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ART. 45. WHEN it is required to find how many times one number contains another, the process is called Division.
Ex. 1. A boy has 32 cents, which he wishes to give to 8 of his companions, to each an equal number; how many must each receive?
2 in 2 2 in 4 2 in 6 2 in 8
2 in 10 2 in 12
ILLUSTRATION. It is evident that each boy must receive as many cents as the number 8 is contained times in 32. We therefore inquire what number 8 must be multiplied by to make 32. By trial, we find that 4 is the number; because 4 times 8 make 32. Hence 8 is contained in 32, 4 times, and the boys receive 4 cents apiece.
2 in 14 2 in 16
2 in 18
2 in 20 2 in 22 2 in 24
The following table should be studied by the learner to aid him in solving questions in Division:
6 in 6
6 in 12
6 in 18 6 in 24 6 in 30 6 in 36 6 in 42
1 time 7 in 2 times 7 in 3 times 7 in 4 times 7 in 5 times 7 in 6 times 7 in 7 times
6 in 48
8 times 7 in
6 in 60 10 times 7 in
10 in 10 10 in 20 10 in 30
5 times 3 in
6 times 3 in
10 times 3 in
6 in 66 11 times 7 in 6 in 72 12 times 7 in
10 in 40
10 in 50 10 in 60 10 in 70 10 in 80
10 in 90 10 in 100
§ V. DIVISION.
4 times 11 in
5 times 11 in
6 times 11 in 7 times 11 in 8 times 11 in 9 times 11 in 10 times 11 in
1 time 10 in
2 times 10 in 120
2. A farmer received 8 dollars for 2 sheep; what was the
price of each ?"
ILLUSTRATION. It is evident, if he received 8 dollars for 2 sheep, for 1 sheep he must receive as many dollars as 2 is contained times in 8; 2 is contained in 8, 4 times, because 4 times 2 are 8. Hence 4 dollars was the price of each sheep.
3. A man gave 15 dollars for 3 barrels of flour; what was the cost of each barrel?
4. A lady divided 20 oranges among her 5 daughters; how many did each receive?
5. If 4 casks of lime cost 12 dollars, what costs 1 cask?
6. A laborer earned 48 shillings in 6 days; what did he receive per day?
7. A man can perform a certain piece of labor in 30 days; how long will it take 5 men to do the same?
8. When 72 dollars are paid for 8 acres of land, what costs 1 acre ? What cost 3 acres?
9. If 21 pounds of flour can be much can be obtained for 1 dollar? How much for 9 dollars?
10. Gave 56 cents for 8 pounds of raisins; what costs 1 pound? What cost 7 pounds?
11. If a man walk 24 miles in 6 hours, how far will he walk in 1 hour? How far in 10 hours?
12. Paid 56 dollars for 7 hundred weight of sugar; what costs 1 hundred weight? What cost 10 hundred weight?
13. If 5 horses will eat a load of hay in 1 week, how long would it last 1 horse?
14. In 20, how many times 2? many times 5? How many times 15. In 24, how many times 3? many times 6? How many times 16. How many times 7 in 21? In 14? In 63 ? In 77? In 70? 17. How many times 6 in 12? In 60? In 42? In 48? In 72? 18. How many times 9 in 27? In 99? In 108?
obtained for 3 dollars, how How much for 8 dollars?
19. How many times 11 in 22? In 110? In 132?
20. How many times 12 in 36? In 120? In 144 ?
ART. 46. The pupil will now perceive, that
DIVISION is the process of finding how many times one number is contained in another.
In division there are three principal terms; the Dividend, the Divisor, and the Quotient, or answer.
The dividend is the number to be divided.
The divisor is the number by which we divide.
The quotient is the number of times the divisor is contained in the dividend.
When the dividend does not contain the divisor an exact number of times, the excess is called a remainder, and may be regarded as a fourth term in the division. The remainder, being part of the dividend, will always be of the same denomination or kind as the dividend, and must always be less than the divisor.
ART. 47. SIGNS. The sign of division is a short horizontal line, with a dot above it and another below; thus, . It shows that the number before it is to be divided by the number after it. The expression 6 ÷ 2 = 3 is read, 6 divided by 2 is equal to 3.
