The ratio of two numbers is 3 : 5. If 6 is subtracted from each, the resulting numbers are in the ratio 1 : 2. Find the differenc…

Question

The ratio of two numbers is 3 : 5. If 6 is subtracted from each, the resulting numbers are in the ratio 1 : 2. Find the difference between the two original numbers.

1. A. 14
2. B. 16
3. C. 10
4. D. 12 (Correct answer)

Correct Answer

Option D — 12

Detailed Solution & Explanation

The correct answer is 12.

Shortcut Trick

• The difference between the parts is $5 - 3 = 2$ units, so the answer is simply 2 × (value of one unit).
• Cross-multiplying the second condition: $2(3x - 6) = 1(5x - 6) \Rightarrow x = 6$.
• Difference $= 2x = \textbf{12}$.

Alternate Method

• Let the numbers be $3x$ and $5x$.
• After subtracting 6: $\dfrac{3x - 6}{5x - 6} = \dfrac{1}{2}$
  • $6x - 12 = 5x - 6$
  • $x = 6$
• The numbers are $18$ and $30$; a check gives $\dfrac{12}{24} = \dfrac{1}{2}$ ✓
• Difference $= 30 - 18 = \textbf{12}$

Additional Information

• The difference between terms of a ratio is itself a fixed multiple of the unit, which is why you rarely need both numbers. Here the gap is always $2x$, whatever $x$ turns out to be.
• Subtracting the same amount from both terms changes a ratio (18:30 = 3:5 becomes 12:24 = 1:2), whereas multiplying or dividing both by the same number leaves it unchanged. That asymmetry is what makes these questions solvable.
• Always verify by substituting back — it costs seconds and catches sign errors in the cross-multiplication.