If there be at last a remainder, write it, with the divisor under it, as a fractional part of the quotient. When the divisor is 1, with one or more ciphers on the right, remove the decimal point in the dividend as many orders to the left as there are ciphers on the right of the divisor. The orders on the left of the point will be the integer of the quotient, and the orders on the right, the fractional part of it.

PROOF-Multiply the integer of the quotient by the divisor, and add to the product the remainder, if any. If the work is correct, this result will equal the dividend.

6. 13505 by 37. 7. 47256 by 6. How many 15. 21 in 1223?

Ans. 2750.

16. 43 in 34165?

17. 18 in 1499?

18. 15 in 75850?

19. 16 in 3251?

Ans. 365. 13. 68750 by 25.
14. 42122 by 103.

times-
Ans. 581.

How much is

20. One fifth of 95340?

21. One tenth of 9534? Ans. 83. 22. One eighty-first of 973? 23. One nineteenth of 3640? Ans. 203. | 24. One thousandth of 31673?

25. When 112 muskets are worth 1344 dollars, what is the value of each? Ans. 12 dollars.

26. How many tons of coal, at 10 dollars per ton, can be bought with 13670 dollars?

If either factor has one or more ciphers on the right, multiply without regard to these ciphers, and annex to the result as many ciphers as are on the right of both the factors.

PROOF.-Review the work, or reverse the order of the factors and multiply. If the work is correct, the result will be the same by both methods.

PROBLEMS.

Multiply and prove

1. 216 by 8. Ans. 1728. | 12. 756 by 72.
2. 405 by 9.
3. 1315 by 6. Ans. 7890. 14. 248 by 19.
4. 116 by 1000.

Ans. 54432.

13. 3216 by 5.

Ans. 4712.

Ans. 13505.

17. 1040 by 11.

15. 160 by 30.

5. 413 by 70. Ans. 28910. 16. 365 by 37.

6. 555 by 4.

7. 4163 by 7. Ans. 29141. 18. 4561 by 603. Ans. 2750283.

345000345 × 1000 345=345 X

1

344655345 × 999

Ans. 1292928.

SOLUTION.-Since 1000 times any number, less once the number, must be 999 times the number, we here abridge the solution by taking the

multiplicand 1000 times, or once too many, by annexing three ciphers, and then subtracting the multiplier. This method of abridgment applies whenever the multiplier is 1 less than 100, 1000, 10000, etc.

=

26. 1316 × 999 what number?

27. 1230 × 9999 what number?

28. What is the product of 1036 by 990?

Ans. 452727.

Ans. 12298770.

1

29. What is the product of 4455 by 105?

4455004455 × 100 222754455 X 5

4677754455 × 105

SOLUTION.-Here the solution may be abridged by writing the product of 4455 by the 1 hundred at once, by annexing two ciphers, and writing

under it the product of 4455 by the 5 ones, and then adding the two partial products.

30. What is the product of 6307 by 1003?

31. If 5362 feet of boards can be sawed in a mill in one day, how many feet can be sawed in it in 313 days?

Ans. 1678306. 32. At 125 dollars a month, how many dollars can be earned in 12 months? Ans. 1500. 33. What will 3158 tons of coal cost at 8 dollars per ton?

3158

8 25264

SOLUTION.-At 8 dollars a ton, 3158 tons of coal will cost 3158 times 8 dollars, which is equal to 8 times 3158 dollars, or 25264 dollars.

8 dollars is the true multiplicand, but since 8 times 3158 gives the same product as 3158 times 8, we, for convenience, in the solution consider both factors as abstract numbers, and make the smaller factor the multiplier.

34. What will 8344 yards of cloth cost at 6 dollars a yard? Ans. 50064 dollars. 35. How many oranges in 47 boxes when each box contains 279 oranges? Ans. 13113. 36. Two factors are 7312 and 7000. What is their product? Ans. 51184000.

