To divide a number by 10, 100, 1000, &c., cut off as many figures at the right of the dividend as there are naughts in the divisor. The remaining figures are the quotient; those cut off, the remainder.

4200 10420 4200 100 42

4200-1000=4, 200 rem.

102. The principle is the same in the case of

visor ending with one or more naughts.

Cut off the naughts at the right of the divisor, and as many figures at the right of the dividend. Divide the remaining figures of the dividend by those of the divisor. If there is a remainder, annex to it the figures cut off from the dividend; if not, these figures are themselves the remainder.

Give the rule for dividing a number by 10, 100, 1000, &c.-102. Divide 9710 by 2400, using the factors of the divisor. What other way is there of arriving at the same result? Give the rule for dividing when the divisor ends with one or more naughts.

MISCELLANEOUS QUESTIONS.-Name the four fundamental rules. Ans. Addition, Subtraction, Multiplication, Division; with these all calculations are performed. What is Addition? Subtraction? Multiplication? Division? What operation enables us to find a whole, when its parts are given? When the whole and one part are given, what operation enables us to find the other part? What is the converse of addition? Of multiplication?

What is the result of addition called? Name the three terms used in subtraction. Ans. Subtrahend, minuend, and difference. Define each of these terms. Name and define the three terms used in multiplication. Name and define the terms used in division. What is meant by the factors of a product? Which term in division corresponds with the product in multiplication? With what do the divisor and quotient correspond? At which side do we begin to add? To subtract? To multiply? To divide?

What does the sign minus denote? On which side of it must the subtrahend be placed? What does a horizontal line between two dots denote? On which side of this sign must the dividend be placed? What does plus denote? What does an oblique cross denote? What is the sign of equality? How is addition proved? Subtraction? Multiplication? Division? In what other way may multiplication be proved? Ans. By dividing the product by the multiplier; if the quotient equals the multiplicand, the work is right.

What is a composite number? Give an example of an abstract composite number; of a concrete composite number. How may we multiply or divide by a composite number? When we divide by factors, how do we find the true remainder? What is the shortest way of multiplying by 10, 100, &c.? How do we divide by 10, 100, &c.? When is division called Short, and when Long? What difference is there in the mode of performing the two operations?

1. Find the sum, then the difference, then the product, of 343 and 8918; divide 8918 by 343.

2. How many times is 20000 contained in the difference between eleven million and eleven billion? Ans. 549450 times.

3. A United States senator receives $3000 a year. If he spends $8 a day, how much of his salary will he save in his six years' term, allowing 365 days to the year? Ans. $480.

4. If a person has an income of $3285 a year, how much is that a day?

5. A mile is 5280 feet. How many steps, of two feet each, will a boy take in walking 5 miles? Ans. 13200 steps.

6. Divide the sum of 168483 and 849717 by the difference between 97234 and 46324, and multiply the quotient by nine times nine.

Ans. 1620.

7. If a man earns $1200 a year, and his yearly expenses are $860, how many years will it take him to lay up $5440?

Ans. 16 years.

8. A farmer buys 75 tons of hay, at $32 a ton. He pays for it in wheat, at $2 a bushel. How many bushels of wheat must he give? Ans. 1200 bushels.

What was the whole cost of the hay? How much wheat, at $2 a bushel, will

pay for it?

9. A merchant began business with $36000. At the end of 9 years he was worth $61875. How much a year had he made?

10. How many pounds of coffee, at 29 cents a pound, will pay for two hogsheads of sugar containing 1160 pounds each, at 19 cents a pound? Ans. 1520 pounds. 11. A person having $2879 in current bills, and $3997 in uncurrent, invests the whole in flour at $9 a barrel; how many barrels can he buy? Ans. 764 barrels.

12. Four partners commencing business put in respectively $8650, $9200, $7950, and $3000. At the end of a year the firm was worth $37875. Required their gain. Ans. $9075.

13. If a man buys 746 barrels of flour for $8206, what must

he sell the whole for, to gain $1 a barrel? How much is that a barrel? Ans. $12 a barrel.

14. A person willed $12000 to his wife, $300 to the poor, and the rest of his property to his six children in equal shares. If he was worth $71370, what was each child's share? Ans. $9845.

What was he worth in all? How much of this did he leave to his wife and the poor? How much remained? Into how many parts must this be divided?

15. A lady worth $48530 leaves her servant $550, her brother four times that amount, and divides the rest of her property equally among her four sons and three daughters. How much does each child receive? Ans. $6540.

How much does she leave to her servant? To her brother? How much to both? How much of her property is left? Among how many is this divided?

16. Three partners divide equally their yearly profit, amounting to $17064. One of them divides his share equally among his four children; what does each child get? Ans. $1422.

17. An army of 4525 men had 103075 days' rations. At the end of 21 days, 500 men were captured. How many days after that did the rations last? Ans. days.

How many rations did 4525 men consume in 21 days? How many rations then remained? After the capture, how many men were left? How long would the rations left support these men?

18. A garrison of 842 men had 63472 days' rations. After 16 days a reënforcement of 158 men arrived. How long after their arrival did the rations last? Ans. 50 days.

19. A person bought 97 acres of land at $51 an acre, and 111 acres at $47 an acre. He paid $9539 cash, and for the balance gave 5 horses; what was each horse valued at? Ans. $125.

What was the cost of the first piece of land? Of the second? Of both? How much cash was paid? What remained due? If 5 horses were valued at this amount, what was each horse valued at?

20. A hogshead containing 63 gallons of molasses was bought for 67 cents a gallon. 7 gallons having leaked out, the rest was sold at 76 cents a gallon. What was the gain? Ans. 35 cents.

21. In an orchard containing 659 trees, 41 trees bear no fruit. If the income from the orchard is $4944, and the apples bring $4 a barrel, how many barrels on an average does each bearing tree produce? Ans. 2 barrels.

22. A railroad forty miles long cost a million of dollars, all but four hundred. What was the cost per mile? Ans. $24990.

23. The dividend of a sum in division is 4719, the quotient 96, the remainder 15. What is the divisor?

Ans. 49. Subtract the remainder from the dividend, and you have the product of the quotient and divisor; then proceed according to § 89.

24. On dividing 734062 by a certain number, I get 807 for the quotient, and 499 remainder. What is the divisor?

25. If 17 cows are worth $816, and each cow is worth as much as 6 sheep, what is the value of one sheep? Ans. $8.

26. An estate of $25101 was left to a family of four brothers and nine sisters. The brothers having given up their share to the sisters, how much did each of the latter receive?

27. A farmer had 100 hens, four of which died; if the remainder laid in one week four basketfuls of eggs, consisting of 120 each, what was the weekly average for each hen?

Relations of Dividend, Divisor, and Quotient.

103. The quotient depends on both dividend and divisor. If one of these is fixed, a change in the other changes the quotient. But, if both dividend and divisor are changed, these changes may neutralize each other, and the quotient remain the same.

246 4

Thus :

Doubling or halving both dividend and

12

3 4

divisor makes no change in quotient:

103. On what does the quotient depend? If either dividend or divisor is fixed, what is the effect of changing the other? If both dividend and divisor are changed, what may follow? With the same divisor, what is the effect of doubling the dividend? Of halving the dividend? With the same dividend, what is the effect of doubling the divisor? Of halving the divisor? What is the effect of doubling or halving both dividend and divisor?