Two numbers are such that their difference, their sum and their product are to one another as 1 : 7 : 24. The product of the two numbers is

## Concept & Logic When a continuous ratio is given for algebraic expressions (like difference, sum, and product), assign a common proportionality constant (like $k$) to convert the ratios into exact equations. ## Step-by-Step Solution * **Given:** Let the two numbers be $x$ and $y$. The ratio is $(x - y) : (x + y) : (xy) = 1 : 7 : 24$. * **Calculation / Deduction:** * Let the exact values be represented using a multiplier $k$: * Difference: $x - y = k$ * Sum: $x + y = 7k$ * Product: $xy = 24k$ * First, use the sum and difference equations to find $x$ and $y$ in terms of $k$: * Add them: $(x - y) + (x + y) = k + 7k \Rightarrow 2x = 8k \Rightarrow x = 4k$. * Subtract them: $(x + y) - (x - y) = 7k - k \Rightarrow 2y = 6k \Rightarrow y = 3k$. * Now, substitute these expressions for $x$ and $y$ into the product equation: * $$ (4k)(3k) = 24k $$ * $$ 12k^2 = 24k $$ * Since the numbers must be non-zero to maintain the ratio, we can divide by $12k$: * $k = 2$ * The question asks for the product of the two numbers, which is represented by $24k$. * Product $= 24(2) = 48$. ## Exam Strategy & Shortcut **Proportion Mapping:** Focus only on the first two parts of the ratio: Difference is $1$ part, Sum is $7$ parts. Larger number = (Sum + Diff)/2 = $(7+1)/2 = 4$ parts. Smaller number = (Sum - Diff)/2 = $(7-1)/2 = 3$ parts. Their product should be $4 \times 3 = 12$ square parts. The given ratio states the product is $24$ parts. So, $12 \text{ parts}^2 = 24 \text{ parts} \Rightarrow 1 \text{ part} = 2$. Actual product is $24 \text{ parts} = 24 \times 2 = 48$. ## Common Pitfall Students often solve for $k=2$ and then mistakenly plug it back in to find $x=8$ and $y=6$, and then forget what the question was asking, potentially wasting time or looking for '8' or '6' in the options. Always read the final question clearly: "The product of the two numbers is..." ## Final Answer **Therefore, the correct answer is 48.**