SUPPLEMENT TO MULTIPLICATION. QUESTIONS. 1. What is Simple Multiplication? 2. How many numbers are required to perform that operation? 3. Collective r together, what are the given numbers called? 4. Separately, what are they called? 5. What is the result, or number sought, called? 6. In what order must the given numbers be placed for multiplication? 7. How do you proceed when the multiplier is less than 12?, 8. When the multiplier exceeds 12 what is the method of procedure? 9. What is a composite number? 10. What is to be understood by the component parts of any number? 11. How do you proceed when the multiplier is a composite number? 12. When there are cyphers on the right hand of the multiplier, multiplicand, either or both, what is to be done? 13. When there are cyphers between the significant figures of the multiplier, how are they to be treated? 14. When the multiplier consists of 9's how may the operation be contracted? 15. How is Multiplication proved? 16. By what method do you proceed in casting out the 9's from any number? 17. How is Multiplication proved by casting out the 9's? 1. What sum of money must be divided between 27 men, so that each may receive 115 dollars. Ans. 3105. Ꭰ EXERCISES. NOTE. The scholar's business in all questions for Arithmetical operations, is wholly with the numbers given; these are never less than two; they may be more, and these numbers in one way or another, are always to be made use of to find the answer. To these, therefore, he must direct his attention, and carefully consider what is proposed by the question to be known. § 4. SIMPLE DIVISION. SIMPLE DIVISION teaches, having two numbers given of the same denomination, to find how many times one of the given numbers contains the other. Thus, it may be required to know how many times 21 contains 7; the answer is 3 times. The larger number (21) or number to be divided, is called the Dividend; the lesser numer (7) or number to divide by, is called the Divisor; and the answer obtained, (3) the Quotient. After the operation, should there be any thing left of the Dividend, it is called the Remainder. This part, however, is uncertain; sometimes there is no remainder. When it does happen it will always be less than the divisor, if the work be right, and of the same name with the dividend. RULE. 1. "Assume as many figures on the left hand of the dividend as contain "the divisor once or oftener; find how many times they contain it, and place the answer as the highest figure of the quotient. 46 2. "Multiply the divisor by the figure you have found, and place the product under that part of the dividend from which it was obtained. 3. "Subtract the product from the figures above it. 4. " Bring down the next figure of the dividend to the remainder and "divide the number it makes up as before." When you have brought down a figure to the remainder, if the number it makes up be still less than the divisor, a cypher must be placed in the quotient, and another figure brought down. Proceed in this operation thus-It being evident that the divisor (5) cannot be contained in the first figure (1) of the dividend, therefore assume the two first figures (12) and inquire how often 5 is contained in 12; finding it to be 2 times, place 2 in the quotient, and multiply the divisor by it, saying 2 times 5 is 10, and place the sum (10) directly under 12 in the dividend. Subtract 10 from 12 and to the remainder (2) bring down the next figure (7) at the right hand, making with the remainder 27. Again inquire how many times 5 in 27; 5 times; place 5 in the quotient, multiply the divisor (5) by the last quotient figure (5) saying 5 times 5 is 25, place the sum (25) under 27, subtract and the work is done. Hence it appears that 127 contains 5, 25 times, with a remainder of 2, which was left after the last subtraction. This Rule, perhaps at first will appear intricate to the young student, although it is attended with no difficulty. His liability to errors will chiefly arise from the diversity of proceedings. To assist his recollection, let him notice that 1. Find how many times, &c. 2. Multiply. The steps of Division are four 3. Subtract. It is sometimes practised to make a point (.) under the figures in the dividend, as they are brought down, in order to prevent mistakes. When the divisor is a large number, it cannot always certainly be known how many times it may be taken in the figures which are assumed on the left hand of the dividend till after the first steps in division are gone over, but the learner must try so many times as his judgment may best dictate, and after he has multiplied, if the product be greater than the number assumed, or that number in which the divisor is taken, then it may always be known that the quotient figure is too large; if after he has multiplied and subtracted, the remainder be greater than the divisor, then the quotient figure is not large enough, he must then suppose a greater number of times, and proceed again. This at first may occasion some perplexity, but he attentive learner after some practice, will generally hit on the right umber. 2. Let it be required to divide 7012 by 52. "Multiply the Divisor and Quotient together, and add the remainder, if there be any to the product; if the work be right, the sum will be equal 'to the dividend." 7 0 1 2 Equal to the dividend. Another and more expeditious way of proving Division is Cast out the 9's from the Divisor and the Quotient, multiply the results and to the product, add the remainder if any after division; from the sum of these cast out the 9's, also cast out the 9's from the Dividend, and if the wo last results agree, the work is supposed to be right. |