Art. 43. RULE FOR LONG DIVISION. 1. Write the divisor at the left of the dividend, draw a curved line between them, and also at the right of the dividend, to separate it from the quotient.

2. Take as many of the left-hand terms of the dividend as will contain the divisor, for a partial dividend; find how many times this will contain the divisor, and write the quotient at the right of the dividend for the first left-hand term of the quotient.

3. Multiply the divisor by the quotient term found, write the product under the partial dividend used, and subtract.

4. To the remainder annex the next term of the dividend for a second partial dividend, and divide, multiply, and subtract, as before.

5. Proceed in this manner until all the terms of the dividend have been used.

NOTE.-When any partial dividend does not contain the divisor, write a cipher in the quotient, and annex another term of the dividend to form a new partial dividend.

Art. 44. When one or more of the right-hand figures of the divisor are ciphers —

1. Cut off the ciphers from the right of the divisor, and an equal number of figures from the right of the dividend.

2. Divide the new dividend thus formed by the new divisor, and the result will be the quotient.

3. Prefix the remainder, if there be one, to the figures cut off from the dividend, and the result will be the true remainder.

Art. 45. To divide any number by 10, 100, 1000, etc.,

Cut off as many figures from the right as there are ciphers in the divisor. The figures cut off are the true remainder.

LESSON VI.

MISCELLANEOUS REVIEW PROBLEMS.

1. The sum of two numbers is 15 and one of the numbers is 6: what is the other?

2. The difference between two numbers is 8 and the smaller number is 9: what is the larger?

3. The product of two numbers is 56 and one of the numbers is 7: what is the other?

4. The quotient of two numbers is 6 and the divisor is 8: what is the dividend?

5. How many barrels of flour, at $8 a barrel, will pay for 24 yards of carpeting, at $2 a yard?

6. How many tons of coal, at $9 a ton, will pay for 15 cords of wood, at $6 a cord?

7. A grocer bought 7 barrels of flour at $6 a barrel: for how much a barrel must be sell it to gain $14 on the lot?

8. If 1 man can build a wall in 36 days, how many men can build it in 4 days?

9. If 6 men can do a piece of work in 8 days, how many men can do it in 12 days?

10. Two vessels start from the same port and sail in the same direction, one sailing 12 miles an hour and the other 9 miles an hour: how far apart will they be in 10 hours?

WRITTEN EXERCISES.

1. The greater of two numbers is 4056 and their difference is 3650: what is the less number?

2. The subtrahend is 34203 and the remainder is 8706: what is the minuend?

3. The divisor is 534 and the quotient 43: what is the dividend?

4. The product of two numbers is 5328 and one of the numbers is 148: what is the other?

5. Multiply 486 + 392 by their difference.

6. Divide the product of 48 and 24 by their differ

ence.

7. A merchant bought 35 yards of cloth at $56, and sold it at $2 a yard: how much did he gain?

8. A drover bought 240 sheep at $8 a head, and then sold 90 of them at $12 a head, 75 at $9 a head, and the rest at $6 a head: how much did he gain?

9. A farmer exchanges 65 bushels of wheat at $2 a bushel, and 35 sheep at $6 a head, for cows at $34 a head: how many cows did he receive?

10. A man's income is $3500 a year; he pays $450 a year for house-rent, $150 for taxes, $350 for hired help, and $45 a month for other expenses: how much has he left at the close of the year?

11. A man bought 80 acres of land at $35 an acre, paid $325 for improvements, and then sold it for $3750: how much did he gain?

12. A grain merchant having 3500 bushels of oats, sold 1650 bushels, and then bought twice as much as he had left: how many bushels did he buy?

13. A man left an estate to his wife and three children; the wife received $4500; the youngest child, $1500; the second child, $1850; and the eldest

child, as much as both of the others less $1350: what was the value of the estate?

14. A and B start together on a journey, A traveling 28 miles a day and B 33 miles: how far apart will they be in 12 days?

15. A and B start together and travel in opposite directions, A going 28 miles a day and B 33 miles: how far apart will they be in 12 days?

QUESTIONS FOR REVIEW.

What is addition? What is meant by sum or amount? What does it contain? What is meant by like numbers? What numbers can be added? What is the sign of addition? What does it show? Give the rule for addition. What is the method of proof?

What is subtraction? The difference, or remainder? The minuend? The subtrahend? What numbers can be subtracted? What does the sum of the remainder and subtrahend equal? What is the sign of subtraction? What does it show? Give the rule for subtraction. What is a method of proof?

What is multiplication? The multiplicand? The multiplier? The product? Of what are the multiplicand and multiplier factors? What is the sign of multiplication? What does it show? How may the product be obtained by addition?

Give the rule for multiplication. How may you multiply when either the multiplicand or multiplier, or both, end in ciphers? How may any number be multiplied by 10, 100, 1000, etc.?

What is division? The dividend? The divisor? The quotient? The remainder? What is the sign of division? What does it show? In what other way may division be expressed? How many times may the divisor be subtracted from the dividend? Of what is division the reverse?

What is short division? When is it used? Give the rule. What is long division? Give the rule. How do you proceed when a partial dividend will not contain the divisor? How may you divide when the divisor ends in ciphers? How may any number be divided by 10, 100, 1000, etc.?

SECTION VI.

PROPERTIES OF NUMBERS.

LESSON I.

Divisor, Greatest Common Divisor, and Factor.

NOTE. The term number used in this section, denotes an integer.

1. What numbers besides itself and 1 will exactly divide 15? 21? 25? 30? 56? 63?

2. What numbers besides itself and 1 will exactly divide 7? 11? 13? 17? 23? 37? 41?

3. What numbers will exactly divide 4? 5? 16? 19? 24? 29? 33? 31? 42?

4. What are the divisors of 10? 28? 31? 33? 43? 49? 53? 55? 70? 90? 99?

NOTE. Since every number is exactly divisible by itself and 1, these divisors need not be given.

5. What number is a divisor of both 9 and 12? 15 and 20? 24 and 27? 42 and 56?

6. What divisor is common to 28 and 35? 27 and 36? 42 and 54? 63 and 81?

7. What is a common divisor of 15 and 30? 45 and 60? 50 and 75? 60 and 84?