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33. 5 sq. yd. 2 sq. ft. — 2 sq. yd. 8 sq. ft. ?
34. 86

sq.
ft. 14

sq.
in. 775

sq.
ft. 121

sq.

in. ? 35. 97

sq.
ft. 111

sq.
in. 85

sq.
ft. 143

sq.

in. ? 36. 49 cu. yd. 14 cu. ft. 31 cu. yd. 23 cu. ft. ? 37. 146 cu. yd. 3 cu. ft. 29 cu. yd. 26 cu. ft. ? 38. 19 yd. 1 qr.

8 yd. 3 qr. ? 44 w. 2 da. 26 w. 5 da.? 39. 6 da. 4 h. 4 da. 6 h.? 22 h. 17 m. 17 h. 22 m. ? 40. 47 m. 23 sec. – 23 m. 47 sec. ? 5 319 - 2 3 2 ?

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The following combinations from the “First Steps in Numbers,” are inserted here for convenience of review. For practical examples applying them, and more full illustrations, see First Steps, Sections 10, 11, 12, and 13

A. 1. How many are 2 X 2? 3 X 2? 4 X 3? 4 X 2 ? 2. 2 x 4? 3 X 3? 5 X 2? 2 x 5? 6 x 2? 2 X 6? 3. 4 X 3? 3 X 4? 9 X 2? 2 X 7? 5 X 3? 3 x 5 ? 4. 4 X 4? 8 x 2? 2 X 8? 9 x 2? 2 x 9? 6 X 3? 5. 3 X 6? 10 X 2? 2 x 10 ? 4 X 5? 5 X 4? 6. 2 X 2 X 2 ? 2 X 3 X 2? 3 X 3 X 2? 7. 2 X 2 X 2 X 2? 2 X 2 X 4? 2 X 3 X 3? 8. 2 X 2 X 5? 2 X 2 X 3? 5 X 2 X 2? 4 X 2 X2?

B. 1. How many times 2 =6? = 18? 20 ? 4? 12? 16 ? 8? 14? 10 ? 2. How

many
times 3 =

12? 18? 6? 15? 9 ?
3. How many times 4 : 20? 4? 12? 16 ? 8?
4. How many times 5 10? 20 ? 5? 15 ?
5. How many times 6 = 18? 6? 12 ?
6. How many times 7 14? 7 ?
7. How many times 8 16? 8 ?
8. How many times 9 9?18?
9. How many times 10 20 ? 10 ?

S

C. 1. 6=3 times what number? 9 3 times what?
2. 12 = 2 times what? 20 = 5 times what?
3. 15 =5 times what?

14 2 times what?
4. 18 9 times what? 8 2 times what?
5. 12 4 times what? 18 = 6 times what?
6. 15 3 times what? 6 = 2 times what?
7. 20 = 10 times what? 14 = 7 times what ?
8. 20 = 2 times what? 18 3 times what?
9. 8 8 times what? 18 2 times what?
10. 16 : 4 times what? 20 = 4 times what?
11. 4. 2 times what? 8 = 4 times what?
12. 12 : 3 times what? 16 : 8 times what?
13. 7=7 times what? 12 6 times what?
14. 10 - 5 times what? 3 3 times what?
15. 10 2 times what?

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SECTION IX.

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Note. Many of the principles illustrated in this section are more fully illustrated in the First Steps. They are inserted here merely for convenience of review.

A. 1. Explanation. When any thing or number is divided into two equal parts, the parts are called halves of the thing or number. One of the parts is called one half, and two of the parts are called two halves of the thing or number.

2. Explanation. When any thing or number is divided into three equal parts, the parts are called thirds of the thing or number. One of the parts is called one third, two of the parts two thirds, and three of the parts three thirds of the thing or number.

3. Explanation. When any thing or number is divided into four equal parts, the parts are called fourths of the thing or number. One of the parts is called one fourth, two of the parts two fourths, three of the parts three fourths, &c.

4. Explanation. In like manner, when any thing or number is divided into five equal parts, the parts are called fifths of the thing or number; when divided into six equal parts, the parts are called sixths; when divided into seven equal parts, they are called sevenths; when into eight equal parts, they are called eighths, &c., &c.

5. What are fifths of a thing?

Ans. Fifths of a thing are parts obtained by dividing a thing into five equal parts.

6. What are ninths of a thing? fourths ? elevenths ? tenths ? halves ? sevenths ? twentieths ? eighths ? thirds ? fourteenths ? nineteenths ?

7. Explanation. Parts obtained by dividing a thing into equal parts, are called fractional parts of the thing divided, and the arithmetical expressions for them are called fractions. The figures, expressing the number of parts into which the thing is supposed to be divided, are called the denominator of the fraction, because they show the size or denomination of each part, while those expressing the number of parts we wish to consider, are called the numerator of the fraction.

