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2. If 28 persons reap a harvest in 36 days, how many will be required to reap it in 21 days! As the answer is to be number of persons, the 28 persons
Persons, Persons. given in the question must be the third term of the pro
21: 36:: 28 48
7: 12 portion; and as the fewer the days the greater must be the
1: 12:: 4 number of workmen, we arrange the first two terms of the
12 proportion so that the second may be the greater; the stating is, therefore, 21 days : 36 days : : 28 persons: the number of persons required.
But, as the first two terms are in the same denomination, we suppress denomination, and use only the abstract numbers 21:36. These we see have a common divisor, 3; we therefore replace them by the quotients 7: 12; but the 7 and the 28 in the third term will divide by 7; we thus get the stating in the simple form 1:12 : : 4 persons; and then proceed as in the margin.
One example more must suffice.
A mass of 106 lb. of Australian gold, found in July 1851, sold at the rate of £3 68. 8d. per ounce: how much did it fetch ? Stating the question as in the margin, putting
£ . d.
1 oz.: 106 lb.:: 3 6 8 : 4240 the greater weight in the second term, because the
12 20 greater sum must be in the fourth, we see that the first and second terms differ in denomination; we
800 12 must, therefore, reduce the second to ounces, before we can regard the stating in the proper form for
12)1017600 800 working with: it then becomes 1 oz. : 1272 oz. : : £3 6s. 8d.
2,0)8480,0 The common denomination ounces is now sup
£4240 pressed, and, for convenience, the money is reduced to pence, the denomination in which the required fourth term must therefore appear: we have then merely to multiply 800 pence by the abstract number 1272; and the required value comes out 1017600 pence; for, as the first term is 1, there is no division.
Instead of making 1272 the multiplier of 800, we make 800 the multiplier of 1272, for convenience; as we know that the number, furnished for the product, is the same in one case as the other; this number, therefore, is so many pence ; which, when reduced to pounds, is £4240.
The following are a few examples for exercise :-
(2.) If 148 gallons cost £119 108., how many gallons will £89 128. 6d. buy ?Ans. 111.
(3.) What is the value of 2 qr. 24 lb. at £5 78. 4d. per cwt. ?-Ans. £3 16s. 8d.
(4.) What is the income of a person who pays £22 78. 5d. for income tax, at the rate of 7d. in the pound :--Ans. £767.
(5.) 441 guineas used to be coined out of 1 lb. of standard gold : how many sovereigns are now coined out of this weight?--Ans. 4640.
(6.) 668. are coined out of 1 lb. of standard silver : what is gained in coining £100 of silver, if the price of the silver be 58. 2d. per oz. ?-Ans. £6 945.
The Double Rule of Three.-The double rule of three is so called because there are at least two single rule-of-three statings implied in it. The following is an example, namely samom
If 12 horses plough 11 acres in 5 days, how many horses will plough 33 acres in 18 days?
This may be divided into two single rule-of-three questions : thus-1st. If 12 horses plough 11 acres, how many will plough 33 acres in the same time? By the former rule,
12 x 33 11:33:: 12 horses :
horses 36 horses. 11
2nd. If 36 horses can perform a work in 7 days, how many can perform the same in 18 days! By the former rule, 18:5:: 36 horses :
36 x 5
horses = 10 horses. 18
It is plain that by these two single rule-of-three operations, the correct answer to the question is obtained; but it is more readily obtained by the following arrangement:
12 x 33 x 5 ::12 horses :
horses 18 :
11 x 18 The fourth term of this compound proportion, as it is called, being got by multiplying the third term by the product of the consequents, and then dividing by the product of the antecedents ; and it is by the same multiplications and divisions that the final result is arrived at in the two distinct statings above. This more compact form of working is described in the following rule :
RULE.-- Put for the third term that one of the given quantities which is of the same kind as the quantity sought, just as in the single rule of three.
Then taking any pair of the remaining quantities like in kind, complete the stating, as if for the single rule of three, paying no regard to the other quantities, or rather considering them to remain the same.
Then take another pair, like in kind, as a new antecedent and consequent to be placed under the former pair; these, with the third term above, completing a second single rule-of-three stating. And proceed in this way till all the pairs are used.
