CONTRACTIONS IN DIVISION.

75. Contractions in Division are short methods of finding the quotient, when the divisors are composite numbers.

CASE I.

76. When the divisor is a composite number.

1. Let it be required to divide 1407 dollars equally among 21 men. Here the factors of the divisor are 7 and 3.

ANALYSIS.-Let the 1407 dollars be first divided into 7 equal piles. Each pile will contain 201 dollars. Let each pile be now divided into 3 equal parts. Each part will contain 67 dollars, and the number of parts will be 21: hence the following

OPERATION.

7)1407

3)201 1st quotient.
67 quotient sought.

RULE.-Divide the dividend by one of the factors of the divisor; then divide the quotient, thus arising, by a second factor, and so on, till every factor has been used as a divisor: the last quotient will be the answer.

NOTE.-It often happens that there are remainders after some of the divisions Ilow are we to find the true remainder?

74.-1. What is the rule for dividing by 25? 2. What is the rule for dividing by 12? 3. What is the rule for dividing by 334 ? 4. What is the rule for dividing by 125?

75. What are contractions in division? What is a composite number?

76. What is the rule for division when the divisor is a composite number?

77. Let it be required to divide 751 grapes into 16 equal

parts.

NOTE.-The factors of the divisor 16, are 4 and 4.

ANALYSIS.-If 751 grapes be divided by 4, there will be 187 bunches, each containing 4 grapes, and 3 grapes over.

The unit

of 187 is one bunch; that is, a unit 4 times as great as 1 grape.

If we divide 187 bunches by 4, we shall have 46 piles, each containing 4 bunches, and 3 bunches over: here, again, the unit of the quotient is 4 times as great as the unit of the dividend.

If, now we wish to find the number of grapes not included in the 46 piles, we have 3 bunches with 4 grapes in a bunch, and 3 grapes besides: hence, 4 × 3 = 12 grapes; and adding 3 grapes, we have a remainder, 15 grapes; therefore, to find the remainder, in units of the given dividend :

I. Multiply the last remainder by the last divisor but one, and add in the preceding remainder:

II. Multiply this result by the next preceding divisor, and add in the remainder, and so on, till you reach the unit of the dividend.

EXAMPLES.

1. Let it be required to divide 43720 by 45.

Divide the following numbers by the factors, for the divisors:

2. 956789 by 7 × 8=56.
3. 4870029 by 8 × 9=72.
4. 674201 by 10 × 11=110.
5. 445767 by 12 × 12=144.

6. 1913578 by 7 × 2 × 3=42. 7. 146187 by 3 × 5 × 7=105. 8. 26964 by 5 × 2 × 11=110. 9. 93696 by 3 × 7 × 11=231.

77. Give the rule for the remainder.

CASE II.

78. When the Divisor is 10, 100, 1000, &c.

ANALYSIS. Since any number is made up of units, tens, hundreds, &c. (Art. 28), the number of tens in any dividend will denote how many times it contains 1 ten, and the units will be the remainder. The hundreds will denote how many times the dividend contains 1 hundred, and the tens and units will be the remainder; and similarly, when the divisor is 1000, 10000, &c.; hence,

RULE.-Cut off from the right hand as many figures as there are ciphers in the divisor-the figures at the left will be the quotient, and those at the right, the remainder.

EXAMPLES.

1. Divide 49763 by 10.
2. Divide 7641200 by 100.

3. Divide 496321 by 1000. 4. Divide 61978 by 10000.

CASE III.

79. When there are ciphers on the right of the divisor. 1. Let it be required to divide 67389 by 700.

ANALYSIS.-We may regard the divisor as a composite number, of

OPERATION.

700)673189

96.. 1 remains. 189 true remain. Ans. 96488.

which the factors are 7 and 100. We first divide by 100 by striking off the 89, and then find that 7 is contained in the remaining figures, 96 times, with a remainder of 1; this remainder we multiply by 100, and then add 89, forming the true remainder 189: to the quotient 96, we annex 189 divided by 700, for the entire quotient: hence, the following

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 remaining 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 8749632 by 37000.

78. How do you divide when the divisor is 1 with ciphers annexed? Give the reason of the rule.

79. How do you divide when there are ciphers on the right of the divisor? How do you form the true remainder?

80. Abstractly, the object of division is to find from two given numbers a third, which, multiplied by the first, will produce the second. Practically, it has three objects:

1. Knowing the number of things and their entire cost, to find the price of a single thing:

2. Knowing the entire cost of a number of things and the price of a single thing, to find the number of things:

3. To divide any number of things into a given number of equal parts.

For these cases, we have from the previous principles (page 57), the following

RULES.

I. Divide the entire cost by the number of the things: the quotient will be the price of a single thing.

II. Divide the entire cost by the price of a single thing: the quotient will be the number of things.

III. Divide the whole number of things by the number of parts into which they are to be divided: the quotient will be the number in each part.

QUESTIONS INVOLVING THE PREVIOUS RULES.

1. Mr. Jones died, leaving an estate worth 4500 dollars, to be divided equally between 3 daughters and 2 sons: what was the share of each?

80. What is the object of division, abstractly? How many objects has it, practically? Name the three objects. Give the rules for the three cases.

2. What number must be multiplied by 124 to produce 40796?

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

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

5. The product of two numbers is 51679680, and one of the factors is 615: what is the other factor?

6. Bought 156 barrels of flour for 1092 dollars, and sold the same for 9 dollars per barrel: how much did I gain?

7. Mr. James has 14 calves worth 4 dollars each, 40 sheep worth 3 dollars each; he gives them all for a horse worth 150 dollars: what does he make or lose by the bargain?

8. Mr. Wilson sells 4 tons of hay at 12 dollars per ton, 80 bushels of wheat at 1 dollar per bushel, and takes in payment a horse worth 65 dollars, a wagon worth 40 dollars, and the rest in cash: how much money did he receive?

9. How many pounds of coffee, worth 12 cents a pound, must be given for 368 pounds of sugar, worth 9 cents a pound?

10. The distance around the earth is computed to be about 25000 miles: how long would it take a man to travel that distance, supposing him to travel at the rate of 35 miles a day?

11. If 600 barrels of flour cost 4800 dollars, what will 2172 barrels cost?

12. If the remainder is 17, the quotient 610, and the dividend 45767, what is the divisor?

13. The salary of the President of the United States is 25000 dollars a year: how much can he spend daily and save of his salary 4925 dollars at the end of the year?

14. A farmer purchased a farm for which he paid 18050 dollars. He sold 50 acres for 60 dollars an acre, and the remainder stood him in 50 dollars an acre: how much land did he purchase?

15. There are 31173 verses in the Bible: how many verses must be read each day, that it may be read through in a year?

16. A farmer wishes to exchange 250 bushels of oats at 42 cents a bushel, for flour at 7 dollars per barrel how many barrels will he receive?

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