It is evident that of two fractions which have the same denominator, the greater has the greater numerator; and also that of two fractions which have the same numerator, the greater has the less denominator. Thus, ⁸/₇ is greater than ⁸/⁹, since the first is a 7th, and the last only a 9th part of 8. Also, any numerator may be made to belong to as small a fraction as we please, by sufficiently increasing the denominator. Thus, ¹⁰/₁₀₀ is ¹/₁₀, ¹⁰/₁₀₀₀ is ¹/₁₀₀, and ¹⁰/₁₀₀₀₀₀₀ is ¹/₁₀₀₀₀₀₀ (108). We can now also increase and diminish the first fraction by the second. For the first fraction is made up of 15 of the 30 equal parts into which 1 is divided. The second fraction is 14 of those parts. The sum of the two, therefore, must be 15 + 14, or 29 of those parts; that is, ½ + ⁷/₁₅ is ²⁹/₃₀. The difference of the two must be 15-14, or 1 of those parts; that is, ½-⁷/₁₅ = ¹/₃₀. 113. From the last two articles the following rules are obtained: I. To compare, to add, or to subtract fractions, first reduce them to a common denominator. When this has been done, that is the greatest of the fractions which has the greatest numerator. Their sum has the sum of the numerators for its numerator, and the common denominator for its denominator. Their difference has the difference of the numerators for its numerator, and the common denominator for its denominator. EXERCISES. 1 1 1 1 53 44 153 18329 --- + --- + --- - --- = ---- ---- - ----- = ------- 2 3 4 5 60 3 427 1282 8 3 4 1834 1 12 253 1 + ---- + ---- + ---- = ---- 2 - --- + ---- = ---- 10 100 1000 1000 7 13 91 1 8 94 3 163 97 93066 --- + --- + ---- = --- --- - ---- = ------- 2 16 188 2 521 881 459001 114. Suppose it required to add a whole number to a fraction, for example, 6 to ⁴/₉. By (106) 6 is ⁵⁴/₉, and ⁵⁴/₉ + ⁴/₉ is ⁵⁸/⁹; that is, 6 + ⁴/⁹, or as it is usually written, (6⁴/₉), is ⁵⁸/₉. The rule in this case is: Multiply the whole number by the denominator of the fraction, and to the product add the numerator of the fraction; the sum will be the numerator of the result, and the denominator of the fraction will be its denominator. Thus, (3¼) = ¹³/₄, (22⁵/₉) = ²⁰³/₉, (74²/₅₅) = ⁴⁰⁷²/₅₅. This rule is the opposite of that in (105). 115. From the last rule it appears that 907 17230907 225 667225 1723 ------ is --------, 667 ----- is ------, 10000 10000 1000 1000 99 2300099 and 23 ------ is -------. 10000 10000