. How many sevenths are 2 sevenths and 3 sevenths? How many are and ? 3. What is the sum of and ? Of and and ğ? 4. What is the numerator of the sum of 6, 10, and? What is the denominator of the sum?

5. How do you add similar fractions, or fractions having a common denominator?

6. What is the fractional unit of ? Of ? Can and be united in one sum? Why?

7. How many twelfths in ? In ? Change and to 12ths. How many 12ths in their sum?

8. To what kind of fraction must and be changed before they can be added? What do they equal? Find their sum.

9. William paid $5 for a knife and $3 for a pair of skates. What did he pay for both?

172. Analysis. — Since William paid $3 for a knife and $3 for a pair of skates, he paid for both the sum of $5 and $3.

The least common multiple of the denominators is 24.

changed to 24ths is ; and 7 changed to 24ths is 24. Now $2 and $31 are $11, or $117. Hence, etc.

10. How do you add dissimilar fractions, or fractions having different denominators?

11. If a farmer sells of his wheat to one man, and { of it to another, what part of his wheat does he sell to both? 12. If John does of a piece of work one day, and of it the next, what part does he perform in both days?

13. A man having of a ton of coal, bought of a ton How much coal had he then?

more.

14. I used of a ton of coal one week, and of a ton the next. How much did I use in the two weeks?

15. A gardener has of an acre of raspberries, and of an acre of strawberries. How many acres of berries has he? 16. In a village lot are ğ of an acre, and in another are of an acre. How much land is in both lots?

17. A owns of a vessel, and B owns 3 of it. What part of the vessel do both of them own?

18. What fraction is that from which, if be taken, the remainder will be ?

19. William saved $ one week, $7 the next, and $ the next. How much did he save in the three weeks?

30. A farmer sold 23 tons of hay to one man, and 3 tons to another. How much did he sell to both?

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173. Analysis. Since the farmer sold 23 tons to one man, and 3 tons to another, he sold to both the sum of 23 tons and 3 tons. 2 tons and 3 tons are 5 tons. changed to 20ths is 15; and

changed to 20ths is 18. Now 18 and 18 are §}, or 117, which added to 5, the sum of the integers, gives 61. Hence, he sold 6 tons. 31. How do you find the sum of two or more mixed numbers? 32. A woodchopper put 3 cords of wood in one pile, and 6ğ cords in another. How much wood was in both?

33. A jeweler paid $243 for a silver watch, and sold it at a gain of $47. What did he get for it?

34. A boy sold 33 bushels of chestnuts one week, and 4,% bushels the next. How many did he sell in both weeks?

35. If 4 yards of cloth are needed for a coat, 23 yards for a pair of pants, and of a yard for a vest, how many yards are needed in all?

36. A boy had $4, earned $3g more, and his father gave him $1. How much had he then?

༔

37. What is the sum of of 18, ¦ of 20, and § of 27? 38. John has $10, Charles has $5%, and Henry has $37 more than both the others. How much have all?

39. If a mechanic's daily wages amounted to $23, and he made $6 by overwork, what did he receive per day?

40. An agent sells 2 acres for one building lot, 3 of an acre for a second, and for a third. How many acres does he sell?

174. Principles.

I. Only fractional units of the same kind of things can be added.

II. Only similar fractions can be added.

Exercises.

Find the sum of 7 and 1.

175. Model. The least common multiple of the denominators

of and is 40.

changed to 40ths is 45, and changed to 40ths

1. What is the unit of 7 eighths? Of 5 eighths? What is the remainder of 7 eighths less 5 eighths?

2. How many ninths are 6 ninths less 3 ninths?

3. What is the numerator of the remainder of What is the denominator of the remainder?

less? less ?

4. How do you subtract one fraction from another fraction having the same denominator?

5. What is the fractional unit of ? Of ? Can be taken directly from? Why?

6. How many 20ths in ? In ? Change and to 20ths. How many 20ths in their difference?

7. Change and to least similar fractions, and find their difference; and ; and 2.

8. From a pile of wood containing of a cord were taken of a cord. How much was left?

of a of a cord. changed

176. Analysis. Since of a cord were taken from cord, there was left the difference between 3 of a cord and The least common multiple of the denominators is 40. to 40ths equals, and changed to 40ths equals 1. cord less of a cord equals 3 of a cord. Hence, etc.

Now of a

9. How do you subtract dissimilar fractions, or fractions having different denominators?

10. Find the difference between 3 of a mile and of a mile. 11. What fraction increased by becomes ? Becomes ? 12. What fraction taken from leaves a remainder of §?

13. Find what fraction must be added to to make. 14. If the sum of two fractions is, and the less, what is the greater? Take the less from the greater.

15. If the sum of two fractions is 11, and the greater is §, what is the less? Take the less from the sum.

16. A grocer bought tea at $ğ a pound, and sold it at $16. How much did he gain or lose on it?

12

17. A man worked of a day, and his son worked ğ. How much longer did the man work than his son?

18. A merchant owned of a vessel, and sold % of it. What part of the vessel did he then own?

19. If James pays of the cost of a pair of skates, and Charles pays the balance, how much does Charles pay? How much less does Charles pay than James?

20. A boy having $3 spent $76. How much remained?

177. Analysis. — $3 equal $2 plus $1, or $2 plus $10, or $210Since he spent $16, he had remaining the difference between $210 and $1, which is $2. Hence, etc.

20

21. If 4 miles be diminished by of a mile, what will remain? 22. Find the remainder of 5 tons diminished by 2% of a ton. 23. What is the remainder of 6 yards less of a yard? 24. I owed my grocer $8%, and handed him a $10-bill. How much change did I receive?

25. A tailor bought 61 yards of cloth, and used 3 yards to make a coat. How much was left?

178. Analysis. — Since a tailor bought 61 yards, and used 33 yards, there remained the difference between 61 yards and 32 yards.