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Geometry.

184. QUADRILATERALS. REVIEW. 1. All the geometrical figures on

Square.

b this page are quadrilaterals; that is, each has four sides.

1

a

[blocks in formation]

185. MISCELLANEOUS REVIEW.

1. The difference of two numbers is 37416; the smaller number is 24337. What is the larger number? *

2. The difference of two numbers is a; the smaller number is b. What is the larger number?

3. James had a certain number of dollars and John had three times as many; together they had 196 dollars. How many had each? (x + 3 x = 196)

4. William had a certain number of marbles; Henry had twice as many as William, and George had twice as many as Henry; together they had 161. How many had each? (x + 2x +4x = 161) †

5. Divide 140 dollars between two men, giving to one man 30 dollars more than to the other. (x + x + 30 = 140) +

6. By what integral numbers is 30, (2 x 3 x 5), exactly divisible besides itself and 1?

7. By what is abc, (a x b X c), exactly divisible besides itself and 1 ?

(1) How many times is a contained in abc ?
(2) How many times is b contained in abc ?
(3) How many times is c contained in abc ?
(4) How many times is ab contained in abc ?
(5) How many times is ac contained in abc ?

(6) How many times is bc contained in abc ? Observe that a number composed of three different prime factors has exact integral divisors.

8. Change to 60ths. Is more or less than 37? 9. Change to 100ths. Change to 100ths. 10. Change 1 to 100ths. Change 5 to 100ths. (P. 334.)

* The difference of two numbers is 5; the smaller number is 4. What is the larger number?

+ See page 78.

FRACTIONS.

Art. 179, continued from page 86. Find the sum of 1. and 2%

6. 18, , and 17 2. 15 and 15

7. 1}, }, and 1 3. 11 and is

8. 11, 4, and 23 4. 15 and 1

9. 15, s, and 12 5. 1 and 1 10. 14, }, and

11 (a) Find the sum of the ten sums.

186. TO SUBTRACT COMMON FRACTIONS. RULE. Reduce the fractions if necessary to equivalent fractions having a common denominator, find the difference of their numerators, and write it over the common denominator.

EXAMPLE

From 2 subtract 375.
(1) The l. c. m. of 25 and 35 is 175.
(2) 3} = 17 35= 11

(3) 17-184%
Compare the following:
77 175ths 35 175ths

= 42 175ths. 77 apples – 35 apples = 42 apples. Find the difference of — 1. and is 3. 2% and }

5. - 1 2. and 18 4. and

6. $- $ (b) Find the sum of the six differences.

Fractions. 187. To subtract one mixed number from another when the fraction in the subtrahend is greater than the fraction in the minuend.

EXAMPLE

From 583= take 32%.
Operation.

Explanation.
583 = 59.24 14 is greater than 24, therefore we take 1 unit
323 = 3214

from the 8 units, change it to 24ths, and add it to

the 9 24ths. Difference 2517 31+ 14 = {. 11 - 14 = }}.

2 units from 7, (8 – 1), units j units. 3 tens from 5 tens 2 tens.

I. Find the difference of 1. 244 and 163

6. 354 and 261 2. 293 and 151

7. 283 and 143 3. 4617 and 1828

8. 36,23 and 81% 4. 525 and 313

9. 652 and 224 5. 473 and 18

10. 347 and 271 a) Find the sum of the ten differences.

II. Reduce to simplest form

1. 54 +33 – 51
2. 62 – 33 +4}
3. 21-11+3}
4. 74 + 33 – 13
5. 61 – 31 +5}
6. 34+41 - 3} – 23 +14
7. 63 – 21-14-140 + 21

8. 51 +43 + 2 +33 +31.
(b) Find the sum of the eight results.

Fractions.
188. To multiply a fraction by an integer.

Multiply 24 by 6.
Operation No. 1.

Operation No. 2. 6 times za are 14 = 14. 6 times 24 = 1 = 14. 1. Observe that by the first operation we obtain 14 ; that in 24 there are 6 times as many parts as there are in 24 and that the parts are of the same size as those in 24

2. Observe that by the second operation we obtain ]; that in there are the same number of parts as there are in z1, and that the parts are 6 times as great as those in 1.

Note 1.– The 7 of }may be regarded as a dividend; the 24, as a divisor, and 3 itself, as a quotient. In ji, we have a dividend 6 times as great as that in }, the divisor remaining unchanged. In ], we have a divisor 1 sixth as great as that in

, the dividend remaining unchanged. Multiplying the dividend or dividing the divisor by any number, multiplies the quotient by the same number.

Note 2. — The 7 of dı may be regarded as the antecedent of a couplet; the 24, as the consequent, and t itself as the ratio. Multiplying the antecedent or dividing the consequent

of

any couplet multiplies the ratio by the same number. RULE. — To multiply a fraction by an integer, multiply its numerator or divide its denominator by the integer.

I. Find the product.

1. P x 4 5. 4x8 9. 12 x 4
2. x 6 6. § x 9 10. 15 x 6
3. 20 x 5 7. *x 8 11. x 5
4. 37 x 7 8. x9 12. já x 7

(a) Find the sum of the twelve products. II. Find the product.

1. 31 x 7 5. 4 x 5 9. 6.3 x 5
2. 53 x 6 6. 17 x 4 10. 213 x 4
3. 777 X 4 7. 53 x 5 11. 45 x 6
4. 3.7 x 5 8. 8} x 4 12. 61 x 7

(b) Find the sum of the twelve products.

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