19. Divide 352107193214 by 210472.

Ans.

--165534 rem.

20. Divide 558001172606176724 by 2708630425. Ans. --24 rem.

CONTRACTIONS IN DIVISION.

CASE I.

§ 36. When the divisor is a composite number.

RULE.

Divide the dividend by one of the factors of the divisor, and then divide the quotient thus arising by the other factor: the last quotient will be the one sought.

EXAMPLES.

Let it be required to divide 1407 dollars equally among Here the factors of the divisor are 7 and 3.

21 men.

Let the 1407 dollars be first divided equally among 7 men. Each share will be 201 dollars. Let each one of the 7 men divide his share into 3 equal parts, each

OPERATION.

7)1407

3)201 1st quotient.

67 quotient sought.

one of the three equal parts will be 67 dollars, and the whole number of parts will be 21; there the true quotient is found by dividing continually by the factors.

2. Divide 18576 by 48=4×12.

3. Divide 9576 by 72=9×8.
4. Divide 19296 by 96=12×8.

Ans. 387.

Ans. 133.

Ans. 201.

§ 37. It sometimes happens that there are remainders after division, for which we have the following

RULE,

The first remainder, if there be one, forms a part of the true remainder. The product of the second remainder, if there be one, by the first divisor, forms a second part. Either of these parts, when the other does not exist, forms the true remainder, and their sum is the true remainder when they both exist together.

EXAMPLES.

1. What is the quotient of 751 grapes, divided by 16?

4)751 4x4=164)187....3

46....3x4=12

3

15 the true remainder.

Ans. 4615.

DEMONSTRATION OF THE RULE.

In 751 grapes there are 187 sets, (say bunches,) with 4 grapes or units in each bunch, and 3 units over. In the 187 bunches there are 46 piles, 4 bunches in a pile, and 3 bunches over. But there are 4 grapes in each bunch; therefore, the number of grapes in the 3 bunches is equal to 4x3=12, to which add 3, the grapes of the first remainder, and we have the entire remainder 15.

Q. What is a composite number? (See $27, page 50.) How do you divide when the divisor is a composite number? When there is a remainder, how do you find the true remainder.

CASE II.

§ 38. When the divisor is 10, 100, 1000, &c.

RULE.

I. Cut off from the right hand of the dividend as many figures as there are O's in the divisor.

II. The left hand figures of the dividend will express the quotient, and the figures cut off the remainder.

EXAMPLES.

1. Divide 3256 by 100.

OPERATION.

100)32 56
Ans. 32100

In this example there are two O's in the divisor, therefore, there are two figures cut off from the right hand of the dividend, and the quotient is 32, and 56÷100

DEMONSTRATION OF THE RULE.

56

The quotient ought to be 10, 100, 1000, &c., times less than the dividend. But the same figure is 10, 100, 1000, &c., times greater or less in value, according to its distance from the unit's place. By cutting off figures from the right hand, the unit's place is removed to the left, and consequently the dividend is diminished 10, 100, 1000, &c., times, according as you cut off 1, 2, 3, &c., figures.

39. When there are ciphers on the right of the divisor.

RULE.

I. Cut off the ciphers by a line, and cut off the same number of figures from the right of the dividend.

II. Divide the remaining figures of the dividend by the significant figures of the divisor, and annex to the remainder, if there be one, the figures cut off from the dividend: this will form the true remainder.

EXAMPLES.

1. Divide 67389 by 700. In this example we strike off the 89, and then find that 7 is contained in the remaining figures, 96 times, with a remainder of 1; to this we annex 89, forming the remainder 189:

OPERATION.

700)67389

96...1 remains. 189 true remain.

Ans. 96488.

to the quotient 96 we annex 189 divided by 700 for the entire quotient.

DEMONSTRATION OF THE RULE.

The number 700=100×7. Hence it is a composite number of which the factors are 100 and 7.

In striking off the two figures 89, from the right of the dividend, we divide it by 100; we then divide the 673 by the other factor 7. We then multiply the remainder 1 by 100 and add 89 to the product, giving 189 for the true remainder, (see § 37.)

2. Divide 8749632 by 37000.

37000)87491632(236

Ans. 911

135803 400700

5. Divide 36599503 by 400700.

Q. How do you divide by 10, 100, 1000, &c.! (see $ 38.) Which part is the quotient? Which part is the remainder? When there are ciphers on the right of the divisor, how do you form the true remainder?

APPLICATIONS IN DIVISION.

OPERATION.

1. Divide 80 dollars equally among four men. Here the 80 dollars is to be divided into 4 equal parts, and the quotient 20 dollars expresses the value of one of the equal parts.

4)80

20 dollars.

2. Four persons buy a lottery ticket; it draws a prize of 10000 dollars: what is each one's share?

3. A person dying leaves an estate of 4500 dollars to be divided equally among 5 children: what is each one's share? Ans. 900 dollars.

4. There are 1560 eggs to be packed in 24 baskets: how many eggs will be put in each basket? Ans. 5. What number must be multiplied by 124 to produce Ans. 329.

40796?

6. How many times can 24 be subtracted from 1416?

Ans. 7. The sum of 19125 dollars is to be distributed among a certain number of men, each is to receive 425 dollars: how many men are to receive the

money ?

Ans.

8. By the census of 1840 the whole population of the 26 States was 16,890,320: if each one had contained an equal number of inhabitants, how many would there have been in each state? Ans. 649,62718

9. If a man walks 12775 miles in a year, or 365 days, how far does he walk each day? Ans. miles. 10. A farmer sells a drove of sheep for 2 dollars a head, and receives 1250 dollars: how many sheep did he sell? Ans. 625.

11. It is computed that the distance to the sun is 95,000,000 of miles, and that light is 8 minutes travelling from the sun to the earth: how many miles does it travel per minute?

Ans.

12. By the census of 1840 it appeared that the City of New York contained 312710 inhabitants; allowing 5 to each house, how many houses were there in the city at that time? Ans. 62,542.

13. A merchant has 5100 pounds of tea, and wishes to pack it in 60 chests: how many pounds must he put in each chest?

Ans.

14. A person goes to a store and buys a piece of cloth containing 36 yards, for which he pays 288 dollars: how much does he pay per yard? Ans. dollars. 15. There are 7 days in a week: how many weeks in Ans. 52 weeks and 1 day over. 16. There are 24 hours in a day: how many days in 2040 hours? Ans. days. 17. Twenty-three persons dined together, their bill was 92 dollars. how much had each one to pay ?

a year of 365 ?

Ans. 4 dollars.