8. Explanation. Fractions are usually expressed by writing the numerator over the denominator, and drawing a line between them.

Thus, five sixths is written f. 5 is the numerator, and 6 is the denominator.

9. How will the fraction one fourth be expressed by figures ?

10. How will 3 fifths be expressed ? 7 eighths ? 5 sixteenths ? 9 tenths ? 8 elevenths? 7 twenty-ninths? 14 sevteenths ? 63 eighty-fourths ? 21 thirty-sevenths?

11. What do you understand by (five eighths) of a thing?

Ans. 5 such parts as would be obtained by dividing a thing into eight equal parts.

12. What do you understand by & (five sixths) of a thing? by it (nine tenths)? by 1 (twenty-one thirty-thirds)? by ? 15? ? ? ? ? ? ? ? ? ? ? 464? 171? 3? ? 367? 831? t}? H? ? 19$? i?

13. What does the denominator and what does the numerator in each of the above fractions show ?

Ans. In the first, &, the denominator 6 shows the parts are such as would be obtained by dividing the thing considered into six equal parts, and the numerator 5 shows that five such parts are taken. In the second, , the denominator 10 shows, &c.

14. How many halves of a thing make the whole thing?

Ans. 2; because halves of a thing are obtained by dividing the thing into two equal parts.

15. How many thirds of a thing make the whole thing ? How many sixths ? ninths ? twelfths ? seventeenths ? thirtyeighths ? sixteenths ? two hundredths? forty-fifths ?

B. Let the pupil prove the truth of the following propositions :

Prop. 1. When the numerator of a fraction is less than its denominator, the value of the fraction is less than 1.

Prop. 2. When the numerator of a fraction is equal to its denominator, the fraction is equal to 1.

Prop. 3. When the numerator of a fraction is greater than its denominator, the value of the fraction is greater than 1.

1. A fraction representing a value less than 1, is called a proper fraction ; as, t, 1t:

2. A fraction representing a value equal to, or greater than, 1, is called an improper fraction; as, g, 14, 14. 3. How will the numerator of a proper fraction

compare

with its denominator ?

4. How will the numerator of an improper fraction compare with its denominator ?

5. Which of the fractions under number 12, letter A, are proper, and which improper ?

C. 1.

3.

8

of what? Ans. 3, or of 1= f of 3; because, if 3 equal things were divided into 5 equal parts, each part would be equivalent to 3 such parts as would be obtained by dividing 1 thing into 5 equal parts. 2. = 17 of what? 1=1t of what? = it of what? of what? of what? I'd

It of what ? 4.

ii of what? f of what ?

to of what? $ 4 of what? of what? 6. } of what ? t of what ? Ž $ of what?

to of what? 8. To what is of 4 equal ?

Ans. 4 of 4 is equal to 9 of 1. 9. To what is of 2 equal ? $ of 3 ? of 5? $ of 7? 10. 5 of 2? of 7? 1 of 3? of 5? of 5? of 6 ? 11. It of 7 ? 1 of 3 ? 1 of 11 ? I' of 9 ? Ty of 6 ? 12. Iz of 11 ? 1 of 6? t of 3? of 8?

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7. io

D. 1. 1 = how many more than 1?
Solution. 1 = - it, and 11 & more than 3. Therefore,

ii more than 11: 2. 1 : how

many more than f? than 7 ? than most?

1

than 6?

than ?

=

3. 1 = how many more than jg? than ? than ti?

4. 1 how many more than ? than t? than ido? than ?

5. 1 how many more than 100 ? than 19 ? than ? than ? 6. 1:

= how many more than 13? than ig? than f ? than 5?

7. 1 how many more than 34 ? than todo? than any ? than 3?

8. How many more than 1 are l? Solution. Since 41=g more than g, it must equal of more than 1.

9. How many more than 1 are g? ? 12? ?? ? 10. How many more than 1 are ií? 17 ? ? ? 13? 11. How many more than 1 are if? 1;? }? ? 17 ? 12. How many more than 1 are 1990 ? ? ?i!? 471? 13. How many more than 1 are ? îi? 14049 ? NOTE. A whole number and a fraction considered together form what is called a mixed number, as 14, 27, 4514, &c.

14. To what mixed number is each of the improper fractions under numbers 9, 10, 11, 12, and 13, equal ?

Ans. = 13, 1=1;, &c. 15. To what mixed number is equal ? ? ? ? ?

12 ?

16. To what mixed number is equal ? 18? 14? 17 ? it?

E. 1. What is the value of 2 + 11?
Solution. += tf = 1, Ans.
2. What is the value of } ++? + 10 + 8?
3. What is the value of 13 + 11? t + 14?
4. What is the value of 15 + 1x? + 2?
5. What is the value of 11+ ? 130 + ?
6. What is the value of 2 + $? f +$?
7. What is the value of 3 tof? 13 + 14k?

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