third figure of the line above instead of the second figure. The reason of this is clear: for, if we were multiplying by 206, when we multiply by the 6 we take the multiplicand 6 times, when we multiply by the 2 we really take the multiplicand, not 20 times, but 200 times. 32. When two numbers are to be multiplied together, it is a matter of indifference, so far as the product is concerned, which of them be taken as the multiplicand or multipher; in other words, the product of the first multiplied by the second, will be the same as the product of the second multiphed by the first. therefore the results are the same, that is, 2x4=4×2 That the product of one number multiplied by another, will be equal to the product of the latter multiplied by the former, may perhaps appear more clearly from the fourwing mode of shewing this equality in the case of the numbers 3 and 5. 3s=1-1-1; 235=1-1-1-1-1-1-1-1-1-1-1-1)-1-1-1 =1-1-1} -1-1 -1-1-1 =15. -1-1-1 -1-1-1 Now, if we regard the ones from left to right, there are 3 over taken 5 times; if we regard them taken from top to bottom, we have 5 ones reperted 3 times; and the number of ones in each case is the same; Le 3x5=5×3: and so in the case of any two other numbers multiplied together. 38. The tr the mutiplicar. prožnet tl fall results in Multiplication may be proved by using pher, and the multipher as multiplicand: If the the same as the product found at first, the results confined our attention to products formed by ~ers only. Products may however arise from more factors; this is termed CONTINUED - denotes the conforted multiplication of s that 2 is to be first multiplied by 3, and the product thus obtained to be then multiplied by 4. The result of such a process would be 24, which is, therefore, the continued product of 2, 3, and 4: we may express it thus, 2 × 3 × 4= 24. (23) Find the product of 234578 by 18, by 29, and also by 53; of 924846 by 67, by 95, and also by 430; 2846067 by 206, by 1008, and also by 907; 8409631 by 21711, by 7009, by 8435, and also by 7980. (24) Find the product of 1754 and 9306; of 47506 and 4500; of 149570 and 15790; of 554768 and 39314; of 815085 and 20048; of 123456789 and 987654321; and of 57298492692 and 700809050321. (25) Multiply 9487352 by 4731246; 4342760 by 599999; 17376872 by 7399078; 38015732 by 400700065; 574585614865 by 2837154309. (26) Multiply six hundred and fifty thousand and ninety, by three thousand and eight; also seventy-six millions, eight thousand, seven hundred and sixty-five, by nine millions, nine thousand and nine. (27) Find the continued product of 12, 17, and 19; of 3781, 3782, and 3783; and of 6565, 6786, and 9898. (28) Multiply 20470 by 1030, and 2958 by 476, explaining the reason of each step in the process. DIVISION. 35. DIVISION is the method of finding how often one number, called the DIVISOR, is contained in another number, called the DIVIDEND. The result is called the QUOTIENT. 36. Division is of two kinds, SIMPLE and COMPOUND. It is called Simple Division, when the dividend and divisor are, both of them, either abstract numbers, or concrete numbers of one and the same denomination. It is called Compound Division, when the dividend, or when both divisor and dividend contain numbers of different denominations, but of one and the same kind. 37. The sign, placed between two numbers, signifies that the first is to be divided by the second. 38. In Division, if the dividend be a concrete number, the divisor may be either a concrete number or an abstract number, and the quotient will be an abstract number or a concrete number, according as the divisor is concrete or abstract. For instance, 5 shillings taken 6 times give 30 shillings, therefore 30 shillings divided by 5 shillings give the abstract number 6 as quotient; and 30 shillings divided by 6 give the concrete number 5 shillings as quotient. SIMPLE DIVISION. 39. RULE. Place the divisor and dividend thus: divisor) dividend (quotient. Take off from the left-hand of the dividend the least number of figures which make a number not less than the divisor; then find by the Multiplication Table, how often the first figure on the left-hand side of the divisor is contained in the first figure, or the first two figures, on the lefthand side of the dividend, and place the figure which denotes this number of times in the quotient: multiply the divisor by this figure, and bring down the product, and subtract it from the number which was taken off at the left of the dividend: then bring down the next figure of the dividend, and place it to the right of the remainder, and proceed as before; if the divisor be greater than this remainder, affix a cypher to the quotient, and bring down the next figure from the dividend to the right of the remainder, and proceed as before. Carry on this operation till all the figures of the dividend have been thus brought down, and the quotient, if there be no remainder, will be thus determined, or if there be a remainder, the quotient and the remainder will be thus determined. Note 1. If any product be greater than the number which stands above it, the last figure in the quotient must be changed for one of smaller value: but if any remainder be greater than the divisor, or equal to it, the last figure of the quotient must be changed for a greater. Note 2. If the divisor does not exceed 12, the division can easily be effected in one line, by means of the Multiplication Table. 40. Ex. Divide 2338268 by 6758. 6758) 2338268 (346 20274 31086 27032 40548 40548 Therefore the quotient is 346. The reason for the Rule will appear from the following considerations. The divisor represents six thousand, seven hundred and fifty-eight: the first five figures on the left-hand side of the dividend represent two millions, three hundred and thirty-eight thousand, and two hundred. Now the divisoris contained in this 300 times; and 6758 × 300=2027400, or omitting the two cyphers at the end for convenience in working, we properly place the 4 under the 2 in the line above; we subtract the product thus found, and we obtain a remainder of 3108, which represents three hundred and ten thousand, and eight hundred. Bring down the 6 by the Rule; this 6 denotes 6 tens or 60, but the cypher is omitted for the reason above stated: the number now represents three hundred and ten thousand, eight hundred and sixty: 6758 is contained 40 times in this, and 6758 × 40=270320; we omit the cypher at the end as before, and subtract the 27032 from the 31086; and after subtraction the remainder is 4054, which represents forty thousand, five hundred and forty. Bring down the 8 by the Rule, and the number now represents forty thousand, five hundred and forty-eight: 6758 is contained 6 times exactly in this number. Therefore 346 is the quotient of 2338268 by 6758. 41. The above example worked without omitting the cyphers would have stood thus: 6758) 2338268 (300+40+6 2027400 310868 270320 40548 40548 hence it appears that the divisor is subtracted from the dividend 300 times, and then 40 times from what remains, and then 6 times from what then remains, and there being now no remainder, 6758 is contained exactly 346 times in 2338268. The truth of the above method might have been shewn as follows: 2338268=2027400+270320 + 40548 6758) 2027400+270320+40548 (300+40+6 2027400 +270320 +270320 +40548 +40548 42. Ex. Divide 56438971 by 4064. 4064) 56438971 (13887 4064 15798 12192 36069 32512 35577 32512 30651 28448 2203 |