To solve the problem:

Problem Details:

• One number is four times another number.
• The sum of their reciprocals is $\frac{1}{4}$.

Let:

• The smaller number be $x$.
• The larger number be $4x$ (since it is four times the smaller number).

The reciprocals are: $\frac{1}{x} \text{ and } \frac{1}{4x}.$

The equation based on the sum of their reciprocals is: $\frac{1}{x} + \frac{1}{4x} = \frac{1}{4}.$

Step 1: Combine the fractions

$\frac{1}{x} + \frac{1}{4x} = \frac{4}{4x} + \frac{1}{4x} = \frac{5}{4x}.$

Thus, the equation becomes: $\frac{5}{4x} = \frac{1}{4}.$

Step 2: Solve for $x$

Multiply through by $4x$ to eliminate the denominator: $5 = x.$

Step 3: Find the numbers

• The smaller number is $x = 5$.
• The larger number is $4x = 20$.

Final Answer:

The numbers are $5$ and $20$.

Would you like more details or further clarifications?
Here are 5 related questions you might find helpful:

1. How can you verify the solution?
2. What happens if the reciprocal sum was different?
3. How would the solution change if the larger number was three times the smaller number?
4. Can you explain the concept of reciprocals and their properties in solving equations?
5. What other real-world problems can this type of equation model?

Tip: Always double-check by substituting your solution back into the original equation!