39

ARITHMETIC.

DECIMALS.

A decimal fraction is one the denominator of which is 10, or some power of 10; thus ·8 =

897: 897 and so on.

=

1000'

Convert to decimals.

89

= 10, ·89= 1009

Divide the numerator by

the denominator, adding cyphers till there is no remainder. 10) 7

7

ADDITION AND SUBTRACTION.

RULE.—Arrange the figures as in simple addition and subtraction, taking care that the decimal points are all under each other.

(a) 6·74+7·976+0087. Ans, 14·7247.

(b) 31 72+7·001+00081+011. Ans. 3873281.

(c) 008+8.76+0087+001.

Ans. 8 7777.

(d) 4·716+417+·0417+004. Ans. 5·1787.

(e) 71 682-57 618. (f) 67·1-31 98. Answers, 14.064 and 35 12.

(g) 8.71-91 87. (h) 11.719-84 72. Answers, -83.16 and -73.001.

(i) 71 976-17981. (j) 67.41-3.876. Answers 70 1779 and 63 534.

MULTIPLICATION.

RULE.-Multiply as with integers, and point off as many figures from the right hand of the product as there are decimal places in both multiplier and multiplicand.

(a) 5 628×00167. (b) 05×·05×·005.

(c) 001600197. (d) 016 × 18.

(e) 19.6×00019.

(f) •017 × 17.

(g) 8010x00019. (h) 867x67.

(i) 2365x2435.

(j) 52.7438×3.141592.

(a) 00939876.

Answers.

(b) ⚫0000125.

(d) ·288. (e) ⚫003724. (ƒ) ·289.

(c) ⚫000003152.

(g) 1.5219. (h) 58089. (i) 05758775. (j) 165 6995001296.

DIVISION OF DECIMALS.

RULE. Bring both the divisor and dividend to whole numbers by removing the decimal point to the right of the last figure, adding O's where required. EXAM.-845)4 225(=845)4225(5. 8.45)4 225(= 4225

8450)42250(*5 42250

84.5)4 225(=84500)422500(05 422500

EXAMPLES.

(a) 00939876÷·00167.

(b) ·0000125÷·0025. (c) 000003152÷00197. (d) ·288÷18. (e) 003724-19 6.

(f) 289÷017. (g) 1·521900019. (h) 58089·867. Answers.-(a), 5.628; (b), 005; (c), 0016; (d), ·016; (e), 00019; (f), 17; (g), 8010; (h), '67.

When the numerator of a vulgar fraction divided by the denominator leaves the same remainder it gives what is called a pure circulating decimal, as = 33, &c., in which one figure constantly recurs from the beginning. Another instance is 2727. If the figures do not begin to recur from the decimal point, they are known as mixed circulating decimals, as they then consist of a period or repetend and an ordinary decimal portion, as 20235, the decimal points over the figures marking off the period.

To convert a pure circulating decimal into a fraction. RULE.—Take the repetend for the numerator, and as many nines as there are figures in the repetend for a denominator. Thus 1818 ·

= 11.

Thus = •111, &c., 2 = = 222, &c., and any pure circulator with the repeating figure may be repressed in a vulgar fraction with that figure for numerator and 9 for denominator. The same may be shewn to be the case with, 7, which give 0101, &c., and 001001, &c., respectively.

Reduce to vulgar fractions (a) 20235; (b) ·30 (c) ·743; (d) ·05'; (e) ·54'; (ƒ) ·729; (g) 6·418.

6745

9999

Ans. (a), ; (b), 305; (c), 143; (d), 5; (e), A; (ƒ), 37; (9), 618.

27.

To convert a mixed circulating decimal to a vulgar fraction.

RULE.-Subtract the figures which do not circulate from the whole decimal, this will give the numerator; and take as many nines as there are circulating figures,

followed by as many O's as there are non-circulating figures; this will form the denominator.

Thus, 18 287 112,87-2 = 1818.

