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Division-Decimals.

133. Find the quotient of 785.6 divided by .5.

Operation

Explanation. .5)785.6'5 First place a separatrix (v) after that figure in

the dividend that is of the same denomination as 1571.3

the right hand figure of the divisor-in this case, after the figure 6. Then divide, writing the decimal point in the quotient when, in the process of division, the separatrix is reachedin this case, after the figure 1.

It was required to find how many times 5 tenths are contained in 7856 tenths. 5 tenths are contained in 7856 tenths 1571 times. There are yet 15 hundredths to be divided. 5 tenths are contained in 15 tenths, 3 times; in 15 hundredths, 3 tenths of a time.

Note.-By holding the thought for a moment upon that part of the dividend which corresponds in denomination to the divisor, the place of the decimal point becomes apparent at once.

5 apples are contained in 7856 apples, 1571 times.
5 tenths are contained in 7856 tenths, 1571 times.

134. Solve and explain the following problems with special reference to the placing of the decimal point :

1. Divide 340 by .8
2. Divide 468.5 by .25
3. Divide 38.250 by 12.5
4. Divide 87 by 2.5
5. Divide 546 by .75
6. Divide .576 by 2.4

7. 86 ; .375
9. 75 : .15

.8)340.00 .25) 468,500 12.5)38.2 50

2.5)87.00 .75)546.00'

2.4).5'76 8. 91.5 2.8 10. 125 +.5

(a) Find the sum of the ten quotients.

Division-United States Money. 135. Divide $754.65 by $.27. Operation.

Explanation. $.27) $754.65' (2795

This means, find how many times 27 54

cents are contained in 75465 cents. 27

cents are contained in 75465 cents, 2795 214

times. 189

PROBLEM. 256

At 27% a bushel, how many bushels of 243

As oats can be bought for $754.65 ?

many bushels can be bought, as $.27 is 135

contained times in $754.65. It is con135

tained 2795 times Therefore, 2795

bushels can be bought. 136. Divide $754.65 by 27. Operation.

Explanation. 27)$754.'65($27.'95

This means, find one 27th of $754.65 54

One 27th of $754.65 is $27.95.

NOTE.-One might find 1 27th of 214

$754.65 by finding how many times $27 189

is contained in $754.65. See p. 52, Note. 256 243

PROBLEM.

If 27 acres of land are worth $754 65, 135

how much is one acre worth? 135

137. Divide $754.65 by .27. Operation.

Explanation. .27($754 .65'($2795

This means find 100 27ths of $754.65. 54

One 27th of $754.65 is $27.95. 100 27ths

of $754.65 is $2795.
214
189

NOTE.-In practice we find one 27th of

100 times $754. 65.
256
243

PROBLEM.

If .27 of an acre of land is worth 135

$754.65, how much is 1 acre worth at the 135

same rate ?

DENOMINATE NUMBERS.

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

Explanation. 4)46 rd. 12 ft. 8 in. This means, find i fourth of 46 rd. 12 ft. 11 rd. 11 ft. 5 in.

8 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. Multiply $275.56 by 2.25. 13. Multiply $275,56 by 21. 14. Can you multiply by a number of dollars ? 15. Can you divide by a number of dollars ?

* The word remainder in this connection suggests incomplete division. the division is complete there can be no remainder.

When

[blocks in formation]

1. Observe that in the above examples we divide each term of the dividend by the divisor.

2. 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.

3. Verify No. 2, by letting a = 3, and b = 5. 4. Verify No. 4, by letting a = 3, b = 5, and c = 7. 141. (6 xa xa xa x a x a) (2 x ax a) = 6 a + 2 a' = 3 a3.

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

142. PROBLEMS.

1. 6a5b 2 a =
2. 4 a*b; 2 a =

3. 8 a 63 = 2 a =
4. 10 a64 = 2 a =

5.
2 a)6 ab + 4 a 62 - 8 a' b3 + 10 a 64

6. Verify Problem 5, by letting a = 3, and b = 5.

Algebraic Division.

143. PROBLEMS.

1. Divide 4 a'x + 8 aRx+ 6 ax3 by 2 ax. 2. Multiply the quotient of problem 1, by 2 ax. 3. Verify problems 1 and 2, by letting a = 2, and x = 3.

4. Divide 3 ab3 + 6 aRb2 + 9 aạb by 3 ab. 5. Multiply the quotient of problem 4, by 3 ab. 6. Verify problems 4 aná 5, by letting a = 3, and b = 5.

7. Divide 2 x'y + x*y– xyby xy. 8. Multiply the quotient of problem 7, by xy. 9. Verify problems 7 and 8, by letting x = 2, and y = 3.

10. Divide 5 a'y- 2 aoy3 + a*ył by a'y. 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 b*x + bx – 3 b*r' by bx. 14. Multiply the quotient of problem 13, by bx. 15. Verify problems 13 and 14, by letting b = 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 was derived.

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

2)8 – 6

4 - 3

16.
2 x )4 x5 – 6 x4 + 8 X3 – 2 x + 6 x.

17. Verify by letting x = 2.

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