37. There were 6 drawers in a desk, 8 compartments in each drawer, and 87 dollars in each compartment. How many dollars did the desk contain?

87 dollars.

48 No. of compartments. 696

348

4176 dollars.

SOLUTION. Since in the desk there were 6 drawers, and each had 8 compartments, there were in the desk 6 times 8, or 48 compartments. Since there were 48 compartments in the desk, and 87 dollars in each, the desk contained 48 times 87 dollars, or 4176 dollars.

32. How many acres of land at 100 dollars each can be bought for 25605 dollars?

33. A field of 101 acres yields 2125 bushels of grain. How much is the yield per acre? Ans. 21 bushels. 34. A man has 14250 dollars, which he wishes to invest in horses at 250 dollars each. How many horses can he purchase? 35. It is proposed to divide a tract of land containing 72000 acres into farms of 320 acres each. How many farms will it make? Ans. 225.

36. In an orchard there are 44520 trees in 212 equal rows. How many trees are there in each row?

37. If the dividend is 126072 and the divisor 612, what is the quotient?

38. The height of a mountain in Asia is height of Mount Washington is 6284 feet. as high as the latter is the former?

28176 feet, and the How many times Ans. 4884

040 6634

TEST QUESTIONS.

The divisor? The

92.-1. What is DIVISION? The dividend? quotient? What in division corresponds to the factors of the product in multiplication? What is the product? What, then, in division may be regarded as the factors of the dividend?

2. What is the SIGN of division? When written between two numbers what does it denote? In what other ways may division be denoted ? 3. Recite the PRINCIPLES of division. Show that division is the reverse of multiplication. Show when the quotient will be an abstract number. When the quotient will be a concrete number.

4. What is a REMAINDER in division? Why are the remainder and dividend similar numbers? How may the remainder be changed to a fractional part of the quotient?

5. Recite the RULE for division. How does long division differ from short division? What is the reason for removing the decimal point to the left in dividing by 10, 100, etc.?

6. What is the PROOF of division?

The reason for it? Since multiplication and division are the reverse of each other, how may multiplication be proved? Show that the product divided by the multiplier gives the multiplicand.

SECTION VIII.

REVIEW PROBLEMS.

MENTAL EXERCISES.

93.-Ex. 1. The factors of a product are 11 and 12. What is the product?

2. The product of 15 by 9 is what number?

3. If a boy can earn 18 dollars in one month, how many dollars can he earn in 10 months?

4. At 14 dollars a ton, what will 13 tons of hay cost?

SOLUTION.-If one ton cost 14 dollars, 13 tons will cost 13 times 14 dollars, or 182 dollars. The multiplication may be conveniently performed thus: 13 times 14 3 times 14+10 times 14; 3 times 14=42; 10 times 14=140; 42+ 140=182.

5. At 16 dollars each, what will 12 garments cost?

6. If 14 men can do a piece of work in 21 days, in how many days can one man do it?

7. John lives 5 miles from a certain place; Andrew, 7 miles farther away; and Benjamin, 8 times as far as Andrew. How far from the place does Benjamin live?

8. How many lengths of 12 feet each are there in a fence which is 132 feet long?

9. A and B start from points 121 miles apart, and travel toward each other, A at the rate of 5 miles per hour, and B at the rate of 6 miles per hour. In how many hours will they meet?

10. When flour is worth 8 dollars a barrel, how many barrels of flour will pay for 24 tons of coal at 5 dollars a ton?

11. If 10 men can do a piece of work in 6 days, in how many days can 4 men do it?

SOLUTION.-If 10 men can do a piece of work in 6 days, 1 man can do it in 10 times 6 days, which are 60 days; and 4 men can do it in one fourth of 60 days, which is 15 days.

12. When 12 tons of coal at 7 dollars a ton pay for 21 pairs of boots, how much are the boots worth a pair?

13. If 8 cords of wood will buy 32 pairs of shoes, how many cords will buy 40 pairs of shoes?