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Division—Denominate Numbers.

138. Divide 46 rd. 12 ft. 8 in. by 4.

Operation. Explanation.

4)46 rd. 12 ft. 8 in. This means, find 1 fourth of 46 rd. 12 ft.

11 rd. lift. 5 in. "„ ,,,„,..,,

One fourth of 46 rd. is 11 rd. with a remainder of 2 rd.; 2 rd. equal 33 ft.; 33 ft. plus 12 ft. equal 45 ft.

One fourth of 45 ft. equals 11 ft. with a remainder of 1 ft.; 1 ft. equals 12 in.; 12 in. plus 8 in. equals 20 in. One fourth of 20 in. equals 5 in. One fourth of 46 rd. 12 ft. 8 in. equals 11 rd. 11 ft. 5 in.

PROBLEM.

The perimeter of a square garden is 46 rd. 12 ft. 8 in. How far across one side of it?

139. Miscellaneous.

Tell the meaning of each of the following, solve, explain, and state in the form of a problem the conditions that would give rise to each number process:

1. Multiply 64 rd. 14 ft. 6 in. by 8.

2. Divide 37 rd. 15 ft. 4 in. by 5.

3. Divide $675.36 by $48.

4. Divide $675.36 by 48.

5. Divide $675.36 by .48.

6. Divide $675.36 by $4.8.

7. Divide $675.36 by 4.8.

8. Divide $675.36 by $.48.
9". Multiply $356.54 by .36.

10. Multiply $356.54 by 3.6.

11. Multiply $356.54 by 36.

12. Can you multiply by a number of dollars?

13. Can you divide by a number of dollars?

Algebraic Division.

140. Examples.
No. 1. No. 2.

4)12 + 5x4-8 a)12a + 2a6 + 4a

3 + 5-2 12 + 26 + 4

No. 3. No. 4.

2)2" + 3 x 22 - 2 6)a63 + c62 + 36

22 + 3 x 2 - 1 a&2 + c6 + 3

1. Prove Nos. 1 and 3, by (1) reducing each dividend to its simplest form, (2) dividing it so reduced, by the divisor, and (3) comparing the result with the quotient reduced to its simplest form.

2. Verify No. 2 by letting a = 3, and 6 = 5.

3. Verify No. 4 by letting a = 3, b = 5, and c = 7.

141. (6xaxaxaxaxa)-*-(2xaxa) = 6as-4- 2a2= 3a8.

Observe that to divide one algebraic term by another we must find the quotient of the coefficients and the difference of the exponents.

142. Problems.

1. 6a'b + 2a= 3. 8a3Z>3 -*- 2a =

2. 4a'6' + 2a = 4. 10a2&4 H- 2a =

5.
2a)6a'6 + 4a462 - 8a363 + 10a2&'

6. Verify problem 5 by letting a = 3 and 6 = 5.

Algebraic Division.
143. Problems.

1. Divide 4a3x + 8aV + 6ax3 by 2ax.

2. Multiply the quotient of problem 1 by 2ax.

3. Verify problems 1 and 2 by letting a = 2 and x = 3.

4. Divide Sab3 + 6a'&2 + 9a3b by 3a6.

5. Multiply the quotient of problem 4 by Sab.

6. Verify problems 4 and 5 by letting a = 3 and & = 5.

7. Divide 2#3y + x2y' xy3 by a;y.

8. Multiply the quotient of problem 7 by xy.

9. Verify problems 7 and 8 by letting x = 2 and y = 3.

10. Divide 5a3y2 — 2a'y3 + a'y* by a2y.

11. Multiply the quotient of problem 10 by a:y.

12. Verify problems 10 and 11 by letting a 1 and y = 2.

13. Divide 3&4a ; + &V - 36V by bx.

14. Multiply the quotient of problem 13 by bx.

15. Verify problems 13 and 14 by letting 6 = 3 and x = 4.

Observe that when the divisor is a positive number, each term of the quotient has the same sign as the term in the dividend from which it is derived.

2)8 - 6 One half of + 8 is + 4; one half of - 6 is - 3.

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1. A triangle has sides and . angles.

2. A right triangle has one right angle; that is, one angle of . . degrees.

3. An isosceles triangle has two angles that are equal and two sides that are equal.

4. An equilateral triangle has equal sides and equal angles.

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. 5. Cut from paper a triangle similar to the one shown in Fig. 5. Then cut it into parts as shown by the dotted lines. Re-arrange the 3 angles of the triangle as shown in Fig. 6. Compare the sum of the 3 angles with two right angles as shown in Fig. 6. Convince yourself that the three angles of this triangle are together equal to two right angles.

6. Cut other triangles and make similar comparisons, until you are convinced that the sum of the angles of any triangle is equal to two right angles.

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1. If in figure 1, the angle a is a right angle, and the angle b is equal to the angle c, the angle b is an angle of how many degrees?

2. If in figure 2, the angle d is an angle of 95° and the angle e is an angle of 40°, the angle / is an angle of how many degrees?

3. If in figure 3, the angle x is an angle of 75°, the angle io is an angle of how many degrees?

4. If in an oblong there are ab square feet, and the oblong is a feet long, it is feet wide. ab +a =

5. If in a rectangular solid there are abc cubic feet, and

the solid is a feet long and b feet wide, it is feet

thick, abc .+- ab =

6. Verify problems 4 and 5 by letting a = 3, b = 4, and c = 2.

7. There is a field that contains 1736 square rods; it is 28 rods long. How wide is the field?

8. There is a solid that contains 4320 cubic inches; it is 24 inches long and 15 inches wide. How thick is the solid?

9. How many square inches of surface in the solid described in problem 8?

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