Question

Eric has a riddle: "I am thinking of a fraction that is equivalent to 25
and the numerator is 18 less than the denominator."

What is Eric's fraction? Explain how you know.

Answer 1

To solve Eric's riddle, we need to find a fraction that is equivalent to 25, and in which the numerator is 18 less than the denominator.

Let's denote the denominator of the fraction as \( d \). According to the riddle, the numerator \( n \) can be expressed as:

\[
n = d - 18
\]

The fraction is then:

\[
\frac{n}{d} = \frac{d - 18}{d}
\]

We know that this fraction is equivalent to 25. This gives us the equation:

\[
\frac{d - 18}{d} = 25
\]

To eliminate the fraction, we multiply both sides by \( d \) (assuming \( d \neq 0 \)):

\[
d - 18 = 25d
\]

Now, let's rearrange the equation to isolate \( d \):

\[
d - 25d = 18
\]
\[
-24d = 18
\]

Now, divide both sides by -24:

\[
d = -\frac{18}{24} = -\frac{3}{4}
\]

To find the numerator \( n \), substitute \( d \) back into the equation for \( n \):

\[
n = d - 18
\]
\[
n = -\frac{3}{4} - 18
\]

To combine these, we can express 18 as a fraction over 4:

\[
n = -\frac{3}{4} - \frac{72}{4} = -\frac{75}{4}
\]

So the fraction is:

\[
\frac{n}{d} = \frac{-\frac{75}{4}}{-\frac{3}{4}} = \frac{75}{3}
\]

Now simplify \( \frac{75}{3} \):

\[
\frac{75}{3} = 25
\]

This verifies that \( \frac{n}{d} = 25 \) holds true. Thus, Eric's fraction is:

\[
- \frac{75}{4} / -\frac{3}{4} = 25.
\]

So, the solution shows that Eric's fraction is equivalent to 25, and the terms fit the criteria specified in the riddle. The fraction itself is \( \frac{-75/4}{-3/4} \).