Two whole numbers are such that the cube of the first number exceeds the cube of the second number by 61 and the ratio of the first number to the second number is 5 : 4. What is the value of the larger number?

Introduction / Context:
This problem combines ratios with cubic expressions. The relationship between the cubes of two numbers is given, along with the ratio of the numbers themselves. The task is to find the larger number. Since cube values grow rapidly, the given difference of 61 strongly suggests small integer values for the numbers, and the ratio condition helps fix them uniquely.

Given Data / Assumptions:

• The cube of the first number minus the cube of the second number equals 61.
• The ratio of the first number to the second number is 5 : 4.
• Both numbers are whole numbers (non negative integers).
• We are asked for the larger of the two numbers.

Concept / Approach:
From the ratio 5 : 4, we express the numbers as 5k and 4k. We substitute these expressions into the given condition involving cubes: (5k)^3 − (4k)^3 = 61. This becomes an equation in k. Because 61 is a prime number and relatively small, we expect k to be a small integer. Solving the resulting cubic equation allows us to find k and then the actual values of the numbers.

Step-by-Step Solution:
Let the larger number be 5k and the smaller number be 4k, since their ratio is 5 : 4. Given that (5k)^3 − (4k)^3 = 61. Compute the cubes: (5k)^3 = 125k^3 and (4k)^3 = 64k^3. So 125k^3 − 64k^3 = 61. This simplifies to 61k^3 = 61. Divide both sides by 61 to obtain k^3 = 1. Thus k = 1 (since we are working with real whole numbers). Therefore the two numbers are 5k = 5 and 4k = 4. The larger number is 5.

Verification / Alternative check:
Substitute the numbers directly into the condition. The cubes are 5^3 = 125 and 4^3 = 64. The difference is 125 − 64 = 61, which matches the given difference. The ratio 5 : 4 also matches the specified ratio of the first number to the second number. Thus both conditions are satisfied, confirming that the larger number is indeed 5.

Why Other Options Are Wrong:
If the larger number were 3, 4, 6 or 7, it would not be possible to find a second whole number such that the cube difference is exactly 61 while also maintaining a ratio of 5 : 4. Testing 6 and 7, for example, leads to cube differences very different from 61 and violates the ratio constraint.

Common Pitfalls:
Some students try to guess both numbers randomly without using the given ratio, which is inefficient. Others expand the cubes but forget to factor out k^3, leading to algebraic mistakes. Recognising that 61k^3 = 61 immediately implies k = 1 makes the problem very simple. Also, checking that the numbers are whole and that both conditions hold is essential.

Final Answer:
The value of the larger number is 5.