Two numbers are in the ratio 2 : 3. If 2 is subtracted from the first and 2 is added to the second, the ratio becomes 1 : 2. The sum of the numbers is

### Concept & Algebraic Formulation When numbers are in a specific ratio, they can be represented as multiples of a common variable. For a ratio of $a : b$, the numbers are $ax$ and $bx$. ### Step-by-Step Solution 1. Let the two numbers be $2x$ and $3x$. 2. According to the given condition, if 2 is subtracted from the first and 2 is added to the second, the new ratio is $1 : 2$. 3. We can write this as an equation: $$ \frac{2x - 2}{3x + 2} = \frac{1}{2} $$ 4. Cross-multiply to solve for $x$: $$ 2(2x - 2) = 1(3x + 2) $$ $$ 4x - 4 = 3x + 2 $$ 5. Isolate $x$: $$ 4x - 3x = 2 + 4 $$ $$ x = 6 $$ 6. The question asks for the sum of the numbers, which is $2x + 3x = 5x$. 7. Substitute the value of $x$: Sum = $5 \times 6 = 30$. ### Exam Strategy & Shortcut **Option Elimination via Multiples:** Since the numbers are $2x$ and $3x$, their sum is $5x$. This means the final answer MUST be a multiple of 5. Looking at the options, only (a) 10 and (d) 30 are multiples of 5. If the sum is 10, $x = 2$, so the numbers are 4 and 6. Applying the condition: $(4-2)/(6+2) = 2/8 = 1/4$ (Incorrect). If the sum is 30, $x = 6$, numbers are 12 and 18. Applying condition: $(12-2)/(18+2) = 10/20 = 1/2$ (Correct!). ### Common Pitfall Students often solve for $x$ and accidentally look for an option that matches $x$ or one of the individual numbers (like 12 or 18) instead of returning to the specific question asked (the sum). ### Final Answer Therefore, the correct answer is **30**.