Thus, taking the expression - (21 - 31): We must remember that the sign of 24 when within the bracket is, according to a remark made in Ex. 3, understood to be plus. In fact, though it is unusual, we might write the expression thus: (+231). Now, remove the bracket, and it becomes 23. 1. Add together (1.), t, t; (2.) §, To, tt; (3.) †, TT, TE. 2. Find the sum of 34, 1711, 21%, 17, §, 33. 3. Add together 23, 3 5 4' 24' 2 of 11, 51 of 1's. 4. Find the difference between (1.) 7 and ; (2.) & and §; (3.) and . 5. Subtract (1.) 63 from 8; (2.) 33 from 41; (3.) 27% from 61. 8. Simplify the expression (211) – (3§ – 73). 9. By how much does 3-1 exceed 218 - ? 10. Take the difference of 67 and 1 from their sum. 11. Add the difference of the same two fractions to their sum. 12. Find the value of the expression 3 of 13 14 Multiplication of Fractions. 22. RULE.-Multiply together the numerators of the fractions for a new numerator, and the denominators for a new denominator. The reason of this rule is easily seen. Let it be required to find the product of 3 and 7, or the value of 3 × 7. Now, what is the meaning of multiplying the ratio 3: 5 by the ratio 7:8? It means evidently that the ratio 3: 5 is to be multiplied by 7, and the result divided by 8. Now (Art. 8) the ratio 3: 5, when multiplied by 7, becomes 3 × 7:5; and (Art. 9) the ratio 3 x 7:5, when divided by 8, becomes 3 x 7:5 x 8; and we have And so on for any number of fractions. Hence the above rule. Ex 1.-Multiply together the fractions,,. Before actually performing the operation of multiplication, it is advisable to strike out any factor common to both numerator and denominator. We see that 5 is common to 5 and 25, 3 common to 3 and 6, 4 common to 12 and 8, and we then have 23. RULE.-Invert the divisor, and proceed as in multiplication. To explain this rule, let us endeavour to divide by §. We may evidently consider the required quotient as nothing else than the ratio : §, and this, by the reasoning of Art. 11, is equivalent to the ratio 7 x 8:9 x 5, and we hence get 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 17 ÷ 63 × 41 ÷ 2 of 24. The given expression= 132 × 47 ÷ (3 × 280) 17 ÷ = Ex. VI. 3 1. Find the sum, difference, and product of 23 and 14. 2. Multiply the sum of the fractions 33, 21% by their difference. 2 3. Simplify the expression { (3,2) ̊ – (114)})} ÷ {3,2 – 144}· 4. Reduce to a simple fraction each of the following expressions (1.) 11 ÷ 74 × 83 ÷ 2 of 204. 5. What is the difference between (81 – 37) and (51 – 4,'a)? 3 + 1 7. Reduce to a simple fraction 3 + 7 +15 8. Simplify the expressions (1.) ΤΣ 61 2-18 13 + 7 (2.) 1 of 14 of 21 × {31 - (2) - 1)}. 9. Find the quotient of 10318 by 301 of 83. 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 34,7, 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 add 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, 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, 1 •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 deci. mal is an exact number of tenths, hundredths, &c., and may, there fore, be transformed into a fraction, having some power of 10 as its |