Multiply the third term by the product of all the consequents, and divide the result by the product of all the antecedents, and the answer will be obtained.
Each given antecedent and consequent is of course to be regarded as an abstract number. It is convenient to indicate merely the several multiplications, at first, to place the divisor under the dividend, in the form of a fraction, as in the above example, and then, before performing the operations, to expunge factors common to numerator and denominator.
EXAMPLE.—If £15 128. pay 16 labourers for 18 days, how many labourers will £35 2s. pay, at the same rate, for 24 days?
As the answer is to be a certain number of labourers, the given 16 labourers is to be the third term; then taking days for the first antecedent and consequent, and money for the second antecedent and consequent, attending to whether either consequent should be greater or less than its antecedent, as in the former rule, the operation is as follows:24
16 x 18 x 702 therefore
4 x 3 x 351
24 x 312
lab. = 27 lab. 13
The 18 is placed in the second term, because fewer labourers are required for 24 days, the work being the same, than for 18; and the £35 2s. is placed in the second term, because more labourers can be paid for that sum than for £15 12s., the time being the same. If the question had been worked by two single rule-of-three statings, we should have had
You see, therefore, that the double rule of three merely compounds the several single proportions into one; it is thus called compound proportion.
I add two examples for exercise in this rule :
1. If 8 persons can be boarded for 16 weeks for £42, how long will £100 support 6 persons at the same rate?—Ans. 50g weeks.
2. If a family of 13 persons spend £64 in butcher's meat in 8 months, when meat is 6d. per lb., how much money, at the same rate of consumption, should a family of 12 persons spend in 9 months, when meat is 63d. per lb. ?—Ans. £72.
In this example, there would be three separate statings, if the question were worked by the single rule of three; these are here to be compounded into one.
Decimals. It was observed at the commencement of this treatise, that in our system of arithmetic numbers are expressed in the decimal notation, and the reason for this designation was stated :—it is simply this—namely, that the unit of any figure in a number is always ten times the unit of the figure in the next place to the right. Thus, in a number consisting of unit-figures-as for instance, in the number 1111-the second unit, beginning with right-hand one, is 10 times the first, the third 10 times the second, the fourth 10 times the third, and so on; or beginning with the first on the left, the second is the tenth part of the first, the third the tenth part of the second, and so on till we come down to the last unit, which is merely one. Now, we may evidently extend this principle still further; and, on the same plan, may represent one-tenth or one, one-tenth of this, or one-hundredth of one, one-thousandth of one, and so on, by simply putting some mark of separation between the integers and these fractions. The mark actually used is a dot, thus: 11111111. The unit next the dot, on the left, is 1; the unit one place from this on the left is 10; the next is 100; the next, 1000; and so on. In like manner, the unit one place from the 1 on the right, is, the next 1, the next, and so on. This being agreed upon, it is easy to interpret such a number as 36-427: it is 36 + 1 + 180 + 1000; each figure, to the right of the point, being a fraction of known denominator; the denominator being 10 for the first figure, 100 for the second, 1000 for the third, and so on. The sum of the fractions represented by the decimal 427, above, is obviously 42; in like manner, the fraction expressed by 2643 is 243; and in general the denominator of the equivalent fraction is always 1 followed by as many zeros as there are decimal places, the numerator being the number itself, when the prefixed dot, or decimal point, as it is called, is suppressed. You will thus easily see that the following are so many identities—namely:
136; 73.641= 7384; 2·07=2170, &c.
Any decimal may therefore be converted into its equivalent fraction at sight; it will be shown presently how any fraction may be converted into its equivalent decimal, though not with the same rapidity.
It is pretty evident that whatever whole number be prefixed to a decimal, the same may be prefixed to the numerator of the fraction which replaces that decimal: thus, taking the values above, we have
24.6246; 136.5413884; 73.64173641; 2·07 = 287, &c.;
for this is only reducing the foregoing mixed quantities to improper fractions.
To reduce a Proper Fraction to a Decimal.
RULE.-Annex a zero to the numerator, and then actually divide by the denominator: if there be a remainder, annex another zero, and continue the division, still annexing a zero, either till the division terminates without remainder, or till as many decimals as are considered necessary are obtained; the quotient, with the decimal point before it, will be the value of the fraction in decimals.
