CASE I.

70. When the divisor is less than 10.

1. Divide 86 by 2.

ANALYSIS.-There are 8 tens and 6 units to be divided by 2. We say, 2 in 8, 4 times, which being tens, we write it in the tens place. We then say, 2 in 6, 3 times, which being units, are written in the units place. Hence, the quotient is 43.

2. Divide 466 by 8.

ANALYSIS.-We first divide the 46 tens by 8, giving a quotient of 5 tens, and 6 tens over. These 6 tens are equal to 60 units, to which add the 6 in the units place. Then say, 8 in 66, 8 times and 2 over: hence, the quotient is 58, and a remainder of 2. This remainder is written after the last quotient figure, and the 8 placed under it; the quotient is read, 58 and 2 divided by 8.

OPERATION.

3. Let it be required to divide 30456 by 8. ANALYSIS.-We first say, 8 in 3 we can not. Then, 8 in 30, 3 times and 6 over; then, 8 in 64, 8 times; then, 8 in 5, 0 times; then, 8 in 56, 7 times.

Hence, we have the following

Rule.

8) 466

58-2 rem.

58 quot.

OPERATION.

8) 30456

3807

Begin

I. Write the divisor on the left of the dividend. ning at the left, divide each figure of the dividend by the divisor, and set each quotient figure under its dividend.

II. If there is a remainder after any division, annex to it the next figure of the dividend, and divide as before. III. If any dividend is less than the divisor, write ◊ for the quotient figure, and annex the next figure of the dividend, for a new dividend.

IV. If there is a remainder, after dividing the last figure, set the divisor under it, and annex the result to the quotient.

Proof.

Multiply the entire part of the quotient by the divisor, and to the product add the remainder if the work is right, the result will be equal to the dividend.

:

23. If it takes 5 bushels of wheat to make a barrel of flour, how many barrels can be made from 65890 bushels?

24. How many barrels of flour, at 7 dollars a barrel, can be bought for 609463 dollars?

25. A vessel sails 8 miles an hour : in how many hours

will it sail 3756 miles?

70. Give the rule for Division, when the divisor is less than 10. How do you prove Division?

26. Suppose a wheel is 7 feet in circumference: how many times would it turn, in going 15840 feet?

27. If a pace is 3 feet, how many paces will a man take in walking 6 miles, or 31680 feet?

28. When John starts, Joseph is 37594 feet ahead; Joseph goes 251 feet a minute, and John goes 260 feet a minute: in how many minutes will John overtake Joseph ?

29. A county contains 207360 acres of land, lying in 9 townships of equal extent: how many acres in each township?

30. An estate, worth 2943 dollars, is to be divided equally among a father, mother, 3 daughters, and 4 sons: what is the portion of each?

31. A railroad, worth 544806 dollars, is owned, in equal shares, by 9 persons: what is the value of the share of each?

CASE II.

71. When the divisor exceeds 9.

OPERATION.

1. Let it be required to divide 7059 by 13. ANALYSIS.-The divisor, 13, is not contained in 7 thousands; therefore, there are no thousands in the quotient. We then consider the 0 to be annexed to the 7, making 70 hundreds, and call this a partial dividend.

The divisor, 13, is contained in 70 hundreds, 5 hundreds times and something over. To find how much over, multiply 13 by 5 hundreds, and subtract the product, 65, from 70, and there will remain 5 hundreds, to which bring down the 5 tens, and consider the 55 tens a new partial dividend.

→→ Thous.

13) 7

Hunds.

6 5

55

5 2

39

39

0

Then, 13 is contained in 55 tens, 4 tens times and something over. Multiply 13 by 4 tens, and subtract the product, 52, from 55, and to the remainder, 3 tens, bring down the 9 units, and consider the 39 units a new partial dividend.

Then, 13 is contained in 39, 3 times. Multiply 13 by 3, and subtract the product, 39, from 39, and we find that the remainder is 0.

OPERATION.

26) 2756 (106

2. Let it be required to divide 2756 by 26. We first say, 26 in 27, once, and place 1 in the quotient. Multiplying by 1, subtracting, and bringing down the 5, we have 15 for the first partial dividend. We then say, 26 in 15, 0 times, and place the 0 in the quotient. We then bring down the 6, and find that the divisor is contained in 156, 6 times.

26

156

156

Hence, if any one of the partial dividends is less than the divisor, write 0 for the quotient figure, and bring down the next figure, forming a new partial dividend.

Rule.

I. Write the divisor on the left of the dividend.

II. Note the fewest figures of the dividend, at the left, that will contain the divisor, and set the quotient figure at the right of the dividend.

III. Multiply the divisor by the quotient figure, subtract the product from the first partial dividend, and to the remainder annex the next figure of the dividend, forming a second partial dividend.

IV. Find, in the same manner, the second and succeeding figures of the quotient, till all the figures of the dividend are brought down.

NOTE 1.—There are five operations in Division: 1st. To write down the numbers; 2d. Divide, or find how many times; 3d. Multiply; 4th. Subtract; 5th. Bring down, to form the partial dividend.

2. The product of a quotient figure by the divisor must never be larger than the corresponding partial dividend; if it is, the quotient figure is too large, and must be diminished.

3. When any one of the remainders is greater than the divisor, the quotient figure is too small, and must be increased.

4. The unit of any quotient figure is the same as that of the partial dividend from which it is obtained. The pupil should always name the unit of every quotient figure.

5. The unit of a remainder is the same as that of the dividend.

Proof.

72. In Division, the divisor shows into how many equal parts the dividend is divided: the quotient is one of these parts, and the remainder is what is left.

Hence, to prove Division,

Multiply the divisor by the quotient, and to the product add the remainder. If the work is right, the sum will be the same as the dividend.

Examples.

1. If 300 be divided into 60 equal parts, what is one of these parts?

2. How many times is 54 contained in 7574?

3. If 295470 be divided into 90 equal parts, what is one of these parts?

4. How many times is 37 contained in 7210449 ?

5. If 62205 dollars be divided equally among a regiment consisting of 957 men, how many dollars will each have?

6. What is one of the equal parts of the number 66708, when divided by 204 ?

71. What is the rule for division, when the divisor exceeds 9? NOTES.-1. How many operations are there in Division? Name them.

2. If a partial product is greater than the partial dividend, what does it indicate.

3. What do you do when any one of the remainders is greater than the divisor?

4. What is the unit of any figure of the quotient? When the divisor is contained in simple units, what will be the unit of the quotient figure? When it is contained in tens, what will be the unit of the quotient figure? When it is contained in hundreds? In thousands?

5. What is the unit of the remainder?

72. In Division, what does the divisor show? What the quotient? What is the remainder? How do you prove Division?