Questions: A positive integer is twice that of another. The sum of the reciprocals of the two positive integers is 1 / 4. Find the two integers. Smaller Integer: Larger Integer:

Solution Steps

To solve this problem, we need to set up a system of equations based on the given conditions. Let the smaller integer be \( x \). Then the larger integer, which is twice the smaller, will be \( 2x \). According to the problem, the sum of the reciprocals of these two integers is \( \frac{1}{4} \). This gives us the equation:

\[ \frac{1}{x} + \frac{1}{2x} = \frac{1}{4} \]

We can solve this equation to find the value of \( x \), and subsequently, \( 2x \).

Let the smaller integer be \( x \). The larger integer, being twice the smaller, is \( 2x \). According to the problem, the sum of the reciprocals of these two integers is given by:

\[ \frac{1}{x} + \frac{1}{2x} = \frac{1}{4} \]

We can combine the left-hand side:

\[ \frac{1}{x} + \frac{1}{2x} = \frac{2}{2x} + \frac{1}{2x} = \frac{3}{2x} \]

Thus, the equation simplifies to:

\[ \frac{3}{2x} = \frac{1}{4} \]

Cross-multiplying gives:

\[ 3 \cdot 4 = 2x \implies 12 = 2x \implies x = 6 \]

Now, we can find the larger integer:

\[ 2x = 2 \cdot 6 = 12 \]

Final Answer