else than the ratio : §, and this, by the reasoning of Art. 11, is equivalent to the ratio 7 × 8:9 × 5, and we hence get Now, is the divisor when inverted, and hence the above rule. We have introduced a bracket on the right side of the first equality, for otherwise the sign ÷ affects only the first fraction 10. On the other side a bracket is unnecessary, for the sign standing before a compound fraction (not two fractions) affects the whole. Ex. 3.-Simplify the expression 11% ÷ 63 × 41÷2 of 24. 1. Find the sum, difference, and product of 2 and 12. 2. Multiply the sum of the fractions 33, 21% by their difference. - 3. Simplify the expression { (3,7) ̊ – (1+4)} + {3, − 144}· 4. Reduce to a simple fraction each of the following expressions (1.) 17 x 83 ÷ 23 of 204. 5. What is the difference between (81 – 37) and (5} − 47',)? 8. Simplify the expressions (1.) — 24 — + 9. Find the quotient of 1038 by 301 of 783. 10. The cost of 7 articles is £65%, what is the cost of each article? 11. Find the cost of 894 articles, when one cost £4. 12. The sum of two quantities is 347, and their difference is 64; required the greater. Reduction of Fractions to Decimals. 24. If we place a decimal point to the right of an integer, and add as many ciphers as we please, it is clear, from Art. 1, that we do not alter its value. And hence a given ratio, as 38, is not altered in value by writing it 3.000 : 8; and further, dividing each of its terms by 8, according to the rule for division of decimals, it becomes 375 : 1. It therefore follows, putting each of these ratios in a fractional form, that We get, therefore, the following rule :— RULE.-Place a decimal point to the right of the numerator, and add as many ciphers as may be thought necessary. Divide the new numerator by the given denominator, according to the rule for division of decimals, and, if necessary, add ciphers to the successive remainders until the division terminates, or until we have obtained as many decimal figures as required. It will be seen that we have arrived at a remainder, 264, exactly the same as the second remainder; and that, therefore, the quotient figures 891 will continually repeat, and that the division will never terminate. We call 891 the recurring period of the decimal, and it is usual to indicate the fact of its recurrence by placing dots over its first and last figures, as above. We have, therefore, as a result, 6 = 016891. NOTE. It is easy to see that no fraction, reduced to its lowest terms, whose denominator contains any prime factor, other than 2 or 5, can be expressed as a terminating decimal. For every terminating decimal is an exact number of tenths, hundredths, &c., and may, therefore, be transformed into a fraction, having some power of 10 as its denominator. Now, if we wish to bring a fraction already in its lowest terms to an equivalent fraction having some power of 10 for its denominator, it can only be done by multiplying its numerator and denominator by some integer; and it is impossible to obtain any power of 10 by multiplication only, from a number which contains any prime factor, other than 2 or 5. Reduction of Terminating Decimals to Fractions. 25. Remembering (Art. 1) that any given terminating decimal may be considered as derived from an integer by diminishing it 10, or 102, 103, &c. fold, we have Hence, 345 is the value of the ratio 345: 103; and we 345 345 have, also, 345 = = 103 1000 The following rule is, therefore, clear :— RULE.-Make a numerator of the integer, formed by taking away the decimal point; and for a denominator put 1, followed by as many ciphers as there are given decimal figures. 26. There are two kinds, and it is convenient to treat them separately. (1.) Pure circulating decimals, where the whole of the decimal figures recur. RULE. Take away the decimal point and the dots, make a numerator of the integer thus obtained, and place under this as denominator as many nines as there are recurring figures. The following example will make this rule clear. Ex. Reduce 207 to a fraction. The value of the decimals is evidently not altered by writing it -207207. Let us remove the decimal point three places to the right, or, what is the same thing (Art. 1), let us multiply the given decimal by 1000. decimal. We then get 207-207 as the value of 1000 times the given Now the number 207-207 includes the integer 207, and the given decimal ·207; and it therefore follows that the integral part 207 is (1000-1), or 999 times the value of the given decimal. Hence, dividing it by 999, we get (2.) Mixed circulating decimals.—Where part only of the figures recurs. RULE.-Take away the decimal point and the dots, subtract from the integer thus obtained the integer formed by the figures which do not recur, and make a numerator of the result. Then, for a denominator, place underneath as many nines as there are recurring figures, followed by as many ciphers as there are figures which do not recur. We shall make this rule clear by the following example:Ex.-Reduce 24573 to a fraction. Let us remove the decimal point two places to the right, thus, by Art. 1, multiplying the given decimal by 100; we then get 100 times the value of 24573 = 24.573. Now, by case (1) above, 24.573 =24838; or reducing to an improper fraction, and noticing that 24 × 999 24000 24, we have Hence, dividing this result by 100, we get = 24573 24 100 24573 24 999 24 99900 (Art. 24.) =. 27. In arithmetical operations involving circulating decimals, and, indeed, any decimals having a large number of decimal figures, it is generally sufficient to obtain a result correct to a given number of decimals. |