Two numbers are in the ratio 7 : 11. If 7 is added to each of the numbers, the ratio becomes 2 : 3. The smaller number is

### Concept & Ratio Transformation When a constant value is added to two numbers in a given ratio, we can establish an algebraic equation by expressing the original numbers in terms of a common variable and setting their new ratio equal to the given condition. ### Step-by-Step Solution * Let the two numbers be $7x$ and $11x$. * According to the problem, $7$ is added to each number, making the new numbers $(7x + 7)$ and $(11x + 7)$. * The new ratio is $2 : 3$. Set up the equation: $$ \frac{7x + 7}{11x + 7} = \frac{2}{3} $$ * Cross-multiply to solve for $x$: $$ 3(7x + 7) = 2(11x + 7) $$ $$ 21x + 21 = 22x + 14 $$ * Rearrange the terms to isolate $x$: $$ 22x - 21x = 21 - 14 $$ $$ x = 7 $$ * The smaller number is $7x$. * Calculate the smaller number: $7 \times 7 = 49$. ### Exam Strategy & Shortcut Since the smaller number is $7x$, the correct answer must be a multiple of $7$. Options (b) $49$ and (d) $77$ are multiples of $7$. Test $49$: if the smaller number is $49$, then $x = 7$, making the larger number $77$. Adding $7$ to both gives $56$ and $84$. The ratio $56:84$ simplifies to $2:3$, which matches the given condition perfectly. ### Common Pitfall A common mistake is solving for $x$ (which is $7$) and forgetting to multiply it back into the original ratio term ($7x$) to find the actual value of the smaller number. ### Final Answer Therefore, the correct answer is **49**.