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 Long 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 dividends.

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.

The pupil should

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

14. Divide 203812983 by 5049.
15. Divide 20195411808 by 3012.
16. Divide 74855092410 by 949998.
17. Divide 47254149 by 4674.
18. Divide 119184669 by 38473.
19. Divide 280208122081 by 912314.
20. Divide 293839455936 by 8405.
21. Divide 4637064283 by 57606.
22. Divide 352107193214 by 210472.

23. Divide 558001172606176724 by 2708630425.
24. Divide 1714347149347 by 57143.

25. Divide 6754371495671594 by 678957
26. Divide 71900715708 by 37149.
27. Divide 571943007145 by 37149.
28. Divide 671493471549375 by 47143.
29. Divide 571943007645 by 37149.

30. Divide 171493715947143 by 57007.

31. Divide 121932631112635269 by 987654321.

NOTES.-1. How many operations are there in long division? Name

them.

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

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?

68. PRINCIPLES RESULTING FROM DIVISION. NOTES.-1st. When the divisor is 1, the quotient will be equal to the dividend.

2d. When the divisor is equal to the dividend, the quotient will be 1.

3d. When the divisor is less than the dividend, the quotient will be greater than 1. The quotient will be as many times greater than 1, as the dividend is times greater than the divisor.

4th. When the divisor is greater than the dividend, the quotient will be less than 1. The quotient will be such a part of 1, as the dividend is of the divisor.

PROOF OF MULTIPLICATION.

69. Division is the reverse of multiplication, and they prove each other. The dividend, in division, corresponds to the product in multiplication, and the divisor and quotient to the multiplicand and multiplier, which are factors of the product: hence,

If the product of two numbers be divided by the multiplicand, the quotient will be the multiplier; or, if it be divided by the multiplier, the quotient will be the multiplicand.

2. The multiplicand is 61835720, and the product 8162315040: what is the multiplier ?

3. The multiplier is 270000; now if the product be 1315170000000, what will be the multiplicand?

4. The product is 68959488, the multiplier 96: what is the multiplicand?

5. The multiplier is 1440, the product 10264849920: what is the multiplicand?

6. The product is 6242102428164, the multiplicand 6795634: what is the multiplier ?

CONTRACTIONS IN MULTIPLICATION.

70. To multiply by 25.

1. Multiply 275 by 25.

ANALYSIS.-If we annex two ciphers to the multiplicand, we multiply it by 100 (Art. 55): this product is 4 times too great; for the multiplier is but one-fourth of 100; hence, to multiply by 25,

OPERATION. 4)27500 6875

Annex two ciphers to the multiplicand and divide the result by 4.

1. Multiply 127 by 25. 2. Multiply 4269 by 25.

EXAMPLES.

3. Multiply 87504 by 25. 4. Multiply 704963 by 25.

71. To multiply by 121

1. Multiply 326 by 121.

ANALYSIS.-Since 12 is one-eighth of 100, Annex two ciphers to the multiplicand and divide the result by 8.

OPERATION.

8)32600

4075

ANALYSIS.-Annexing two ciphers to the mul OPERATION.

tiplicand, multiplies it by 100: but the multiplier

is but one-third of 100: hence,

Annex two ciphers and divide the result by 3.

EXAMPLES.

1. Multiply 889626 by 33. 2. Multiply 740362 by 331.

3)67500

22500

| 3. Multiply 5337756 by 331. 4. Multiply 2221086 by 331.

68. When the divisor is 1, what is the quotient? When the divisor is equal to the dividend, what is the quotient? When the divisor is less than the dividend, how does the quotient compare with 1? When the divisor is greater than the dividend, how does the quotient compare with 1? 69. If a product be divided by one of the factors, what is the quotient?

73. To multiply by 125.

1. Multiply 375 by 125.

ANALYSIS.-Annexing three ciphers to the multiplicand, multiplies it by 1000: but 125 is but one-eighth of one thousand: hence,

Annex three ciphers and divide the result by 8.

OPERATION.

8)375000

46875

3. Multiply 970406 by 125. 4. Multiply 704294 by 125.

74. By reversing the last four processes, we have the four following rules:

1. To divide any number by 25;

Multiply the number by 4, and divide the product by 100. 2. To divide any number by 121.

Multiply the number by 8, and divide the product by 100. 3. To divide any number by 331:

Multiply the number by 3, and divide the product by 100. 4. To divide any number by 125:

Multiply by 8, and divide the product by 1000.

EXAMPLES.

1. Divide 3175 by 25.
2. Divide 106725 by 25.
3. Divide 2187600 by 25.
4. Divide 2426225 by 25.
5. Divide 1762405 by 25.
6. Divide 4075 by 12.
7. Divide 3550 by 121.
8. Divide 592624 by 121.

9. Divide 880300 by 121. 10. Divide 22500 by 331. 11. Divide 654200 by 331. 12. Divide 7925200 by 33. 13. Divide 4036200 by 33. 14. Divide 93750 by 125. 15. Divide 3007875 by 125. 16. Divide 6758625 by 125.

70. What is the rule for multiplying by 25? 71. What is the rule for multiplying by 124? 72. What is the rule for multiplying by 334? 73. What is the rule for multiplying by 125?