Division is also indicated by writing the dividend above a short horizontal line and the divisor below, thus g. The expression 3 is read, 6 divided by 2 is equal to 3.
There is a third method of indicating division by a curved line placed between the divisor and dividend. Thus the expression 6)12 shows that 12 is to be divided by 6.
EXERCISES FOR THE SLATE.
ART. 48. The method of operation by Short Division, or when the divisor does not exceed 12.
Ex. 1. Divide 7554 dollars equally among
Ans. 1259 dollars.
* Quotient is derived from the Latin word quoties, which signifies how often, or how many times.
Art. 46. What is division ? What are the three principal terms in division? What is the dividend? What is the divisor? What is the quotient? What the remainder? What will be the denomination of the remainder ? How does it compare with the divisor? Art. 47. What is the first sign of division, and what does it show? What is the second, and what does it show? What is the third, and what does it show? - Art. 48. What is short division?
In performing this question, we first inquire how many times 6, the divisor, is contained in 7, the first figure of the dividend, and find it to be 1 time, and 1 remaining. We write the quotient figure, 1, directly under the 7, and then imagine the 1 remaining to be placed before the next figure of the dividend, which is 5, thus forming the number 15. We then inquire how many times 6, the divisor, is contained 15, and find it to be 2 times, with a remainder of 3. The 2 being written under the 5, we next imagine the 3 (remainder) to be placed before the next figure of the dividend, which is 5, and we have 35, which, being divided by 6, gives 5 for the quotient, with 5 for a remainder. Writing down the quotient figure as before, and imagining the remainder to be placed before the 4 in the dividend, we have 54, which, divided by 6, gives 9 as a quotient, which we write under the figure, and which completes the operation, giving 1259 as the quotient, or the number of times which the dividend 7554 contains the divisor 6.
Divisor 6)755 4 Dividend. 1259 Quotient.
ART. 49. From the foregoing illustration we deduce the following
RULE.-1. Write the divisor at the left hand of the dividend, with a curved line between them, and draw a horizontal line under the dividend.
2. Then inquire how many times the divisor is contained in the lefthand figure of the dividend, or figures if more than one is necessary to contain it, and place the result below the line, directly under the last figure of the dividend taken, as the first figure of the quotient. If there be no remainder, proceed in the same manner with each of the subsequent figures of the dividend.
3. But if there be a remainder after dividing the first or any subsequent figure of the dividend, regard that remainder as prefixed to the next figure of the dividend, and then inquire how many times the divisor is contained in the number thus formed, and place the quotient figure underneath, as before. Proceed in this way, until all the figures of the dividend are divided.
4. If, in any instance, the divisor is greater than the figure to be divided in the dividend, and therefore cannot be contained in it, a cipher must be written in the quotient, and the undivided figure must be regarded as prefixed to the next figure of the dividend.
5. If there is a remainder after dividing the last figure of the dividend, it may be placed at the right hand of the quotient and marked Remainder, or be written over the divisor, with a horizontal line betreen them, and annexed to the quotient.
QUESTIONS. How are the numbers arranged for short division ? At which hand do you begin to divide? Why not begin at the right, where you begin to multiply? Where do you write the quotient? If there is a remainder after dividing a figure, what is done with it?-Art. 49. What is the rule for short division?
ART. 50. First Method of Proof.-Multiply the divisor and quotient together, and to the product add the remainder, if there is any, and, if the work is right, the sum thus obtained will be equal to the dividend.
NOTE. It will be seen, from this method of proof, that division is the reverse of multiplication. The dividend answers to the product, the divisor to one of the factors, and the quotient to the other.
EXAMPLES FOR PRACTICE.
17. Divide 5678956 by 5. 18. Divide 1135791 by 7. 19. Divide 1622550 by 8. 20. Divide 2028180 by 9. 21. Divide 2253530 by 12. 22. Divide 1877940 by 11.
Sum of the quotients,
Art. 50. How is short division proved? Of what is division the reverse? To what do the three terms in division answer in multiplication? What, then, is the reason for this proof of division?