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WRITTEN ARITHMETIC.

V

SECTION 4.

ABBREVIATIONS.

When there are ciphers on the right hand of a divisor, cut them off, and omit them in the operation; also cut off and omit the same number of figures from the right hand of the dividend. Finally, place the figures cut off from the dividend, on the right of the remainder.

1. How many times 900 are there in 741 725 ? 900)741725

We divide 7417 by 9; there remains 1, to which we annex the 824 125 25, making the true rem. 125. 2. How many times 70 are there in 8 563 512 ? 3. How many times 300 are there in 6374 ? 4. How many times 5000 are there in 46 578 ? 5. How many times 40 are there in 80 603 ? 6. How many times 600 are there 675 700? 7. How many times 8000 are there in 16 000 ? 8. Divide 65 237 by 50, and prove the operation. 9. Divide 567 289 by 400, and prove the operation. 10. How many times 570 are there in 35 871 ? 11. How many times 280 are there in 6423? 12. How many times 4200 are there in 91621? 13. How many times 9060 are there in 287 000 ?

When the divisor is 10, 100, 1000, &c., cut off as many figures from the right hand of the dividend, as there are ciphers in the divisor; the other figures of the dividend will be the quotient, and the figures cut off will be the remainder.

14. How many times 10 are there in 240?

15. How many times 10 in 435; and how many over? 16. How many times 100 are there in 4000 ? 17. How many times 100 in 748; and how many over ? 18. 100 cents are equal to 1 dollar. dollars are there in 5400 cents?

How many

19. In 642 cents, how many dollars are there; and

now many cents over?

20. In 1937 cents, how many dollars are there; and how many cents over?

When factors of the divisor can be found, (that is, when two numbers can be found, which, being multiplied together, produce the divisor,) you may divide the dividend by one of the factors, and the quotient thence arising by the other: the last quotient will be the true one.

21. In a certain school there are 36 scholars, among whom 540 quills are to be equally divided. How many will 1 scholar receive?

Let us suppose the school to be divided into 4 classes, allowing 9 scholars to be in each class. Then we will find how many quills 1 class will receive, and from this number, find how many 1 scholar will receive.

4)540 number of quills for the school.
9)135 number of quills for 1 class.

15 number of quills for 1 scholar.

Observe in the above example, that the divisors 4 and 9, are the factors of 36: and, if we had divided first by the 9, and then by the 4, our last quotient would have been the same it now is.

22. Divide 11 376 by 72; using the factors of 72. 23. If 1024 dollars be divided equally among 64 men, how many dollars will 1 man receive?

24. How many times is 42 contained in 1176?

25. If 27 yards of cloth cost 216 dollars, how many dollars does 1 yard cost?

26. Suppose 1952 to be a dividend, and 32 the divisor; what is the quotient?

To obtain the true remainder, where factors have been used as divisors, multiply the last remainder by the fir divisor, and to the product add the first remainder.

27. Suppose 622 to be a dividend, and 35 the div sor; what is the quotient; and what the remainder? 28. Suppose 99 to be a dividend, and 25 the divisor; what is the quotient; and what the remainder?

29. Suppose 4862 to be a dividend, and 81 the divisor; what is the quotient; and what the remainder? 30. Divide 1739 bv 56.

Questions to be answered Orally.

(1) When we say, '3 is contained in 20, 6 times, and 2 over,' which of these numbers have we for the dividend? Which for the divisor? Which for the

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quotient? Which for the remainder ? (2) What is meant by the dividend? (3) What is meant by the divisor? (4) What is meant by the quotient? (5) What is meant by the remainder? (6) Can the remainder ever be equal to, or greater than the divisor? Why? (7) Suppose you have a number of dollars to divide among a number of men; which number do you make the dividend;—and which the divisor? -If there be a remainder, will it be so many dollars, or so many men? (8) Recite the rule for division. (9) How do you proceed when there are ciphers on the right hand of the divisor? divide by 10, 100, 1000, &c.? divide by means of factors?

(10) How do you (11) How can you (12) When you have

divided by the factors of the divisor, how do you find the true remainder? (13) How do you prove an operation in division?

Perform the following examples by either of the foregoing methods, which may be found convenient.

31. Suppose it takes 7 bushels of apples to make a barrel of cider, how many barrels of cider can be made from 945 bushels of apples?

32. Suppose an acre of land properly cultivated, to produce 38 bushels of corn; how many acres must be cultivated to produce 4902 bushels?

33. If 50 dollars will pay for an acre of land, how many acres can be bought for 6900 dollars?

34. How many days will a ship be in sailing from New York to Liverpool; allowing the distance to be 3000 miles, and the ship to sail 100 miles a day?

35. A vintner wishes to put 6615 gallons of wine into hogsheads that will hold 63 gallons apiece;-how many hogsheads must he have?

how

36. If you had 118 dollars, how many hats could you pay for, at 5 dollars apiece; and what number of dollars would you have left?

37. Suppose a drover has 2130 dollars; how many oxen can he pay for, at 47 dollars apiece; and how many dollars will he have left?

38. In 668 360 yards of cloth, how many pieces, and how many bales; there being 35 yards in each piece, and 56 pieces in each bale?

39. If 4810 dollars be shared equally among 130 men, how much will each man receive?

40. A farmer planted 2072 trees in 14 equal rows. How many did he plant in a row?

41. A gentleman wishes to spend 136 days in performing a journey of 3264 miles. How many miles must he travel each day?

42. If a man whose property is valued at 21 148 dollars be worth 17 times as much as his neighbor, how much is his neighbor worth?

RETROSPECTIVE OBSERVATIONS.

In the course of the last four chapters, you have practised four kinds of operations on numbers: viz. Addition, Subtraction, Multiplication, and Division. These operations should be perfectly understood-the effect of each should be distinctly perceived; for, it is on their proper application, that the solution of all questions in arithmetic depends.

Addition is the operation by which two or more numbers are united in one sum.

Subtraction is the operation by which the difference between two numbers is found.

Multiplication is the operation by which a number is produced, equal to as many times one given number, as there are units in another given number.

Division is the operation by which we find how many times one number contains another, and, by which we divide one given number into as many equal parts, as there are units in another given number.

Questions to be answered Orally.

(1) How many kinds of operations are practised on numbers? (2) What are they called? (3) What is Addition? (4) What is Subtraction? (5) What is Multiplication? (6) What is Division? (7) Propose a question that you would solve by addition. (8) Propose a question that you would solve by subtraction. (9) Propose a question that you would solve by multiplication. (10) Propose a question that you would solve by division. (11) How can a question in multiplication be solved by addition? (12) How can a question in division be solved by subtraction?

SECTION 5.

MISCELLANEOUS EXAMPLES.

1. The population of the world has been estimated to be as follows. North America, twenty-six millions; South America, twelve millions; Europe, two hundred and twenty millions; Asia, five hundred millions; Africa, thirty-eight millions; Australia, four millions. What is the whole number?

2. In 1830, the national debt of the United States was 48 565 406 dollars; in 1831 it was 39 123 191 dollars. How much was paid in one year?

3. The national debt of England cannot be less than 1 900 000 000 dollars. How many years would it take to pay this debt, allowing ten millions of dollars to be paid annually?

4. What would be the expense of laying a rail-way from Louisiana to Maine; the distance being 1800 miles, and the rail-way costing 14 000 dollars a mile?

5. In how many days could a passage be effected from Maine to Louisiana, on the proposed rail-way; allowing a car to run 25 miles an hour, day and night?

6. How many days would it take a man to ride on horseback from Maine to Louisiana, riding 5 miles an hour, and 10 hours a day?

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