Rule for Case III.-Multiply without regarding the ciphers on the right of the factors; then annex to the product as many ciphers as are at the right of both factors. V. DIVISION. ART. 36. 1. If you divide 6 apples equally between 2 boys, how many will each have? ANALYSIS.-It will require two apples to give each boy 1; hence, each boy will have as many apples as there are times 2 apples in 6 apples, that is 3. How many times 2 in 6?-Ans. 3. Why? Because 3 times 2 are 6. Why? 2. If you divide 8 peaches equally between 2 boys, how many will each have? Ans. 4. 3. How many times 2 in 10-? Why? REVIEW.-33. What is a composite number? What are its component parts or factors? Give an example. ber, Rule? Illustrate this method. of a number differ from its parts? How multiply by a composite numREM. In what respect do the factors Give an example. 34. If one cipher is placed on the right of a number, how are the orders changed? If two ciphers? How multiply by 10, 100, 1000, &c.? DEFINITIONS. The process by which the preceding examples are solved, is called Division; hence, Division is the process of finding how many times one number is contained in another. The number by which to divide, is the divisor. The number denoting how many times the divisor is contained in the dividend, is the quotient. ART. 37. How many times 3 in 12? Ans. 4 times. Here, 3 is the divisor, 12 the dividend, and 4 the quotient. Since 3 is contained in 12 four times, 4 times 3 are 12; that is, the divisor and quotient multiplied, produce the dividend. Hence, since 3 and 4 are factors of the product 12, the divisor and quotient correspond to the factors in Multiplication; the dividend, to the product. BY MULTIPLICATION, Dividend. Factors. Product. 3X4 = 12 BY DIVISION, 12 divided by 3 Or Divisor. Quotient. = 4 Hence, Division is the process of finding one of the factors of a product, when the other factor is known. ART. 38. If 7 cents be divided equally among 3 boys, each boy would receive 2 cents, and there would be 1 cent left, or remaining undivided. The number left after dividing, is called the remainder. REMARKS.-1. Since the remainder is a part of the dividend, it must be of the same denomination. If the dividend be dollars, the remainder will be dollars if pounds, the remainder will be pounds. : 2. The remainder is always less than the divisor; for, if it were equal to, or greater than it, the divisor would be contained at least once more in the dividend. 3. If the dividend denotes things of one denomination only, the operation is called Simple Division. REVIEW.-36. What is Division? What is the number by which to divide? What the number to be divided? What the number denoting how many times the divisor is contained in the dividend? ART. 39. A boy has 8 cents: how many lemons can he buy at 2 cents each? ANALYSIS.-He can buy 4, because 4 lemons at 2 cents each, will cost 8 cents. If he did not know that 4 times 2 are 8, the operation would be thus: The boy would give 2 cents for 1 lemon, and then have 6 cents left. After giving 2 cents for the 2d lemon, he would have 4 cents left; Then giving 2 cents for the third, he would have 2 cents left; Lastly, after giving two cents for the fourth, he would have nothing left: having taken 2 cents 4 times from 8 cents, and each time received one lemon. Left, 2d lemon 8 cents. 1st lemon 2 cents. 6 cents. 2 cents. Left, 4 cents. 2 cents. Left, 2 cents. ... 2 cents. 3d lemon 4th lemon The natural method of performing this operation is by Subtraction; but, When it is known how many times 2 can be subtracted from 8, instead of subtracting 2 four times, say, 2 in 8 four times, and 4 times 2 are 8; which, subtracted from 8 once, nothing is left. The last method is by Division, and it differs from the first in this that the subtractions, instead of being performed separately, are all made at once. Hence, Division may be termed a short method of making many subtractions of the same number. The divisor is the number subtracted; the dividend REVIEW.-37. Three in 12, 4 times; what is 3 called? 12? 4? To what is the product of the divisor and quotient equal? To what do they correspond? To what does the dividend correspond? What is division? 38. When there is a number left after dividing, what is it called? REM. Of what denomination is the remainder? Why? Why is the remainder always less than the divisor? When the dividend denotes things of one denomination only, what is the operation called? 39. To what natural method do the operations in division belong? IlWhat is the lustrate by an example. What may division be termed? divisor? Dividend? Quotient? the number from which the subtractions are made; the quotient shows how many subtractions have been made. ART. 40. DIVISION IS DENOTED BY THREE SIGNS: 2d. 3d. 12÷÷3 means that 12 is to be divided by 3. Use the 1st sign when the divisor does not exceed 12; draw a line under the dividend, and write the quotient beneath. If the divisor exceeds 12, draw a curved line on the right of the dividend: place the quotient on the right of this. The sign,÷, in the Table, is read divided by. PRINCIPLES AND EXAMPLES. ART 41. 1. I wish to put 15 hats into boxes, each containing 3 hats: how many boxes do I need? 1ST SOLUTION.-I need as many boxes as 3 hats are contained times in 15 hats; that is, 5 boxes. Hats. Hats. 3) 15 (5, boxes. 15 2. Having 15 hats, I wish to separate them into 5 equal lots: how many hats will there be in each lot? 5) 15(3, hats in each. 2D SOLUTION.-Putting one hat into Hats. Hats. each lot will require 5 hats; hence, there will be as many hats in each lot, as there are times 5 hats in 15. 15 The first solution shows, that by Division a number can be separated into parts containing a certain number of units, and the number of parts found. The second solution shows, that by Division a given number can be separated into a certain number of equal parts, and the number of units in each part found. REVIEW.-10. How many signs are used to denote division? What is the first? Secord? Third? Illustrate their meaning. 41. What does the first solution show? What the second? REM. How does it appear that the divisor and dividend are both of the same denemination? Is the quotient an abstract or a concrete number? What does it show? What may it represent? |