Product of 21. 1 3 3 3 4 6 2 2 2. Multiply 9382716 by 31. 3. Multiply 1234567 by 41. 4. Multiply 4364369 by 51. 5. Multiply 6937845 by 61. 6. Multiply 364812 by 71. 7. Multiply 482436 by 81. 8. Multiply 2468 by 91. Prod, 224588. Prod, 290864196. Note.-When in either of the two preceding cases, cyphers intervene between the figures, the same mode of operation may be adopted, if care be taken to give each figure its true place. Ex. 1. Multiply 6456 by 105. 1. Bought 52 horses at $75 each; what did they cost? Ans. $3900. 2. What cost 84 tons of hay at $15 per ton? Ans. $1260. 3. If a man can travel 43 miles in one day, how far can he travel in 60 days? Ans. 2580 miles. 4. There are 144 square inches in one square foot. How many square inches are there in 67 square feet? Ans. 9648. 5. If there be 18 panes of glass in one window, how many are there in a house which has 56 such windows? Ans. 1008. 6. Bought 342 bales of linen, each containing 56 pieces of 25 yards each. How many yards did I buy? Ans. 478800. and 7. There is an orchard consisting of 126 rows of trees, in each row there are 109 trees. How many apples are there in the orchard, allowing an average of 1007 on a tree? 13830138. Ans: 8. A certain State contains 50 Counties; each County, 35 towns; each town, 300 houses, and each house, 8 persons. What is the population of the State? Ans. 4200000. QUESTIONS. What is the nature of Multiplication? How many numbers are employed in the operation? What are they called, and what is peculiar to each? What is the number obtained called? Of what rule is multiplication an abbreviation? Illustrate. What are the multiplicand and multiplier called when spoken of together? What is the value of each figure in the product when you multiply by a unit figure only? Units multiplied by units give what? Units by tens? Units by hundreds? When the multiplying figure is tens, what effect will it have on the value of the product of each figure in the multiplicand? and what will be the effect if the multiplying figure be hundreds? Give farther illustration of the value of the product figures. What is case 1st? What is the rule? Case 2d? The rule? Case 3d? The rule? What is a composite number? What are the component parts of a number? Case 4th? The rule? Case 5th? The rule? Case 6th? The rule? SIMPLE DIVISION. We now come to the reverse of the preceding rule. There we had two factors given to find their product. Here we have given the product, or what corresponds to it, and one of the factors, and are required to obtain the other factor. Multiplication, as was shown, could be performed by repeated additions; Division may be performed by repeated subtractions. Suppose it be required to ascertain how many times 4 is contained in 12. It may be done by taking 4 from 12, till nothing remains, or till a number less than 4 remains. Thus, 12 8=12-4. 4 4-12-4+4. 4 012-4+4+4. The operation shows three 4's may be taken from 12. 4 is therefore contained in 12 three times. This is, however, a slow mode of operation. A more expeditious one must be sought; and, preparatory for it, the scholar is required to learn the following table: MENTAL EXERCISES IN DIVISION. Bought 4 apples for 8 cents; what was the price of one? 8-4 how many? Bought 3 oranges for 12 cents; what was the price of one? 12-3 how many? At another time bought 5 for 15 cents; what was the price? 15-5=how many? Paid 24 cents for 6 peaches; what did one cost?,24÷6 how many? What is the cost of one orange when 8 cost 24 cents? when 4 cost 24 cents? when 3 cost 24 cents? 24-8 how many? 24÷4 how many? 24÷3=how many? Paid 35 cents for 7 melons; what was the value of one? Paid the same money for 5; what was the value of one? 35÷ 7 how many? 35÷5-how many? Bought 8 silks for 16 cents; what was one worth? What is the value of one when I pay 24 cents for 8? when I pay 32 cents? 40 cents? 48 cents? 56 cents? 16-8-how many? 24-8=how many? 32÷8 how many? 40÷8 how many? 48÷8=how many? 56-8 how many? 64÷8 how many? 72÷÷8= how many? Paid 18 shillings for 9 yards of calico; what was the price per yard? What is the price per yard when the same quantity costs 27 shillings? 45 shillings? 36 shillings? 72 shillings? 63 shillings? 18-9-how many? 27÷ 9 how many? 45-9-how many ? 36÷9 how many? 72+9=how many? 63-9=how many? What is the cost of one lemon, if 10 cost 30 cents? 70 cents? 60 cents? 50 cents? 80 cents? 40 cents? 90 cents? 30-10=how many? &c. What is the cost of one sheep, when 8 cost 24 dollars? What, if 8 cost 64 dollars? 88 dollars? 72 dollars? 48 dollars? 80 dollars? 40 dollars? 32 dollars? 24-8=how many? 64-8-how many? &c. What is the value of one bushel of wheat, when 12 bushels are worth 24 dollars? What, when the 12 are worth 36 dollars? 72 dollars? 96 dollars? 48 dollars? 108 dollars? 60 dollars? 132 dollars? 84 dollars? 120 dollars? 144 dollars? 24-12-how many? 36÷12 how many? &c. Bought 8 pounds of raisins for 72 cents; what was the price per pound? If 9 pounds of rice cost 81 cents, what will one pound cost? A man traveled 96 miles in 8 days; how far was that per day? A horse ran 54 miles in 6 hours; how far was that per hour? How many pounds of sugar at 9 pence per lb. can be bought for 54 pence? for 63 pence? for 81 pence? for 108 pence? for 99 pence? for 72 pence? for 45 pence? for 36 pence? The scholar must now be prepared to apply what he has learnt from the preceding table and questions, to division on a more extensive scale. He will have noticed that for each operation two numbers are given; viz. a number to be divided, which is called the dividend; and a number by which to divide, called the divisor. The number obtained is called the quotient; a word which signifies, how many; because this number always determines how many times the divisor is contained in the dividend. After the operation is performed there is frequently a number left. This is called the remainder, and is always less than the divisor. When the division is performed, if there be no remainder, the quotient multiplied by the divisor will always produce the dividend; and if there be a remainder, the dividend will be produced by multiplying as before, and adding the remainder to the product. Hence division is proved by multiplication. The scholar will readily perceive that these two rules are the reverse of each other. The operations in division will be illustrated under two general heads; viz. Short Division, and Long Division. I. SHORT DIVISION. When the divisor does not exceed 12, the process is abbreviated by keeping the computation in the mind, and writing down only the quotient figures. RULE.-1st. Write down the dividend, and place the divisor on the left, with a curve line drawn between them. 2d. Take as many figures on the left of the dividend, as will contain the divisor once or more, and write the figure expressing the number of times, directly under those divided. 3d. If in dividing there be a remainder, imagine the next figure in the dividend to be placed on the right hand of it. This will form a new number, which may be divided as before. Continue the same process till all the figures of the dividend have been disposed of, and the number obtained will be the quotient required. 4th. If in taking any figure of the dividend the number produced be not sufficient to contain the divisor once, a cypher must be placed in the quotient, and another figure of the dividend taken. |