If the numerator and the denominator of a proper fraction are each decreased by the same positive number (greater than zero), the resulting fraction is:

(a) Always less than the original fraction
(b) Always greater than the original fraction
(c) Always equal to the original fraction
(d) Such that nothing can be claimed definitely

Question

If the numerator and the denominator of a proper fraction are each decreased by the same positive number (greater than zero), the resulting fraction is:

(a) Always less than the original fraction
(b) Always greater than the original fraction
(c) Always equal to the original fraction
(d) Such that nothing can be claimed definitely

ONLiNE UPSC · Accepted Answer

Let the original proper fraction be: a/b, where 0 < a < b Now subtract a positive number k from both numerator and denominator (i.e., k > 0 and a > k, b > k New fraction = (a − k) / (b − k) Let’s compare (a − k) / (b – k) with a/b We want to know if: (a − k) / (b – k) < a/b Cross-multiplying: (a – k) * b < a * (b − k) ab – kb < ab – ak −kb < −ak kb > ak b > a Which is true, since a < b (because it's a proper fraction). Therefore, the new fraction is always less than the original. Hence, option a is correct.

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E Question 1	⏱ 0
If the numerator and the denominator of a proper fraction are each decreased by the same positive number (greater than zero), the resulting fraction is: (a) Always less than the original fraction (b) Always greater than the original fraction (c) Always equal to the original fraction (d) Such that nothing can be claimed definitely