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Algebra-Parentheses. 154. When an expression consisting of two or more terms is to be treated as a whole, it may be enclosed in a parenthesis. | 12 + (5 + 3) = ?

7 a + (3 a + 2 a) = ? 12 + 5 + 3 = ?

7 a + 3 a + 2 a = ? Observe that removing the parenthesis makes no change in the results. 12 - (5 + 3) = ?

5 7a – (3 a + 2 a) ? 12-5-3 = ?

77 a 3 a -- 2 a = ? Observe the change in signs made necessary by the removal of the parenthesis. 12 - (5-3) = ?

S 7a - (3 a 2 a) = ? 12- 5 + 3 = ?

3 a + 2 a = ? Observe the change in signs made necessary by the removal of the parenthesis.

A careful study and comparison of the foregoing problems will make the reasons for the following apparent:

I. If an expression within a parenthesis is preceded by the plus sign, the parenthesis may be removed without making any changes in the signs of the terms.

II. If an expression within a parenthesis is preceded by a minus sign, the parenthesis may be removed; but the sign of each term in the parenthesis must be changed; the sign + to -, and the sign to +.

7 a

155. PROBLEMS. (1) Remove the parenthesis, (2) change the signs if necessary, (3) combine the terms. 1. 15- (6 + 4) =

6. 15 b - (12 b - 46) = 2. 18+ (4 – 3)

7. 18 C + (9C-30) = 3. 27 - (8 + 3) =

8. 24 d - (5 d + 3 d) = 4. 45 + (12 - 3) =

9. 36 x - (5x + 4x) = 5. 75 – (18 + 27) 10. 45 y + (8 y + 7 y) =

Algebra-Parentheses. 156. Multiplying an expression enclosed in a parenthesis. 6(7 + 4) = ?*

6(7 a + 4b) = ?
6(a + b) = ? Ans. 6 a + 66.
a(b + c) = ? Ans, ab + ac.

Observe that in multiplying the sum of two numbers by a third number, the sum may be found and multiplied; or each number may be multiplied and the sum of the products found.

1. In the last three examples given above, let a = 5, 6 = 3, and c = 2; then perform again the operations indicated, and compare the results with those obtained when the letters were employed. 6(7 - ) = ?

6(7a - 46) = ?
6(a - b) = ? Ans. 6 a – 6 6.
a(b-c) = ? Ans. ab ac.

Observe that in multiplying the difference of two numbers by a third number, the difference may be found and multiplied; or each pumber may be multiplied and the difference of the products found.

2. In the last three problems given above, let a = 5, b = 3, and c = 2; then perform again the operations indicated and compare the results with those obtained when the letters were employed.

157. PROBLEMS. If a = 5, b = 3, and c = 2, find the value of the following:

1. 3(a + b)2(b + c).
2. 4(a +26)-3(b-C).
3. 2(a - b) + 2(2 b - c).

* This means, that the sum of 7 and 4 is to be multiplied by six; or that the sum of six 7's and six 4's is to be found.

+ This means, that the difference of 7 and 4 is to be multiplied by six; or that the difference of six 7's and six 4's is to be found.

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1. The sum of the angles of any triangle is equal to right angles or degrees. *

2. In a right triangle there is one right angle. The other two angles are together equal to

3. In a right triangle one of the angles is an angle of 40°. How many degrees in each of the other two angles ? Draw such a triangle.

4. Convince yourself by drawings and measurements that every equilateral triangle is equiangular.

EQUILATERAL

EQUIANGULAR

TRIANGLES.

TRIANGLES.

5. Note that in every equiangular triangle each angle is one third of 2 right angles. So each angle is an angle of

degrees. 6. If any one of the angles of a triangle is greater or less than 60, can the triangle be equiangular? Can it be equilateral ?

7. If angle a of an isosceles triangle measures 50°, how many degrees in angle b? in angle c?

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* See p. 59.

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159. MISCELLANEOUS REVIEW. 1. I am thinking of a right triangle one of whose angles measures 32°. Give the measurements of the other two angles. Draw such a triangle.

2. I am thinking of an isosceles triangle; the sum of its two equal angles is 100°. Give the measurement of its third angle. Draw such a triangle.

3. Let a equal the number of degrees in one angle of a triangle and b equal the number of degrees in another angle of the same triangle; then the number of degrees in the third angle is 180° – (a + b). If a equals 30, and b equals 45, how many degrees in the third angle? Draw such a triangle.

4. Name four multiples of 16.
5. Name three common multiples of 16 and 12.
6. Name the least common multiple of 16 and 12.

7. Find the sum of all the prime numbers from 101 to 127 inclusive.

8. Find the prime factors of 836.

9. With the prime factors of 836 in mind or represented on the blackboard, tell the following:

(a) How many times is 19 contained in 836 ? (6) How many times is 209, (11 x 19), contained in 836? (©) How many times is 418,(19x11x2), contained in 836?

160. PROBLEMS.

Find the 1. c. m.

1. Of 18 and 20. 6. Of 36, 72, and 24.
2. Of 13 and 11. 7. Of 45, 81, and 27.
3. Of 24 and 32. 8. Of 33, 55, and 88.
4. Of 16 and 38. 9. Of 45, 65, and 85.

5. Of 46 and 86. 10. Of 3, 5, 7, and 11. (a) Find the sum of the ten results.

DIVISIBILITY OF NUMBERS.

161. Numbers exactly divisible by 2; by 2}; by 3}; by 5; by 10.

1. An integral number is exactly divisible by 2 if the right-hand figure is 0, or if the number expressed by its right-hand figure is exactly divisible by 2.

EXPLANATORY NOTE. — Every integral number that may be expressed by two or more figures may be regarded as made up of a certain number of tens and a certain number (0 to 9) primary units, thus: 485 is made up of 48 tens and 5 units ; 4260 is made up of 426 tens and 0 units ; 27562 is made up of 2756 tens and 2 units. But ten is exactly divisible by 2 ; so any number of tens, or any number of tens plus any number of twos, is exactly divisible by 2.

2. Tell which of the following are exactly divisible by 2, and why: 387, 5846, 2750, 2834.

3. Any number, integral or mixed, is exactly divisible by 27, if the part of the number expressed by figures to the right of the tens' figure is exactly divisible by 21.

4. Show why the statement made in No. 3 is correct, employing the thought process given in the “Explanatory Note” above.

5. Tell which of the following are exactly divisible by 21, and why: 485, 470, 365, 4721, 38471.

6. Any number, integral or mixed, is exactly divisible by 33, if the part of the number expressed by figures to the right of the tens' figure is exactly divisible by 3}. Show why.

7. Tell which of the following are exactly divisible by 3}, and why: 780, 283}, 5763, 742, 80.

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