We take ,151 first, which is 151 thousandths (for the denominator of any decimal is always of the same order as the lowest order taken). This divided by 36 gives 4 thousandths, because the lowest dividend taken is thousandths. in the quotient it must have two prefixed thus ,004. as quotient. This 4 is 4 order in the part of the Therefore when it is put ciphers and a separatrix We now subtract from the dividend 36 times, ,004 or ,144. (See rule for Decimal Multiplication.) It is desirable in such cases to place ciphers and a separatrix in the remainder, to make them stand in their proper orders. To the remainder (,007) bring down the 2 tenths of thousandths, making 72 tenths of thousandths. This divided by 36 gives 2 tenths of thousandths as quotient, which is set in that order. 36 times 2 tenths of thousandths (or ,0072) being subtracted, nothing remains. Sometimes we must add ciphers to the dividend before we can begin to divide. For example, let,369 be divided by 469, and we proceed thus, We find that ,369 cannot be divided by 469, so we add a cipher to it, making it 3690 tenths of thousandths. This divided by 469 gives 7 as quotient, which is 7 tenths of thousandths, (,0007) because the lowest order of the dividend is of that order. We now subtract 469 times,0007 (which is,3283) from the dividend, and ,0407 remain. Of what order is the denominator of any decimal? To this remainder we add a cipher, and change it from 407 tenths of thousandths to 4070 hundredths of thousandths. This divided by 469 gives 8 as quotient, which is 8 hundredths of thousandths, because the lowest order in the dividend is hundredths of thousandths. We now subtract 469 times 8 hundredths of thousandths (or ,03752) from the dividend and ,00318 remain. We could continue dividing, by adding ciphers to the remainders, but it is needless. Instead of this we can set the divisor under the remainder as in common division, thus. It is not needful to retain the separatrix and ciphers when thus writing a remainder, because when put in the quotient, it is not considered as the part of a whole number, but as a part of the lowest order in the decimal, by which it is placed. Thus when this is put with the above quotient, we read the answer thus, 78 hundredths of thousandths, and 21% of another hundredth of thousandth. Let the following sums be performed and explained as above. Decimal Division when the Divisor is a Decimal. When the divisor is a decimal, we proceed as in dividing by a Vulgar Fraction, viz. We multiply by the denominator, and divide by the nume rator. Thus if we are to divide 24 by,4, we are to find how many 4 tenths there are in 24. We first multiply 24 by the denominator 10, to find how many one tenths there are, and then divide by the numer ator 4, to find how many 4 tenths there are. 24 is multi What is the rule for decimal division, when the divisor is a decimal? plied by ten, thus; 24'0, and has the inverted separatrix, to show that it is not 240 whole numbers, but tenths. We now have found that in 24 there are 240 one tenths, we now divide by 4, to find how many 4 tenths there are. The answer is 60, which according to the rule, must be of the same order as the lowest order in the dividend, or 60 tenths, and must be shown by the inverted separatrix thus (6'0.) This may be changed to whole numbers by reverting the separatrix thus (6,0.) When the dividend is a decimal, we can multiply by removing the separatrix. Thus let 8,64 be divided by,36. Here we are to multiply by 100, to find how many one hundredths there are in the dividend, and then divide by 36 to find how many 36 hundredths there are. We multiply by 100, by removing the separatrix two orders toward the right, and then dividing by 36, we have 24 as answer, which is 24 units, because the dividend is units, as appears below. 36)864,(24 72 144 144 000 If the divisor is a mixed decimal, we change it to an improper decimal, and then proceed as before, multiplying by the denominator and dividing by the numerator. Thus let 10,58 be divided by 4,6. We first change the divisor into an improper decimal thus, 4'6 (46 tenths.) We now are to multiply the 10,58 by 10, to find how many one tenths there are, and then divide by 46, to find how many 46 tenths there are. If the divisor be a mixed decimal, what is the process? We multiply by 10 by removing the separatrix thus, 105,8, and proceed as follows. 46)105,8(2,3 92 13,8 13,8 000 Here we divide 105 units by 46, and the quotient fig. ure is 2 units. We then subtract 46 times 2 units from the dividend, and 13 units remain. To this bring down the 8 tenths. This is divided as if whole numbers, but the quotient 3 is 3 tenths, because the lowest order in the dividend is tenths. It is set in the quotient with the separatrix before it, and then 46 times,3 (or 13,8) is taken from the dividend, and nothing remains. Let the following sums be performed, and explained as above. The following then is the rule for Decimal Division. RULE FOR DECIMAL DIVISION. If the divisor is a whole number, divide as in common division, placing each quotient figure in the same order as the lowest order of the dividend taken. If the divisor is a decimal, multiply by the denominator,' and divide by the numerator, placing each quotient figure in the same order as the lowest order of the dividend taken. If the divisor is a mixed decimal, change it to an improper decimal, and then proceed to multiply by the denominator and dipide by the numerator. What is the rule for decimal division? N. B. The rule for multiplying and dividing Federal Money, is the same as for Decimals. EXAMPLES. How many times is $2,04 contained in $9,40? 8,41 As it is found to be invariably the case that the decimal orders in the divisor and quotient always equal those of the dividend, the common rule for decimal division is formed on that principle, and may now be used. COMMON RULE FOR DECIMAL DIVISION. Divide as in whole numbers. Point off in the quotient enough decimals to make the decimal orders of the divisor and quotient together equal to those of the dividend, counting every cipher annexed to the dividend, or to any remainder, a decimal order of the dividend. If there are not enough figures in the quotient, prefix ciphers. as In pointing off by the above rule, let the teacher ask these questions. How many decimals in the dividend? How many in the divisor? How many must be pointed off in the quotient, to make as many in the divisor and quotient, as there are in the dividend? EXAMPLES. At $,75 per bushel, how many bushels of oats can be bought for $14,23? How much butter at 16 cents a pound, can be bought for $20 ? A half cent can be written thus, $,005 (for 5 mills is half a cent, or 5 thousandths of a dollar.) What is the common rule for decimal division ? |