tient thus obtained by another, and so on if there be more than two factors, until every factor has been made a divisor. The last quotient will be the quotient required.

TO FIND THE TRUE REMAINDER.

121. If remainders occur in successive division, it is evident that the true remainder must be the least number, which, subtracted from the given dividend, will render all the divisions

exact

1. Divide 5855 by 168, using the factors 3, 7, and 8, and find the true remainder.

=

Dividing 1951 by 7, we have 278 for a quotient, and a remainder of 5. Hence, 5 subtracted from 1951 would render the second division exact. But to diminish 1951 by 5 would require us to diminish 1951 × 3, the dividend of the first exact division, by 5 × 3 15, (93, III); and we therefore write 15 for the second part of the true remainder. Dividing 278 by 8, we have 34 for a quotient, and a remainder of 6. Hence, 6 subtracted from 278 would render the third division exact. But to diminish 278 by 6 would require us to diminish 278 × 7, the dividend of the second exact division, by 6 × 7; or 278 × 7 × 3, the dividend of the first exact division, by 6 x 7 x 3 = 126; and we therefore write 126 for the third part of the true remainder. Adding the three parts, we have 143 for the entire remainder.

Hence the following

RULE. I. Multiply each partial remainder by all the preceding divisors.

II. Add the several products; the sum will be the true re mainder.

10. Divide 10197 by 120 2 x 3 x 4 x 5.

11. Divide 29792 by 144

12. Divide 73522 by 168

=

Ans. 29.

Ans. 364.

8 x 7 x 3.

Ans. 5789.

= 9 x 7 x 2.

7 x 5 x 3.

Ans. 5832.

Rem. 13.

= 3 x 5 x 7.

103.

117.

3 x 8 x 6.

128.

4 x 6 x 7.

106.

13. Divide 63844 by 135 = 3 x 5 x 9.

124.

14. Divide 386639 by 720 = 2 × 3 × 4 × 5 × 6.

122. When the divisor is a unit of any order.

If we cut off or remove the right hand figure of a number, each of the other figures is removed one place toward the right, and, consequently, the value of each is diminished tenfold, or divided by 10, (57, III). For a similar reason, by cutting off two figures we divide by 100; by cutting off three, we divide by 1000, and so on; and the figures cut off will constitute the remainder. Hence the

RULE. From the right hand of the dividend cut off as many figures as there are ciphers in the divisor. Under the figures so cut off, place the divisor, and the whole will form the quotient.

123. When there are ciphers on the right hand of

the divisor.

1. Divide 25548 by 700.

OPERATION.

7/00) 255/48

36 Quotient. 3 20 rem
3 x 100+ 48 = 348 true rem.

ANALYSIS. We resolve 700 into the factors 100 and 7. Dividing first by 100, the quotient is 255, and the remainder 48. Dividing 255 by 7, the final quotient is 36, and the second remainder 3. Multiplying the last remainder, 3, by the preceding divisor, 100, and adding the preceding remainder, we have 300+ 48348, the true remainder, (121). In practice, the true remainder may be obtained by prefixing the second remainder to the first. Hence the

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

II. Divide the remaining figures of the dividend by the remaining figures of the divisor, for the final quotient.

III. Prefix the remainder to the figures cut off, and the result will be the true remainder.

EXAMPLES COMBINING THE PRECEDING RULES.

1. How many barrels of flour at $8 a barrel, will pay for 25 tons of coal at $4 a ton, and 36 cords of wood at $3 a cord?

Ans. 26.

2. A grocer bought 12 barrels of sugar at $16 per barrel, and 17 barrels at $13 per barrel; how much would he gain by selling the whole at $18 per barrel?

3. A farmer sold 300 bushels of wheat at $2 a bushel, corn and oats to the amount of $750; with the proceeds he bought 120

head of sheep at $3 a head, one pair of oxen for $90, and 25 acres of land for the remainder How much did the land cost him per

acre?

Ans. $36.

4. Divide 450+(24—12) × 5 by (90÷6) + (3 × 11) — 18.

5. Divide 648 × (32 × 23) (4375 ÷ 175) × 42 + 32.

Ans. 17.

9

(291015) by 2863 ÷ Ans. 7129.

6. The product of three numbers is 107100; one of the numbers is 42, and another 34. What is the third number?

Ans. 75. 7. What number is that which being divided by 45, the quotient increased by 72 +1, the sum diminished by the difference between 28 and 16, the remainder multiplied by 6, and the product divided by 24, the quotient will be 12? Ans. 450.

8. A mechanic earns $60 a month, but his necessary expenses are $42 a month. How long will it take him to pay for a farm of 50 acres worth $36 an acre?

9. What number besides 472 will divide 251104 without a remainder?

Ans. 532.

10. Of what number is 3042 both divisor and quotient?

Ans. 9253764.

11. What must the number be which, divided by 453, will give the quotient 307, and the remainder 109? Ans. 139180.

12. A farmer bought a lot of sheep and hogs, of each an equal number, for $1276. He gave $4 a head for the sheep, and $7 a

head for the hogs; what was the whole number purchased, and how much was the difference in the total cost of each?

Ans. 232 purchased; $348 difference in cost.

13. According to the census of 1850 the total value of the tobacco raised in the United States was $13,982,686. How many school-houses at a cost of $950, and churches at a cost of $7500, of each an equal number, could be built with the proceeds of the tobacco crop of 1850? Ans. 1654, and a remainder of $6386.

14. The entire cotton crop in the United States in 1859 was 4,300,000 bales, valued at $54 per bale. If the entire proceeds were exchanged for English iron, at $60 per ton, how many tons would be received?

15. The population of the United States in 1850 was 23,191,876. It was estimated that 1 person in every 400 died of intemperance. How many deaths may be attributed to this cause in the United States, during that year?

16. In 1850, there were in the State of New York, 10,593 public schools, which were attended during the winter by 508464 pupils; what was the average number to each school?

Ans. 48.

17. A drover bought a certain number of cattle for $9800, and sold a certain number of them for $7680, at $64 a head, and gained on those he sold $960. How much did he gain a head, and how many did he buy at first?

Ans. Gained $8 per head; bought 175. 18. A house and lot valued at $1200, and 6 horses at $95 each, were exchanged for 30 acres of land. At how much was the land valued per acre ?

19. If 16 men can perform a job of work in 36 days, in how many days can they perform the same job with the assistance of 8 more men? Ans. 24.

20. Bought 275 barrels of flour for $1650, and sold 186 barrels of it at $9 a barrel, and the remainder for what it cost. How much was gained by the bargain? Ans. $558.

21 A grocer wishes to put 840 pounds of tea into three kinds of boxes, containing respectively 5, 10, and 15 pounds, using the