How do you proceed when the divisor is a composite number? Does it make any difference which factor we first divide by? When there are several remainders, explain how the true remainder is obtained.

EXAMPLES.

1. Divide 839 by 120. We will resolve 120 into the three factors, 4x5×6= 120. Now, proceding agreeably to the rule, we have the annexed operation.

OPERATION.

4)839

5)209 3 first rem.
6)41 4 second rem.
6 5-third rem.

Now, to obtain the true remainder, we have this

5x5+4=29. Again, 29 x 4+3=119.

Had there been more than three factors, the operation would have been equally simple, but a little more lengthy. 2. Divide 8217 by 35=5×7. Ans. 234 with 27 rem. 3. Divide 33678 by 15-3×5. Ans. 2245 with 3 rem. 4. Divide 9591 by 72=8×9. Ans. 133 with 15 rem. 5. Divide 10859 by 49=7×7. Ans. 221 with 30 rem.

CASE IV.

30. When the divisor ends with one or more ciphers. We have seen (ART. 4,) that a number is multiplied by 10 by annexing a cipher; it is multiplied by 100 by annexing two ciphers; by 1000 by annexing three ciphers, &c. Conversely, a number is divided by 10 by cutting off one figure from the right; it is divided by 100 by cutting off two figures from the right, &c.

EXAMPLES.

1. Divide 2475 by 20. Having cut off the 5 from the right of the dividend, and the 0 from the right of the divisor, which is, in effect, dividing both dividend and

OPERATION.

2|0)24715

123 15 remainder.

divisor by 10, we proceed to divide 247 by 2, (ART. 25.) We obtain 123 for a quotient and 1 for a remainder. This remainder is 1 ten, since it is a part of the 7 of the dividend which occupies the ten's place; annexing the 5 units, which was cut off, to the 1 ten which remained, we have 1 ten and 5 units, or 15 for the true remainder.

NOTE. This case may be comprised under CASE III., ART. 29. Thus, taking the preceding example, the divisor 20=2×10. Dividing 2475 first by 10, which division is effected by cutting off the right-hand figure, 5, we have 247 for the first quotient, and 5 for the first remainder. Next, dividing 247 by 2, we find 123 for the quotient sought, and 1 for the second remainder.

Now, by the rule under the case referred to, we find the true remainder to be 1x10+5=15.

Hence, the true remainder is 20 × 100+94—2094.
From the abové operations we deduce this

RULE.

Cut off from the right of the dividend as many figures as there are ciphers at the right of the divisor; divide what remains by the divisor without the ciphers at its right. To the final remainder annex the figures cut off from the divi dend, for the true remainder.

How do you proceed when there are ciphers at the right of the divisor?

3. Divide 7123545 by 421000. Ans. 16 and 387545 rem 4. Divide 1212121212 by 42000.

Ans. 28860 and 1212 rem. 5. Divide 123456789 by 12300. Ans. 10037 and 1689 rem. 6. Three men are to share equally in the sum of 1236 dollars. How many dollars will each have? Ans. 412 dolls. 7. Divide 1245 acres of land equally between five brothers. Ans. Each has 249 acres.

8. It is about 95000000 miles from here to the sun. Now, admitting that it requires 8 minutes for light to pass from the sun to the earth, how many miles does it pass one minute? Ans. 11875000 miles.

in

9. Allowing 22 bricks to be sufficient to make one cubic foot of masonry, how many cubic feet are there in a work which requires 100000 bricks?

Ans. 4545 cubic feet and 10 trick remaining

10. The circumference of the earth is about 25000 miles. How long would it require for a person to travel around it, if he could pass uninterruptedly at the rate of 200 miles per day? Ans. 125 days.

11. In 1845 the extent of post-roads in the United States was 143940 miles, and the amount paid for the transportation of the mail during the same year was 2905504 dollars. How much was the average expense per mile?

Ans. Between 20 and 21 dollars. 12. The distance of Uranus from the sun is about 1860624000 miles. How many hours would it require to pass this distance at 18 miles per hour? Also, how many days, and how many years, counting 24 hours to the day, and 365 days to the year?

It would require 103368000 hours.

13. How many barrels of apples, at 3 dollars per barrel, can I buy for 2568 dollars? And if one tree produce 8 barrels, how many trees will be required to yield the required amount ?

Ans.

{

856 barrels.

107 trees.

31. QUESTIONS INVOLVING THE FOUR GROUND RULES.

1. A person owes to one man 375 dollars, to another he owes 708 dollars, to a third man he owes 911 dollars. How much does he owe to the three men? Ans. 1994 dollars.

2. A farmer has sheep in five fields; in the first, he has 917; in the second, 249; in the third, 413; in the fourth, 1000; and in the fifth, he has 197. How many sheep has he in the five fields? Ans. 2776 sheep.

3. A person owes to one man 302 dollars, to another man he owes 707 dollars, and has owing to him 2000 dollars. How much will remain after paying his debts? Ans. 991 dollars.

4. A farmer receives for his wheat 103 dollars, for is corn 60 dollars, for his butter 511 dollars, for his cheese 1212 dollars, for his pork 601 dollars. He pays towards a new farm 1000 dollars, for a new wagon 50 dollars, for hired help on his farm 290 dollars, for repairing house 173 dollars. How much money has he remaining?

Ans. 974 dollars.

5. A person wills 1200 dollars to his wife, 300 dollars for charitable purposes, and what remains is to be equally divided among 6 children. Allowing his property to amount to 8562 dollars, how much would each child have? Ans. 1177 dollars.

E. A man gave 13558 dollars for a farm; he then sold 73 acres, at 75 dollars per acre; the remainder stood him in at 59 dollars per acre. How many acres did he purchase?

Ans. 210 acres.

7. Four boys divide 336 apples as follows: the first takes one sixth of the whole; the second takes one fourth of what was left; the third takes one half of what was then left; the fourth has the remainder. What number of apples did each boy have?

56.

The first had