4. A bicycler rode 27 miles on Monday, 333 miles on Tuesday, 37 miles on Wednesday, and 423 miles on Thursday. How far did he ride in the four days?

5. A dry-goods merchant sold a lady 18 yards of flannel, 217 yards of silk, and as many yards of calico as of both the other goods. How many yards in all did he sell?

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

4. If you have $7 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 8 and 1.

Process.

Explanation.

Since and are like fractions, having a common denominator, 11, their difference is 8 elevenths

- 5 elevenths, or 3 elevenths.

2. What is the difference between 9 and 4?

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.

20. 561 - 293.

7. 365-273. 14. 884-5319. 21. 6511-301.

13. Answer the following inquiries :

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

PROBLEMS.

1. 31 yards, 45 yards, and 12 yards were cut off from a piece of silk containing 30 yards. How many yards remained?

2. A man spent for other expenses.

of his income for rent, 1 for food, and 1 What part of his income remained?

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 §. 5. If I pay my grocer $183, my coal dealer $274, and my tailor $223, how much will I have left out of four 20-dollar bills?

13

6. of a pole is in the mud, 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 1342-61% +3 111⁄2 +87 — 7 3 · 83 10180 425 is a correct equation.

=

501+1=?

4. 5267-357-511=?

5. 231623 + 10119 =?

6. 453 (32+10) = ?

7. LXXVII. — + CLXIX. — 11 =?

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. How much is 5 times §?

5. Multiply by 10.

by 3.

6. Have you been multiplying numerators or denominators?

7. Then what effect has multiplying the numerator?

8. What is 3 times? What are the lowest terms of §?

9. Then 3 times . How could you have obtained 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 by 4.

Process.

1 X 4 =

Explanation.

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, .

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