Two numbers are such that the square of one is 224 less than 8 times the square of the other. If the numbers be in the ratio of 3 : 4, the numbers are

## Concept & Logic When two numbers are given as a ratio, they can be expressed using a common variable multiplier (e.g., $3x$ and $4x$). The problem translates to an algebraic equation relating their squares. $$ (4x)^2 = 8(3x)^2 - 224 $$ ## Step-by-Step Solution * **Given:** The ratio of the numbers is $3:4$. Let the numbers be $3x$ and $4x$. * **Calculation / Deduction:** * Square of the first number = $(3x)^2 = 9x^2$ * Square of the second number = $(4x)^2 = 16x^2$ * According to the condition, the square of one is 224 less than 8 times the square of the other. Let's test the relation: $16x^2 = 8(9x^2) - 224$ * $16x^2 = 72x^2 - 224$ * $56x^2 = 224$ * $x^2 = 4$ * $x = 2$ * The numbers are $3(2) = 6$ and $4(2) = 8$. ## Exam Strategy & Shortcut **Option Elimination:** Instead of solving the quadratic equation, directly test the options. Option (a) 6 and 8: Square of 8 is 64. 8 times the square of 6 is $8 \times 36 = 288$. The difference is $288 - 64 = 224$. This matches the condition perfectly, saving you valuable time. ## Common Pitfall Students often misinterpret "square of one is 224 less than 8 times the square of the other" and set up the equation backwards, subtracting from the wrong variable. Always use common sense or option verification to confirm the logical relationship. ## Final Answer **Therefore, the correct answer is 6, 8.**