Constant Quotient.

$ 57. The quotient of two numbers remains the same, when those numbers are both multiplied, or both divided, by the same number.

For example, 6÷2 gives 3, and 5 times 6÷÷5 times 2, that is, 30÷10, also gives 3. By reversing the process, we find 30-10 equal to of 30 of 10.

RULE VII.

$ 58. To Divide by a number not exceeding 12; or by such Number with Os annexed.

1. Take figures enough in the left of the dividend to contain the divisor, and set down the number of times the divisor goes therein, noting the excess, if any.

2. Take the next figure of the dividend, with the preceding excess, if any, prefixed, and set the number of times the divisor is found therein on the right of the first quotient; if the divisor will not go therein, put a 0 in the quotient, and include the next figure in dividing; and so on.

3. Ciphers in the right of the divisor are omitted in dividing; but as many figures must be omitted in the right of the dividend, and annexed to the remainder. If there be no other remainder, these figures will form the remainder.

4. When the divisor is 10 or 100, &c., take for the remainder as many figures from the right of the dividend as there are Os in the right of the divisor, and the other figures of the dividend for the quotient.

5. Under the remainder, if any, set the given divisor, and annex the part so found to the quotient.

EXAMPLES.

1. To divide 7805 by 3.

3)7805

2601/

We say, 3 in 7, twice, and 1 over; prefixing the 1 to the next figure 8 of the dividend, we have 18; 3 in 18, 6 times; 3 in 0, 0 time; 3 in 5, once, and 2 over. This excess 2 is the remainder,-under which setting the divisor 3, we have to annex to the quotient (§ 56).

The quotient 2601 shows that the dividend 7805 contains the divisor 3, 2601 times, and of the divisor, besides.

The quotient is also of 7805 (§ 51.)

2. To divide 13127 by 120.

Omitting the 0 in the right of the divisor, and the 7 in the right of the dividend, we say,

12 in 13 once, and 1 over; prefixing the 1 over to the next figure of the dividend, we say 12 in 11, 0 time; including the next figure 2 of the dividend, we say 12 in 112, 9 times, and 4

over.

Annexing the 7, omitted, to the 4, we have the remainder 47, under which setting the given divisor 120, we have to annex to the quotient.

3. To divide 14723 by 100, we take the two figures 23 from the right of the dividend, for the remainder, and the other figures 147 for the quotient.

Hence the complete quotient is 147,23

100°

The figures of the dividend first taken in dividing, have a local value 10 times, or 100 times, &c., their simple value, according as one, or two, &c., figures follow them on the right (§ 16); and the first quotient figure must therefore have 10 times, or 100 times, &c., its simple value. The proper local value is assigned to the quotient figure, by the succeeding quotient figures, these being always just as many as the succeeding figures of the dividend. In like manner each quotient figure receives its proper value.

The excess belonging to any particular place in the dividend, is so many tens in the next place on the right (§ 11); and is made tens to the next figure by prefixing it to that figure.

In the second example, we divided 12 in 1312, instead of 120 in 13127. But 120 is 10 times 12, and 13127 is 13120+7, or 10 times 1312,7; and 12 in 1312 gives the same quotient as 10 times 12 in 10 times 1312 (§ 57).

In the same way it may be shown that any number of Os may be omitted in the right of the divisor, if an equal number of fignres be omitted in the right of the dividend.

The Operation Proved.

§ 59. Division may be verified or proved, by multiplying the divisor and quotient together, and adding the remainder, if any, to the product; the result must be equal to the dividend.

EXERCISES.

1. How many barrels of apples, at be bought for 150 dollars?

dollars per barrel, may

The number of barrels that may be bought, is the number of times that 2 dollars is contained in 150 dollars (§ 53).

Ans. 75 barrels.

2. At 3 dollars per yard, how many yards of broadcloth may be purchased for 387 dollars? Ans. 129 yards. 3. How many cords of wood, at 4 dollars per cord, may be purchased for 600 dollars? Ans. 150 cords.

4. At 5 dollars apiece, how many superfine beaver hats may be purchased for 3700 dollars? Ans. 740 hats.

5. How many dozen of shoes, at 6 dollars per dozen, may be purchased for 750 dollars ? Ans. 125 dozen. 6. There being 7 days in a week, it is required to find how many weeks there are in 728 days? A s. 104 weeks.

7. If one box will hold 80 pair of shoes, how many of such boxes will be required to contain 1840 pair? Ans. 23 boxes. 8. At the rate of 900 dollars each, how many dwelling houses could be built for 11700 dollars? Ans. 13 houses. 9. At 11 dollars each per month, how many laborers could be hired a month for 2530 dollars? Ans. 230 laborers. 10. At 120 dollars apiece, how many fine horses could I purchase for the sum of 4200 dollars? Ans. 35 horses.

11. What quantity of flour could be bought for when the price is 5 dollars per barrel ?

dollars,

One dollar would buy } of a barrel; hence 3 dollars would buy of a barrel.

Or, 3 dollars would buy the same part of a barrel that 3 dollars is of 5 dollars: 3 dollars is 3 of 5 dollars (§ 47).

Ans. of a barrel.

Ans. of a ton.

