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2. Divide 4567894 by 3700. OPERATION.

If, in this second example, we 37100)45678/94(1234 regard the divisor 3700 as a 37 composite number, whose fac

86 tors are 100 x 37=3700, the ex

74 ample will also properly come under Case III; according to

111 which, the 94 cut off from the

168 right of the dividend is to be

148 considered the first remainder,

2094=rem. and the 20 is the last remainder. Hence, the true remainder is 20 X 100+94=2094. From the above 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 whai 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 in one minute ?

Ans. 11875000 miles. 9. Allowing 22 bricks to be sufficient to make one cubic foot of masonry, how many cubic feet are there in å work which requires 100000 bricks?

Ans. 4545 cubic feet and 10 brick 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 listance 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. . Ans. 5 6 5 4307000 days. 16 6 "

11800 years. 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 ?

> 107 trees.

Ans 5 856 barrels.

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 dol. lars. How much will remain after paying his debts ?

Ans. 991 dollars. 4. A farmer receives for his wheat 103 dollars, for beis 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. €. 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 ? [The first had 56.

The second had 70.

The third had 105. (The fourth had 105.

8. An estate of 8100 dollars was divided among 9 children in the following way: the first had 100 dollars and one tenth of the remainder; after this the second had 200 dollars and one tenth of the residue ; again, the third had 300 dollars and one tenth of the remainder, and so on; each succeeding child had 100 dollars more than the one immediately preceding, and then one tenth of what still remained. What was the share of each?

Ana S They shared equal; each

I had 900 dollars. 9. A and B each owe C: A owes 1472 dollars, which is less than what B owes him, and yet the difference between A's and B's debts is 719 dollars. How much does B owe C?

Ans. 2191 dollars. 10. Admitting the earth to move 68000 miles per hour, how far will it move in one day; and how far in a year of 365 days? And I 1632000 miles in one day.

| 595680000 miles in one year. 11. If the President of the United States expends daily 60 dollars, how much will he be able to save at the end of the 365, out of his salary of 25000 dollars ?

Ans. 3100 dollars. 12. An army, consisting of 4525 men, have 103075 loc.ves of bread. At the end of 21 days, 500 men are killed in a battle Now, if each man in each day eat one loaf of bread, how many days after the battle will the bread sustain the army?

Ans. 2 days. 13. Two locomotives start from the same place, and move in the same direction; the first goes 25 miles each hour, the second only 15 miles. After the first has passed a distance of 100 miles it commences a backward motion, maintaining the same velocity, until it meets the second locomotive. How many hours after starting will they meet? And at what distance will they meet from the starting point? And They will meet in 5 hours,

I at a distance of 75 miles. 14. One hundred miles of railroad track are to be laid with heavy rail, requiring 116 tons to the mile. After receiving iron at 52 dollars per ton to lay 58 miles, the price per ton was increased so as to make the whole cosi of the entire road 612944 dollars. What was the lattir price per ton of the iron ?

Ans. 54 dollais.

FRACTIONS.

32. A fraction is a part of a unit.

Several methods are used to express fractions or parts of units, which give rise to several distinct kinds of fractions. Those usually employed in arithmetic are VULGAR or COMMON FRACTIONS, and DECIMAL FRACTIONS.

What is a fraction? What two methods are usually employed to express fractions ?

VULGAR FRACTIONS.

33. Vulgar fractions consist of two distinct parts or terms, the one written above the other, with a straight horizontal line between them, as in division, (Art. 23.) The number above the line is called the numerator. The number below the line is called the denominator. The

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