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decimal place. It is plain, that in any multiplication we may begin from the left instead of from the right of the multiplier, and that we may continue the process with the successive figures towards the right, provided we arrange the partial products so that they shall stand in their proper order below one another, each one step further towards the right, thus,
661:24 1702 The vertical line, as before, cuts off the superfluous work, but it is to be observed that no part of that work need be set down, if we attend when multiplying to the effect that would be produced on each of the abbreviated partial products by carriages from the omitted figures.
The best method, however, of performing this kind of contracted multiplication, is as follows:
RULE.—Write the multiplier in an inverted order, and place it under the multiplicand so that what was its units' figure shall fall under the last place of decimals in the multiplicand which we have occasion to retain in the final product, adding ciphers to the multiplicand, if necessary, so that every figure of the multiplier shall have a figure or cipher above it. Multiply in the ordinary way from right to left, but commence each line of products with that figure of the multiplicand which stands next to the right of the one placed directly over the figure used as multiplier, not setting down the result of this multiplication, but carrying the nearest* number of tens from it to the next figure. Let the first figures of the several products be placed vertically under one another. Add as usual, and point off the required number of decimal places from the result.
The reason of this process will readily appear on inspection of the work of the last example.
Example: (1.) To multiply as before, 72.563 by 9.154, retaining only two decimal places in the result.
726 362 29
Product by 9 with carriage from 9 x3
5 x 5 4
4 x 2
* In order to ensure the utmost attainable correctness of the result, the nearest number of tens should always be carried from the first omitted figure of the multiplicand, as in the work of the example above given. When the carriage is from such a number as 27, which ends with a figure greater than 5, 3 should be carried rather than 2, since 27 is nearer to 30 or 3 tens than to 20 or 2 tens. Similarly, as regards 6 and 8, 1 should be carried, as each of these is nearer to 10 than to 1 When the carriage is from any number, such as 25, ending with 5, it is immaterial whether or not we increase the tens to be carried, 25 being exactly half way between 20 and 30; in this case the amount of carriage must be left to the judgment of the operator.
To secure the accuracy of the last decimal figure, it is advisable always to work for, at least, one place beyond that intended to be kept in the result : if the result is to be true to a tenth, the hundredths, or second decimal figure should be obtained ; if to a hundredth, the thousandths, or third decimal figure should be obtained, and so on.
Example : (2.) Required the product of 159•6 by 1.254, true to one, two, and three decimal places respectively.
200.1384 The third result is the same as that which the multiplication would afford, if it were performed at full-length in the ordinary way; there is therefore no saving of work in this instance, when the third decimal figure is required to be exact.
Should the figures of the multiplier extend beyond the left of the multiplicand, it is necessary to continue the process so long as there is either multiplication or carriage from a previous figure : thus,
2615 314 21 3
29.53 Here, there is no figure above the 7 of the multiplier, but there is a carriage of 3 (or 4) to be set down in the right hand column, arising from 7 X5 =35, which gives either 3 or 4 tens to be carried.
The test of the correctness of the work by a casting out the nines” cannot, of course, be applied in the case of ontracted multiplication, since the hole of the figures are not obtained. To verify a result, under these circumstances, the only, trustworthy method is to reverse the positions of the multiplier and multiplicand and go through the process in that order. The tendency to repeat the same errors is obviated by changing the first arrangement of the factors.
Officers will find it exceedingly serviceable in business to practise contracted multiplication of decimals; the labour of reducing gallons of spirits to proof, of computing the areas of malt utensils, or the amount of floor charges by the pen, and many other operations involving the use of decimals, may be greatly abridged and all the requisite accuracy ensured, by reversing the figures of the multiplier and working agreeably to the foregoing rule for one place of decimals more than it is necessary to preserve in the result. Proof quantities of spirits, for instance, are expressed in no smaller denomination than tenths of a gallon, and it is useless,
therefore, in any case of reduction to proof to bring out the result further than the second place of decimals, as this will sufficiently provide for the correctness of the tenths.
Division of Decimals.-RULE (applicable to every case.)—If the divisor and dividend have not the same number of decimal places, annex ciphers to equalise the number of places. Then, disregard or strike out the decimal points, as well as all ciphers preceding the first significant figures—that is, treat the numbers as integersand if the dividend be greater than the divisor, proceed as in common division, continuing the operation by the aid of ciphers, until no remainder is left, or as many quotient figures have been obtained as are thought necessary. For every cipher so used in continuing the division, mark off one place of decimals from the right of the quotient.
Where the dividend is less than the divisor, the quotient will be entirely fractional. In this case, after equalising the decimal places, &c., annex ciphers to the dividend until it will contain the divisor, and for each cipher so annexed, except the last, place a preliminary cipher in the quotient. Continue the division to any desired extent, as before :
Examples: (1.) Divide ·4368 by .0078.
Here the number of decimal places in dividend and divisor are equal. It is only requisite, therefore, to throw away the points and operate as in ordinary division.
