Question

A positive integers is twice another. The difference of the reciprocals of the two positive integers is (1/18). Find the two integers.

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Madelyn

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Let the integers be \( x \) and \( y \), where \( y = 2x \).

The reciprocals of the integers are \( \frac{1}{x} \) and \( \frac{1}{y} \), respectively.

The difference of the reciprocals is given by:

\frac{1}{x} - \frac{1}{y} = \frac{1}{18}

Substitute \( y = 2x \) into the equation:

\frac{1}{x} - \frac{1}{2x} = \frac{1}{18}

Simplify the left side:

\frac{2 - 1}{2x} = \frac{1}{18}

This reduces to:

\frac{1}{2x} = \frac{1}{18}

Solve for \( x \):

2x = 18

x = 9

If \( x = 9 \):

y = 2x = 18

So the two integers are:

x = 9, \quad y = 18

The reciprocals of the integers are \( \frac{1}{x} \) and \( \frac{1}{y} \), respectively.

The difference of the reciprocals is given by:

Substitute \( y = 2x \) into the equation:

Simplify the left side:

This reduces to:

Solve for \( x \):

If \( x = 9 \):

So the two integers are:

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