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times 670, and their difference 55 times 81. Find the less number.

17. I bought 87 acres of land at $50 an acre, and paid $3150 in cash, and the balance in labor at $240 a year. How many years of labor did it take?

18. A farmer has 1000 head of cattle in five fields. In the first he has 315 head; in the second, 175 head ; in the third, 300 head ; and in the fourth, the same number as in the fifth. How many has he in the fifth ?

19. If a man sells 19 bushels of potatoes at $.55 a bushel, 23 bushels of oats at $.53 a bushel, and with the proceeds buys 8 yards of broadcloth, how much does he pay a yard for the broadcloth ?

20. If a newsboy buys papers at $.08 a dozen, and sells them at $.01 apiece, how much can he clear in March, if he averages 120 papers a day?

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Analysis reasons from the given number to one, and from one to the required number.”

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1. A man bought 13 horses for $2405. What would he pay

for 37 horses at the same rate ?

Process Indicated.

$2405 X 37

= sum received.

13 Process.

Explanation. 13 horses cost $2405.

1. Since 13 horses cost $2405, 1 horse 1 horse costs $185.

will cost $2405 ; 13, or $185.

2. Since 1 horse costs $185, 37 horses will 37 horses cost $6845,

cost $185 X 37 or $6845.

Let the pupil indicate the solution by using the appropriate signs.

2. If 25 pounds of sugar cost $2.50, what will 36 pounds cost?

3. If I exchanged 40 barrels of flour for 61 yards of cloth at $4 a yard, how much did I get per barrel for the flour?

Indicate the process and explain. 4. A carriage maker sold 15 carriages for $1875. How much would he receive for 25 carriages, selling them at the same rate ?

5. 190 bushels of corn cost $100.70. At what rate must it be sold to gain 13 cents a bushel ?

6. If 93 oranges cost $5.58, what will 75 oranges cost?

7. If 12 yards of cloth cost $48.00, what will 7 yards cost?

8. If 16 horses cost $1952, what will 22 horses cost at $6 less a head ?

9. A. paid $27,144 for a farm, at the rate of 15 acres for $3510. How many acres did he buy?

10. If 46 acres of land produce 2484 bushels of corn, how many bushels will 120 acres produce ?

a

INDICATED SOLUTIONS. As we have already attempted to show, the solution of any problem should first be indicated by means of signs, and afterwards carried to completion as the signs direct.

In completing a solution indicated, a parenthesis or a vinculum must be removed first. The other signs, whether within a vinculum or not, may be safely used in the following order: X,-,-, +. (12 ; 3) X 2 = 4 x 2 = 8; but 12 = 3 X 2 = 12 x

; = : 6 2.

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1. Perform the operations indicated in (48 X 2 – 84 - 6 X 2) + 7 — 3.

1. (48 X 2 — 84 ; 6 X 2) + 7 — 3.
2. By removing sign X, (96 84 - 12) + 7 — 3.
3. By removing sign • , (96 — 7) + 7 — 3.
4. By removing sign 89 + 4.

5. By removing sign +, 93. 2. Find the value of 48 x 2 84 ; 6 x 2 + 7 - 3.

3. If 6 men can do a piece of work in 10 days, how long will it take 5 men to do the work ?

We may let x stand for the required number of days, and write an equation thus : Process.

Explanation.
ANALYSIS.—Since 6 men require 10 days,

1 man will require 6 X 10 days. Hence, 5
10 x 6
: 12

6 x 10 5 men will require days. Performing

5

10 X 6 the operations indicated, we have 12 days.

5 60

12 days. 5

X

X =

4. If 12 men can build a school-house in 25 days, how long will it take 25 men to build it?

5. How many pounds of butter, at $.23 a pound, must be given for 5 pounds of raisins at $.11 a pound, 2 pounds of tea, at $.63 a pound, and a barrel of sugar, at $9 ?

Suggestion :
5 X $.11 + 2 X $.63 +- 9

$.23 6. Find the value of:

1. 28 X 6 = 14 + 9 X 8 - 12 + 42 - 7 X 3.
2. 99 X (8 + 51) X 10 — (7 X 104 + 26).
3. (105 = 21 + 80 = 5) X (81 + 36 = 9).
16 X 3125 — 127 X 0 + (380 = (100 = 50)

239
(125 X 30) = (25 X 25) x (32—21) — 55 = 5
5.

11

4.

7. A lady paid a store bill of $784, giving 30 twentydollar bills, 4 one-dollar bills, and the remainder in five-dollar bills. How many five-dollar bills did she use?

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8. Two trains leave New York for Chicago, 900 miles, at the same hour, one averaging 30 miles an hour, the other 45 miles an hour. How long will the second train be in Chicago before the first arrives ?

9. How many men will it take to do a piece of work in 26 days that 39 men can do in 76 days?

10. How long can 125 men subsist on an amount of food that will last 3 men 4500 days ?

11. A quantity of provisions lasts an army of 2500 men 72 days. How long would it last 18,000 men ?

12. I bought a carriage for $140, a horse for $125, and a set of harness for $18; kept them a month at an expense of $17.25, and then sold the team for $300. Did I gain or lose, and how much ?

13. A pedler sells beets, six in a bunch, at 10 cents a bunch, and gains one cent on each bunch. Find the cost per C.

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14. I paid $86.40 for 1440 blocks of granite. What was the price per M?

Suggestion :

$86.40

x 1000 1440

?

15. If 8 acres of land cost $656, what will 35 acres cost at $1 more per acre ?

16. From 126 + (16 + 4) x 2 take (48 + 2) + 34 X 6 • (17 — 15).

GENERAL PRINCIPLES OF DIVISION.

If 24 is the dividend (D.), 4 the divisor (d.), and 6 the quotient (Q.), we have

6, Q. 4, d.

24, D.

We will now notice the effect upon Q, if we multiply and divide D. and d. by 2, as follows:

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Analysis reasons from particular instances to general principles.”

Reasoning from the particular instances above, we derive the following

PRINCIPLES.
1. Multiplying D. multiplies Q.
2. Multiplying d. divides Q.
3. Dividing D. divides Q.
4. Dividing d. multiplies Q.
5. Multiplying both D. and d. does not change Q.
6. Dividing both D. and d. does not change Q.

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Let D.

1728 and d. 144. Find Q., and illustrate each of the above six principles.

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