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RULE. --Cut off the ciphers from the right of the divisor, and as many figures from the right of the dividend.

Divide the rest of the dividend by the rest of the divisor.

Annex to the remainder the figures cut off; the result will be the true remainder.

Divide the following:
3. Divide 1869 by 100.
4. Divide. 12345 by 200.
5. Divide 89325 by 700.
6. Divide 35968 by 900.

7. Divide 2465 by 1000. 8. Divide 13692 by 4000. 9. Divide 83005 by 1100. 10. Divide 75684 by 1500.

11. How many miles of railroad at $50000 a mile can be constructed for $38968457?

12. How many schooners carrying 8300 bushels of wheat will it require to carry 984364 bushels?

13. The area of the State of New York is 47000 square miles, or 30080000 acres. How many acres in a square mile?

103. RELATION OF DIVIDEND, DIVISOR, AND QUOTIENT.

The value of the quotient depends upon that of the dividend and divisor. If one of these is changed, while the other remains the same, the quotient will be changed. If both are changed, the quotient may not be changed.

The changes may be illustrated as follows:

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CHANGED EQUATIONS.
1. 128 :8=16 1. Multiplying the dividend

by 2 multiplies the quotient

1. Dividend

changed.

by 2.

2. 32 :8= 4

2. Dividing the dividend by 2 divides the quotient by 2.

1. 64; 16 =

4

2. Divisor

changed.

1. Multiplying the divisor by 2 divides the quotient by 2.

2. Dividing the divisor by 2 multiplies the quotient by 2.

2. 64 : 4 = 16

3. Both

changed.

1. 128 : 16 = 8 Multiplying or dividing

both dividend and divisor by

2 does not change the quo2. 32 ; 4=8 tient.

From these illustrations the following principles are deduced:

104, PRINCIPLES.—1. Multiplying the dividend or dividing the divisor, multiplies the quotient.

2. Dividing the dividend or multiplying the divisor, divides the quotient.

3. Multiplying or dividing both dividend and divisor by the same number, does not change the quotient.

ANALYSIS AND REVIEW.

105. Analysis is the process of solving problems by tracing the relation of the parts.

In analyzing we commonly reason from the given number to one, and then from one to the required number. 1. If 8 yards of cloth cost $16, what will 12 yards cost ?

ANALYSIS.—Since 8 yards cost $16, 1 yard 8 yards=$16. will cost one-eighth of $16, or $2; and since 1 " =$ 2.

1 yard costs $2, 12 yards will cost 12 times 12 --$24. $2, or $24. 2. If 8 horses cost $2400, what will 6 horses cost? 3. If 8 lemons cost 40 cents, what will 11 lemons cost? 4. How much will 12 hats cost, if 8 hats cost $16?

PROCESS.

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

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

7. If 12 barrels of flour are worth $132, what are 22 barrels worth?

8. If it requires 576 feet of boards to build 18 rods of fence, how many feet will be required to build 13 rods?

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

10. If I exchanged 18 barrels of flour for 61 yards of cloth at $4 a yard, how much did I get per barrel for the flour?

106. The Parenthesis, (), shows that the numbers included within it are to be subjected to the same operation.

Thus, (5 +6—2) X3 shows that 5 +6—2, or 9, is to be multiplied by 3. 107. The Vinculum,

, may be used instead of the parenthesis.

Thus, instead of (5+6—2) X3, we may write 5 + 6 - 2X3.
Find the value of the following:
11. (12 +7-9) X 5.
12. (13–6+8) X 6.
13. (11-2+5) X 8.
14. (3+4) X9—(3+6)=3.
15. (5+7-3) X 3+(3+5—4)- 4.
16. (36-7) X 5+ (102+6) -:-9.
17. (99 — 3) --8-- (86 +10) + 12+(3+6) + 3.
18. (45+3)=6+ (10 + 15)-(7-2) + 6.

19. A man dying, left the following tracts of land to be divided equally among his five children. The first tract contained 1118 acres; the second, 3 times as much lacking 193

acres.

acres; the third, twice as much as the other two lacking 105

What was each one's share? 20. A gentleman bought 1516 head of cattle at $39 per head. During the summer 97 died of disease, but he sold the remainder so as to gain on the whole number $1819. How much did he get for his cattle per head?

21. If a young man who has a salary of $30 per week, pays $7.25 for his board and $4.25 for other expenses, how long will it take him to save $1500?

22. A man bought a horse for $115 and after keeping him three months, sold him for $155. If he paid $30 for his keeping and received $50 for the use of him during that time, how much did he gain?

23. A speculator purchased a certain number of bushels of wheat for $8735. He sold it for $9215 and in so doing gained $.25 per bushel. How many bushels did he buy?

24. I bought 25 barrels of flour for $200. For what must it be sold per barrel to gain $50? What will be the gain per barrel?

25. A tailor having $585 wished to purchase with this an equal number of yards of two kinds of broadcloth. One kind was worth $6 a yard, the other $7 a yard. How many yards of each kind could he buy?

26. Two men leave the same place at the same time and travel in opposite directions, one at the rate of 48 nuiles per day, the other at the rate of 52 miles per day. How far apart will they be at the end of 5 days?

27. If 20 men can do a piece of work in 31 days, how many days will be required to do an equal amount of work if 11 additional men are employed?

28. A farmer wished to obtain $120. He sold 16 barrels of apples at $3.50 per barrel, and enough barley at $.80 a bushel to make up the sum required. How many bushels of barley did he sell?

29. Mr. B. bought 140 acres of land for $17500, and sold enough at $120 per acre to amount to $9600. The rest of the land he sold at cost. How many. acres did he sell at cost, and what was the entire loss?

30. A man pays $628 a year for groceries, $350 for house rent, $262 for clothes, twice as much for traveling expenses as for house rent, $175 for annual premium for life insurance, and saves in 4 years enough money to purchase 130 acres of land at $53 an acre. What is his yearly income?

31. In October, 1871, the great fire in Chicago burned over an area of 2124 acres. The estimated loss occasioned by the fire was $196000000. What was the average

loss
per

acre? 32. A boy has a velocipede which he can run at the rate of 140 rods in 4 minutes. How many minutes will it take him to run it 630 rods?

33. A farmer has 1000 head of cattle in 5 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 ?

34. A man gave away $45000 in three equal amounts. One share he gave to his son, one share to his daughter, and the rest to his grandchildren, giving them $1500 apiece. How many grandchildren had he?

35. In the Centennial Exhibition, at Philadelphia, a section of a cable in process of construction for the new suspension bridge at New York was shown. It was composed of 6000 galvanized steel wires, and its ultimate strength was 22,300,000 pounds. What weight was each wire capable of sustaining ?

36. The main building of the Centennial Exhibition at Philadelphia, the largest building in the world, contained on the ground floor an area of 872320 square feet, on the upper

floors in projections 37344 square feet, in towers 26344 square feet. If there are 43560 square feet in an acre, how many acres did the floors of the building contain ?

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