For example: let it be required to express in decimals; the operation is that in the margin. That: = 375 is easily proved; for =3888; consequently, dividing numerator and denominator by 8, we
have = 37375, from the very nature of decimals. If an improper fraction had been chosen, the operation would clearly have been just the same, only there would have been an integer prefixed to the decimal: thus, the operation for would have been as here annexed, showing that y=2.375. We need not take the trouble of actually annexing the zeros, as here it is enough that we proceed as if they were inserted, as in the marginal work, for reducing to a decimal; where it is plain, from the remainders, that 54 would recur continually; so that equal to a recurring decimal; the recurring period being 54. As a final example, let it be required to convert fs into a decimal. When one 0 is annexed to the 8, the divisor 113 will go no times; therefore, the first decimal place is to be occupied with a 0. Annexing now a second 113)8 (07079, &c. 0, the next decimal figure is 7, and the work proceeds as in the margin: the noughts being suppressed, though conceived to be annexed to the 8, and brought down one at a time, as in ordinary division. The quotient shows that 307079, &c.: the decimals may be carried out to any extent; but if we stop the work here, the error cannot be so great as '00001; that is, it is less than 100000 but it is obvious that, by continuing the work, we can make the error as small as we please.
The following are a few examples for exercise :
(1.) 42=1.875. (2.) 16='4375.
(4.) 303125. (5.) 1076923, &c. (6.) To24-00488, &c.
Addition and Subtraction of Decimals.-The rules for these fundamental operations are in reality the same as those for integers. We must here be careful not only to place units under units, tens under tens, and so on, but also to place tenths under tenths, hundredths under hundredths, &c.: that is, the decimal points must all range under one another in the same vertical line. This attended to, the operations are just the same as those with integers. See the operations in the margin.
Multiplication of Decimals.-Multiplication requires no special rule. The multiplier is to be placed under the multiplicand, just as if both were integers, no regard being paid to the decimal points. The only thing to be attended to is the marking off the proper number of decimal places in the product; and this is a very easy matter. We have seen that a number involving decimals is, in fact, a fraction with that number, the decimal point being suppressed, for numerator, and 1, followed by as many ciphers as there are decimal places, for denominator. Two such fractions multiplied together, being the product of the numerators divided by the product of the denominators, will therefore be a fraction of which the denominator is 1, followed by as many ciphers as there are in both factors. Consequently, in the multiplication of decimals, as many decimal places are to be marked off in the product as there are decimal places in both factors.
The example in the margin will suffice for illustration. As there are three decimals in the multiplicand, and two in the multiplier, five are marked off in the product.
Division of Decimals.-This operation, like that of multiplication, is the same for decimals as for integers; and the way to estimate the number of decimal places in the quotient is suggested by the plan adopted in multiplication.
All the decimals employed in the dividend, including, of course, whatever ciphers may have been added to it to carry on the division, are to be counted. We have then only to provide so many in the quotient, that when added to the
number of them in the divisor, we may have just as many as in the 2.35)23-621(10-0515
If the quotient figures, though all be considered as decimals, be too few in number to make up, with those in the divisor, the number in the dividend, then ciphers sufficient for this purpose are to be prefixed to the quotient figures, and the decimal point to be placed before them. See the second example in the margin. In the first of these examples, six have been used in the dividend, and as there are two in the divisor, there must be four in the quotient, which is therefore 10.0515. The last decimal, 5, is a little too great, but it is easy to see that if we had made it 4, the error in defect would have exceeded the present error in excess; and in limiting the number of decimals, we always make the last figure as near the truth as possible. In the second example, five decimals have been used in the dividend; and as there is but one in the divisor, four are required in the quotient; and to make up this number, a cipher is prefixed. The quotient is, therefore, 0265, as far as the decimals have been carried: that is, to four places.
32-4) 86 (0265
The following examples will serve for exercise in these two rules:(1.) 321096×2465
(3.) 56.10833 ÷ 17.371 = 3·23.
Extraction of the Square Root.-If a number be multiplied by itself, the product is called the second power, or the square, of that number. If this also be multiplied by the same number, the product is called the third power, or the cube of that number: and so on for the fourth power, fifth power, &c. This raising of