78)4368(56 Answer, 56
0 (2.) Divide 1 by •015 to three places of decimals. In this instance, three ciphers must be added to the dividend to give it the same number of places as the divisor. We have then to divide 1000 by 15, carrying the division to 3 decimal places.
15)1000(66 666 Answer.
33000000 In obtaining this quotient, three ciphers are added to the dividend before it will contain the divisor: the first significant figure of the quotient must stand therefore in the third place of decimals.
It must be obvious that the ciphers on the right of the divisor may be suppressed, and the work of the process greatly shortened, by using only the original divisor, provided one place of decimals be pointed off in the quotient for every cipher suppressed in the divisor. hus,
As six ciphers have been omitted in the divisor, six decimal places must be assigned in the quotient, making it 001463.
PARTICULAR RULE.—When the number of decimal places in the dividend exceeds the number in the divisor, an easy method of determining the value of the quotient is to deduct the number of places in the divisor from those in the dividend, and let the remainder be the number of decimal places in the quotient. For, as in every division the divisor multiplied by the quotient, and increased by the remainder, if any, equals the dividend, so the number of decimal places in the quotient must, by the rule of multiplication of decimals, be the difference between the number in the dividend and divisor. Thus,
1•6)91.808(5738 Here there are 2 more places in the dividend than in the divisor, consequently two places should be pointed off from the right of the quotient figures, giving 57.38.
An application of the rule for the division of vulgar fractions will illustrate the reason of the rules for the division of decimals. Let it be required to divide •054 by •76. These reduced to a common denominator are respectively •054 and •760 or 58, and 260 But
1567 • 1000
1560 X 40=* It appears, then, that when the number of decimal places have been made equal and the decimal points thrown away, the rest of the operation is the same as that of reducing a vulgar fraction to a decimal. (See page 80). In the present example, let three ciphers be annexed to both numerator and denominator (dividend and divisor) and let each term be then divided by 760. 760)54000
=:071 the quotient of the divisor of .054 by 76, to 3
places of decimals. To determine the value of the first quotient figure by inspection. The only difficulty in the division of decimals, is to assign the proper position of the decimal point, and it is frequently useful to know before dividing what will be the value of the first figure of the quotient, whether it will be a whole number or a decimal, and in what place of decimals, &c.
An easy, general method of determining by inspection of the numbers, what will be the character of the quotient, is supplied by the consideration, that the first figure of the quotient, in every instance of the division of one decimal by another, must be of the same local value as that figure of the dividend under which stands the units' place of the product of the divisor by the first quotient figure.
Example : If it be required to divide 15.6 by 5289 4 ; arranging the divisor over the dividend, so that the right hand figure of the divisor shall stand over that figure or place of the dividend, to which the product of the divisor by the first quotient figure would reach, it is seen that the units’ place of the divisor coincides with the third decimal place of the dividend ; and from this it is inferred, that the first quotient figure should have the same value, that is, stand in the third place of decimals, thus,
15.600 As a second example, take the division of .379 by •006.
.379 Here, the units' place of the divisor is marked by a cipher, and as that place stands over what would be the hundreds' figure of the dividend—if it had a whole number
prefixed—the quotient must also have two figures before the decimal point. In other words, its first figure must have the value of hundreds.
A convenient mode of applying this rule is as follows :
Insert the pen in that part of the divisor where the decimal point should stand in order that the divisor may go into the dividend, as in the first step of ordinary division, that is, be contained in the dividend some number of times less than ten. If the pen be on the right of the decimal point, the quotient will consist of a whole number, with or without decimals attached : if on the left, of decimals only: and the value of the first quotient figure will appear from the number of figures or places between the pen and the decimal point inclusive. Thus, when the interval comprises one figure, the first quotient figure will be so many tens or tenths : when the interval comprises two figures, the quotient will commence with hundreds or hundredths, and so on. (1)
(3) Examples : 9.5 divided by 11:
•08 by 90
76 by •16 Modified position of
decimal point in
15 Value of the first quotient figure. tenths
hundreds (1st decimal place.) (4th decimal place.) (3rd place of whole
numbers.) A little practice in the application of this method will not only enable officers to dispense with any other rule for the division of decimals, but also give them great expertness in the valuation and management of decimals generally.
To divide a decimal by 10, 100, 1000, &c.—RULE.—Move the decimal point as many places to the left as there are ciphers in the divisor.
Example : 21564-327 divided by 10, 100, 1000, 10000, and 100000, respectively) becomes
2156.4327 215.64327 21.564327 2:1564327
.21564327 Contracted division of decimals.—RULE.—Divide in the ordinary way, until the number of figures remaining to be found in the quotient is less than the number of figures in the divisor. When this is the case, instead of bringing down more figures from the dividend or annexing ciphers to the remainders, cut off at each step a figure from the right of the divisor, and in multiplying the divisor so curtailed, by the additional quotient figures, carry the nearest number of tens* from the figure last cut off. Example : Divide 7.9482 by 5•643 to 6 decimal places. Contracted process.
Ordinary or full length process.
2 3052 48000
34 500 1
642 For na explanation of the meaning of this phrase-the nearest number of tens-see the article on multiplication, (page 86 note.)