Two numbers are in the ratio 3 : 5. If 9 is added to each, the new ratio becomes 3 : 4. The smaller number is:

Explanation

Let the two numbers be 3k and 5k because their original ratio is 3 : 5. After 9 is added to each number, the new ratio is given as 3 : 4, so (3k + 9)/(5k + 9) = 3/4. NCERT's Class VII exemplar on comparing quantities states that equal ratios form a proportion and that cross products in proportions are equal, which justifies cross-multiplying here. Thus, 4(3k + 9) = 3(5k + 9), giving 12k + 36 = 15k + 27. Rearranging gives 3k = 9, so k = 3. The smaller number is therefore 3k = 9.

Why the other options are wrong

• (A) 15 is the larger number, since 5k = 5 x 3 = 15, not the smaller number asked for.
• (B) 12 does not fit the original 3 : 5 form after solving the proportion, because the scale factor is k = 3, not 4.
• (D) 18 would require the smaller number 3k to equal 18, but the proportion gives k = 3 and hence 3k = 9.

Concept

This tests ratio and proportion in Reasoning & Mental Ability: convert a given ratio into algebra, then use the equality of two ratios. It recurs in RAS-style arithmetic because a short proportion often decides the answer faster than trial and error.

Source

• NCERT Exemplar Problems, Class VII Mathematics: Comparing Quantities

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