990

EXAMPLES. (a), 078703; (b), 7·1439; (c) 4·1236; (d), 6·0432; (e), 3·0909; (ƒ), 6·245; (g), ·5681; (h), 6·925; (i), 2·06; (j), 6·54950.

629

275

2475

Ans. (a), 2; (b), 7-27; (c), 434; (d), 6107; (e), 3101; (f), 611; (g), 23; (h), 6835; (i), 215;

(5), 61221.

2222

10

Find the value of ⚫678

of 1 ton.

⚫678 tons.

25 999

Reduce 37 to the decimal of 4s. 7d.

6 72 lbs., and so on. Ans. 13cwt. 2qrs. 6.72lbs. EXAMPLES.-(a), 3s. 7 d. to f. of 9s. Od. ; (b), 3s. 11 d. to f. of 6s. 74d.; (c), 1 ton 5 cwt. to f. of 6 ton 1 qr.; (d), 791⁄2 lbs. to f. of 67 lbs. ; (e), 146875 of lb. avoirdupois; (f), 625 cwt.; (g), 625 gal.; (h), 14265 year; (i), 4 ton 2 cwt. to f. of 31 ton 17 cwt.; (j), £3 16s. 8d. to f. of 1 guinea; (k), 6s. 71⁄2d.. to f. of 1 guinea; (1), £ 78 to f. of guinea.

Ans. (a), 22; (b), 195; (c), 181; (d), 1,25: (e), 2.35 oz.; (f), 70 lbs.; (g), 5 pints; (h), 52.06725 days; (i), 7; (j), 341; (k), 5535; (1), 38.

82

637

1. If 67 lbs. cost £3 41, what will 6 03 lbs. cost? Ans. £30 69.

2. If 4 97 cwt. cost £14.795, what will 34 79 cwt. cost? Ans. £103.565.

3. If 16 7 yds. of cloth cost 31.9d., what will 150.3 yds. cost? Ans. 287-1d.

4. If a wall 6 9 yds. long, 1.7 yds. high, and 7.3 yds.. wide, take 64 men 9.5 days, of 7.6 hours each, how

many men will a wall take to build which is 62 1 yds. long, 85 yds. high, 14-6 yds. wide, in 4.75 days, of 3.8 hours each ? Ans. 2304 men.

5. If 37 men can earn £64 79 in 18 4 days, of 13.9 hours each, how much will 74 earn in 9 2 days, of 6.95 hours each? Ans. £32 395.

RATIOS.

"The ratio of one quantity to another is that number which expresses what fraction the former is of the latter." Thus the ratio of 24 to 36 (24:36) is 2, which fraction expresses what fraction 24 is of 36. Of this ratio the first figure in the phrase, 24:36, is known as the antecedent, and the second the consequent. If two ratios be taken which are equal to each other, that is, which will reduce to the same fraction, these will make a proportion. Thus 2:36 and 4:72 are equal ratios, since each ratio reduces to This proportion is thus expressed: 2 is to 36 as 4 is to 72, or 2:36:: 4:72. In any proportion the numerical value of the product of the means = the numerical value of the product of the extremes; thus 2 x 72=4×36, or 144-144; for the proportion may be expressed fractionally as 1, in which the extremes are 18 and 1, and the means are 18 and 1, the products of which must be equal. If therefore any 3 terms out of a proportion be known we can find the third; thus if a:b::c:d where x is required dx = bc,:.x= a:x::c:d then cx ad.. x = ; if

bc
d'

с

=ad if a:b::c:a then bc=

a:b::x:d then bx=ad.. x=

x=

bc

a

a.ab. In simple proportion we generally arrange the terms so that the unknown quantity may form the fourth term; then multiplying the means (2nd and 3rd terms), and dividing the product by the 1st 2nd term × 3rd term term, we obtain x; thus x=

1st term

in

which, of course, we may cancel any common factors out of the numerator and denominator.