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4. A bicycler rode 27 miles on Monday, 33} miles on Tuesday, 37* miles on Wednesday, and 427 miles on Thursday. How far did he ride in the four days ?

5. A dry-goods merchant sold a lady 184 yards of flannel, 21% yards of silk, and as many yards of calico as of both the other goods. How many yards in all did he sell ?

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SUBTRACTION OF FRACTIONS.

INDUCTIVE STEPS.

2. 3

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1. From 5 subtract .

- what? 3. If you have $3 (of a dollar) and spend $, how much have you left ?

4. If you have $} and spend $1, how do you find the remainder?

5. What kind of fractions can be subtracted without reduction.

6. What kind require reduction ? Reduction to what? 7. Give four brief directions for such reduction. 8. What introductory step is sometimes necessary ?

PRINCIPLES.

1. Only like fractions can be subtracted.

2. Unlike fractions can be reduced to like fractions and then subtracted.

EXERCISES.

1. Find the difference between and :

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Explanation. Since and are like fractions, having a common denominator, 11, their difference is 8 elevenths 5 elevenths, or 3 elevenths.

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2. What is the difference between 1 and ?
Process.

Explanation.
Since it and are unlike frac-

tions, we reduce them to eightieths, * -*= 46 — 3= 13 용

making them like fractions. 16 - }

= 45 – 3%= %.
3. Subtract 75 from 11g.
Process.

Explanation.
We subtract integers and fractions sepia-

rately. cannot be taken from ; but 1, 75 75

taken from 11, equals ; + 3 1; } 39 33

10 - for -7 = 3. Uniting the two results, we have 33, the remainder.

11ģ

=

1014

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RULE.

1. Reduce unlike fractions to a common denominator.

2. Write the difference of the numerators over the common denominator.

3. Subtract integers and fractions separately, and unite the results.

4. From take . From take }.
5. From take t. From take .
6. From } take it. From 1 take fr.
7. From ji take 15. From 24 take 372.
8. From take 15. From 17 take 14:
9. From 1034 take }} From 112 take 754.

.
10. From 606take 704. From 5063 take 4181.
11. What is the value of :
1. 5 - 35

7. 1198

1493
-
360

8. 5893 – 674

3. 3 - 175 .

9. 754 – 6341 . - 1

10. 721 5814
- - 300.

11. 4912 361
6. 3% - TBT
262 T:

12. 42

100

39 85

58

85 2. 237

900

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4. 13

5. 77

336
97

29

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25

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12. Find the value of : 1. 51 — 204

8. 48

222 15. 76-31 - 6876 2. 66 — 364 9. 358 - 294.

- 16. 2445 - 9 3. 61 - 593.

10. 443

- 273. 17. 4311 — 2838. 4. 38 - 374. 11. 483 -9. 18. 1011 - 938. 5. 59 — 3214. 12. 731 — 27. 19. 2834 – 1635. 6. 111 — 3114. 13. 22} --- 7. 20. 561 – 294 . 7. 36% -- 27. 14. 881 - 53:14. 21. 651 - 304.

} 13. Answer the following inquiries :

1. 5 + - 12 = ?

ģ $
2. $ + 19 - 1} = ?
3. 21% – 226 + 34 = ?
4. 53 - 416 + 34 = ?
5. 1311 + 193 -- 264 ?
6. 95 — 311 + 496 = ?
7. 4036 1156 + 3 = ?
8. 37 - 23% + 24 = ?
9. 34 + 914 + 6 ?
10. 36 — 24 – 4 – 61 ?
11. 20 – 81 - 610 - 5
-

?
12. 200 — 304

304 — 1712
1712 – 2018 = ?

:
13. 39 Too – 18 – H = ?

=
14. 53 – 45 2 – 9= ?

-
15. } + 1 + 1 - 4-1 - To

NotE.—The teacher will suggest the shortest method of answering the above inquiries.

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49

19

17

200 27

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20

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To = ?

PROBLEMS. 1. 31 yards, 45 yards, and 124 yards were cut off from a piece of silk containing 301 yards.

301 yards. How many yards remained ? 2. A man spent of his income for rent, 1 for food, and }

What part of his income remained ?

8

for other expenses.

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3. A farmer sold of his corn to one man, & to another, and had 50 bushels remaining. How much corn had he at first?

4. Show that the fraction is greater than and less than 4.

5. If I pay my grocer $18, my coal dealer $274, and my tailor $221, how much will I have left out of four 20-dollar bills?

6. ^ of a pole is in the mud, Í of it is in the water, and the rest of it is in the air. What part of it is in the air ? 7. Show that 13 2015 — 616 + 3 — 18 + 8} — 43

28 – 1013 442 is a correct equation. 8. Find the second members of these : 1. 450 + (12 X 5) - 8646

501 +4=?
2. 81 + 3} — 6=?
3. 59 (274 - 4) = ?
4. 524 + 67 — 3530 – 51 =?
5. 231 — 623 + 10114 – } = ?
6. 453 (323 + - 10) = ?
7. LXXVII. – 17 + CLXIX. — 11:1 =?

+

12

5

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MULTIPLICATION OF FRACTIONS.

INDUCTIVE STEPS.

1. How much is 2 times 3 dollars ?
2. How much is 2 times 3 sevenths ?
3. How much is 2 times 4 ?
4. How much is 5 times ?
5. Multiply i by 10. is by 3.

6. Have you been multiplying numerators or denominators ?

7. Then what effect has multiplying the numerator? 8. What is 3 times f? What are the lowest terms of f?

1

9. Then 3 times ] $How could you have obtained f more directly than by multiplying the numerator?

10. What effect, then, has dividing the denominator?

11. Is that effect in agreement with principle 4, page 68 ? Why?

PRINCIPLE. Multiplying the numerator or dividing the denominator multiplies the fraction.

EXERCISES. 1. Multiply to by 4. Process.

Explanation. To X 4 =

1

According to the principle we may multiply the

numerator or divide the denominator. Since the denominator, 16, is divisible by 4, we divide and obtain the result, 7.

2. Multiply 4 by 17.

Process. 4 X 1 = 2 = 111

4 X 77

28

Explanation. Since the denominator, 17, is not exactly divisible by 4, we multiply the numerator by 4 and obtain the result, íj = 117.

RULE.

To find the product of an integer and a fraction divide the denominator or multiply the numerator by the integer.

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