The sum of two numbers is 40 and their difference is 4. The ratio of the numbers is

## Concept & Formula When required to find the ratio of two numbers given their sum ($S$) and difference ($D$), you can use the direct formula to bypass calculating the actual numbers. $$ \text{Ratio} = \frac{S + D}{S - D} $$ ## Step-by-Step Solution * **Given:** The sum of two numbers ($x + y$) is $40$. Their difference ($x - y$) is $4$. * **Calculation / Deduction:** * Add the equations to find the larger number ($x$): * $2x = 44 \Rightarrow x = 22$ * Subtract the equations to find the smaller number ($y$): * $2y = 36 \Rightarrow y = 18$ * The ratio of the numbers is $x : y = 22 : 18$ * Simplify the ratio by dividing by $2$: * $11 : 9$ ## Exam Strategy & Shortcut **Direct Ratio Formula:** Use the formula derived in the concept section for immediate results. Ratio $= \frac{40 + 4}{40 - 4} = \frac{44}{36}$. Divide the numerator and denominator by $4$ to get $11/9$, or $11:9$. This eliminates the intermediate variables entirely. ## Common Pitfall Failing to simplify the final ratio. An unsimplified ratio like $22:18$ will not appear in standard exam options, potentially causing panic. Always reduce fractions to their simplest terms. ## Final Answer **Therefore, the correct answer is 11 : 9.**