NOTE. In Ex. 44 write the part of the dividend left after division as a remainder; in Ex. 45 express it as a fraction; and in Ex. 46 extend the division to two decimal places. 31. Principles of Multiplication and Division. A man whose property is in uncertain investments, and whose relatives are scattered over different parts of the world, leaves a will which divides all his property equally among all his relatives. An estimate is made of the probable value of the property and of the probable number of relatives. From these estimates, by what process can the probable amount to be received by each relative be obtained? The property to be divided is what element of the operation? 1. Suppose that the value of the property is found to be twice the estimated value. How will the amount received by each relative compare with the estimated amount? Multiplying the dividend has, then, what effect on the quotient? 2. Suppose that the value of the property is found to be only one-half the estimated value? How will the amount received by each relative compare with the estimated amount? Dividing the dividend has, then, what effect on the quotient? 3. Suppose that the number of relatives is found to be twice the estimated number. How will the amount received by each relative compare with the estimated amount? Multiplying the divisor has, then, what effect on the quotient? 4. Suppose that the number of relatives is found to be only one-half the estimated number. How will the amount received by each relative compare with the estimated amount? Dividing the divisor has, then, what effect on the quotient? 5. Suppose that the amount of property is found to be twice the estimated amount, and the number of relatives to be twice the estimated number. How will the amount received by each relative compare with the estimated amount? Multiplying both dividend and divisor has, then, what effect on the quotient? 6. Suppose that the amount of property is found to be only one-half the estimated amount, and the number of relatives to be only one-half the estimated number. How will the amount received by each relative compare with the estimated amount? Dividing both dividend and divisor by the same number has, then, what effect on the quotient? The boys of a certain school determine to secure a base-ball outfit. An estimate is made of the cost of the outfit and of the probable number of boys who will contribute to its purchase, From these estimates, by what process can the probable amount to be contributed by each boy be obtained? The cost of the outfit is what element of the operation? The cost to each boy? 1. Explain two ways by which the cost to each boy may be wice the estimated cost. In what two ways, then, can a quotient be multiplied? 2. Explain two ways by which the cost to each boy may be only one-half the estimated cost. In what two ways, then, can a quotient be divided? 3. Explain how the cost to each boy may be exactly the estimated cost. (1) Although the cost of the outfit is twice the estimated cost. (2) Although the cost of the outfit is only one-half the estimated cost. (3) Although the number of boys is twice the estimated number. (4) Although the number of boys is only one-half the estimated number. How, then, can we counteract the effect on the quotient (1) Of multiplying the dividend by a number? (2) Of dividing the dividend by a number? (3) Of multiplying the divisor by a number? For convenience of reference, we express the preceding laws in the following PRINCIPLES OF DIVISION. 1. Multiplying the dividend multiplies the quotient. 2. Dividing the dividend divides the quotient. 3. Multiplying the divisor divides the quotient. 4. Dividing the divisor multiplies the quotient. 5. Multiplying both dividend and divisor does not change the quotient. 6. Dividing both dividend and divisor does not change the quotient. The preceding principles may be expressed in the following CONDENSED PRINCIPLES. 1. A change in the dividend, by multiplication or division, produces a like change in the quotient. 2. A change in the divisor, by multiplication or division, produces an opposite change in the quotient. 3. A like change in both dividend and divisor, by multiplication or division, does not change the quotient. A man about to visit a neighboring city decides to make extensive additions to his library. He estimates the number of books he will need to purchase and the average price of each book. From these estimates, by what process can he determine the probable amount to be expended? The cost of each book is what element of the operation? The total cost? 1. Suppose that the average cost of each book is twice the estimated cost. How will the total cost compare with the estimated cost? Suppose that the number of books that he decides to purchase is twice the estimated number. How will the total cost compare with the estimated cost? Multiplying either factor has, then, what effect upon the product? 2. Suppose that the cost of each book is but one-half the estimated cost. How will the total cost compare with the estimated cost? Suppose that the number of books he decides to purchase is but one-half the estimated number. How will the total cost compare with the estimated cost? Dividing either factor has, then, what effect on the product? 3. Suppose that the cost of each book is twice the estimated cost, but that he decides to purchase only one-half the estimated number; or suppose that the cost of each book is only one-half the estimated cost, but that he decides to purchase twice the estimated number. In either case, how does the total cost compare with the estimated total cost? Multiplying one factor and dividing the other by the same number has, then, what effect upon the product? For convenience of reference we express the preceding laws in the following PRINCIPLES OF MULTIPLICATION. 1. Multiplying either factor multiplies the product. 2. Dividing either factor divides the product. 3. Multiplying one factor and dividing the other by the same number does not change the product. 32. To Multiply an Integer by 10, 100, etc. In the number 675, the 5 signifies what? the 7? the 6? How does the position of the 5 compare with its previous position? How does the position of the 7 compare with its previous position? How does the position of the 6 compare with its previous position? How does the position of each figure compare with its previous position? How, then, does the value of each figure compare with its previous value? How, then, does the value of the entire number compare with its previous value? What effect would the addition of two 0's to an integral number have Upon the position of each figure? Upon the value of each figure? Upon the value of the entire number? Explain in the same way the several effects of the addition to an integral number Give, then, a rule for multiplying an integral number By 10. By 100. By 1,000. By 100,000. In using each of these multipliers, how does the number of o's added to the multiplicand compare with the number of 0's in the multiplier? Give, then, a general rule for multiplying by 10, 100, 1000, etc. 33. To Multiply a Decimal by 10, 100, etc. In the number 6.75. the 5 represents what? the 7? the 6? Remove the decimal one place to the right, thus changing 6.75 to 67.5. How does the position of the 5 compare with its previous position? How does the position of the 7 compare with its previous position? How does the position of the 6 compare with its previous position? How does the position of each figure compare with its previous position? How, then, does the value of each figure compare with its previous value? How, then, does the value of the entire decimal compare with its previous value? |