12. At the rate of 15 dollars per ton, what quantity of hay could be bought for 7 dollars? 13. If land sell at the rate of 50 dollars per acre, what quantity of land could be purchased for 17 dollars?

Ans. 1 of an acre. 14. A ferryboat is valued at 100 dollars. What share or interest in the boat could be purchased for 37 dollars ?

100

Ans. 37 of it. 15. A person who undertook a journey of 425 miles, having traveled 89 miles; what part of the journey has he accomplished? Ans. of it. 16. A manufactory is estimated to be worth 12000 dollars. What interest in it could be purchased for 2143 dollars?

17. What quantity of cloth, at 10 dollars a yard, ought I to have for 503 dollars?

The divisor being 10 we omit the in the right of the dividend, 503, and thus find 10 in 503, 50 times, with remainder 3.

The quotient is therefore 503•

The 50 times 10 in 503 gives 50 yards, and the 3 dollars over will buy of a yard. Ans. 50,3 yards.

18. At 3 dollars per cord, what quantity of wood might be bought for 175 dollars? Ans. 58 cords. 19. At the rate of 7 miles an hour, how many hours would a stage coach be in running 975 miles? Ans. 139 hours. 20. Allowing a workman to build 9 rods of fence in a day, how many days would he require to build 1603 rods?

21. How many tons of hay, at 11 purchased for 3071 dollars?

22. How many acres of land, at 40 bought for 7309 dollars?

23. How many tons of steel, at 120 bought for the sum of 1577 dollars?

Ans. 178 days. dollars per ton, may be

Ans. 279 tons. dollars per acre, may be

Ans. 182 acres. dollars per ton, may be Ans. 13 tons.

24. Allowing 30 days to make a month, how many months would there be in 200 days+177 days? Ans. 127 months. 25. A has 2345 dollars, and B 3000 dollars. What quantity of land can the two together purchase, at the rate of 50 dollars per acre? Ans. 106 acres.

26. A farmer sold beef for 130 dollars, and pork for 200 dollars. With the proceeds of these sales, he wishes to purchase corn at 3 dollars per barrel; what quantity can he buy? Ans. 110 barrels.

27. A merchant sold cloth for 423 dollars, cotton for 125 dollars, and silk for 300 dollars; and invested the proceeds in sugar at 12 dollars per barrel. How many barrels of sugar did he buy? Ans. 70 barrels.

28. If 5 acres of ground sell for 163 dollars, what is the price per acre?

One acre being of 5 acres, is worth is 163 dollars-5 (§ 51 and 54).

of 163 dollars; which Ans. 32 dollars.

29. If a man travels 200 miles in 6 days, at what rate does he travel per day?

Ans. 33 miles.

30. If 7 head of horses sell for 850 dollars, what will be the average sum received for each? Ans. 121 dollars.

31. Allowing 1703 acres of land to be divided into 8 farms of equal size, what will be the number of acres in each?

Ans. 212 acres.

32. If 100 yards of li en cost 57 dollars, what is the price per yard?

One yard costs

of 57 dollars, or 57 dollars÷100; which is, Ans. of a dollar.

57% of a dollar (§ 47 and 51).

100

а

57

33. If 50 bushels of corn sell for 13 dollars, what is the

selling price per bushel?

34. If 3 plantations of equal value what sum would be received for each ? 35. If 8 yards of Irish linen amount price per yard does the linen sell?

50

Ans. 1 of a dollar. sell for 8491 dollars, Ans. 2830 dollars. to 7 dollars, at what Ans. of a dollar.

36. If the construction of 4 bridges on a turnpike road, cost 14803 dollars, what is the average cost of each ?

Ans. 3700 dollars. 37. Allowing 600 acres of ground to produce 24000 bushels of wheat, what would be the produce per acre?

Ans. 40 bushels.

38. Allowing 900 barrels of flour to sell for 9900 dollars, what would be the selling price per barrel? Ans. 11 dollars. 39. A capital stock of 225000 dollars is held in 1000 equal shares. What is the amount of each share?

Ans. 225 dollars. 40. A farmer has 70 acres of land worth 2450 dollars, and 110 acres worth 4500 dollars. What is the whole worth, and what is each tract worth per acre? The whole, 6950 dollars;

41. A drover bought 54 head of cattle at one time, and 66 head at another time, the whole amounting to 1800 dollars. What was the average cost per head ? Ans. 15 dollars.

42. A planter bought 7 mules at 35 dollars apiece, 4 at 40 dollars apiece, and 9 at 37 dollars apiece. What sum was paid for all, and what was the average sum paid for each?

Ans. 738 dollars for all; and 3628 for each. 43. In how many days could 10 men accomplish the same amount of work that one man could do in 349 days?

10 men could do the work in of the time in which 1 man would do it; that is, in of 349 days. Ans. 34 days.

44. In how many days ought 7 masons to build a wall which one mason could build in 175 days? Ans. 25 days.

45. How long ought 20 men to subsist on a stock of provisions which would suffice one man 433 days? Ans. 2113 days. 46. How long ought 12 horses to be fed on a quantity of oats which would be sufficient for 1 horse 185 days? Ans. 15 days.