Two numbers are such that the ratio between them is 4 : 7. If each is increased by 4, the ratio becomes 3 : 5. The larger number is

## Concept & Logic When a constant is added to two numbers in a given ratio, cross-multiplication of the new fraction provides a direct linear equation to solve for the base multiplier. $$ \frac{ax + k}{bx + k} = \frac{c}{d} $$ ## Step-by-Step Solution * **Given:** The initial ratio is $4:7$. Let the numbers be $4x$ and $7x$. The constant added is $4$, making the new ratio $3:5$. * **Calculation / Deduction:** * Set up the equation based on the new ratio: * $$ \frac{4x + 4}{7x + 4} = \frac{3}{5} $$ * Cross-multiply to solve for $x$: * $5(4x + 4) = 3(7x + 4)$ * $20x + 20 = 21x + 12$ * Bring like terms together: $20 - 12 = 21x - 20x$ * $x = 8$ * The question asks for the larger number, which corresponds to $7x$. * Larger number = $7 \times 8 = 56$. ## Exam Strategy & Shortcut **Cross-Product Method:** Write the initial ratio $(4:7)$ and final ratio $(3:5)$ vertically. Write the change $(+4)$ below them. Cross multiply ratios: $(7 \times 3) - (4 \times 5) = 21 - 20 = 1$ unit. Cross multiply lower part: $(5 \times 4) - (3 \times 4) = 20 - 12 = 8$. So, $1$ unit $= 8$. Larger number $= 7 \text{ units} \times 8 = 56$. ## Common Pitfall A common error is solving for $x = 8$ and immediately looking for $8$ in the options, or mistakenly calculating the smaller number ($4 \times 8 = 32$) instead of the requested larger number. Always underline what exactly the question is asking for. ## Final Answer **Therefore, the correct answer is 56.**