To find out how many rock songs Fernando should have on his road trip playlist, we first need to determine the ratio of country songs to rock songs on his summer playlist.
Fernando has:
- 100 country songs
- 24 rock songs
The ratio of country songs to rock songs is: \[ \text{Ratio} = \frac{\text{Country songs}}{\text{Rock songs}} = \frac{100}{24} \]
We can simplify this ratio by dividing both the numerator and denominator by their greatest common divisor (GCD), which is 4: \[ \frac{100 \div 4}{24 \div 4} = \frac{25}{6} \]
This tells us that for every 25 country songs, Fernando has 6 rock songs.
Now, we know that on his road trip playlist, he has 25 country songs. To find the number of rock songs, we can set up a proportion based on the established ratio of 25 country songs to 6 rock songs: \[ \frac{25 \text{ country songs}}{6 \text{ rock songs}} = \frac{25 \text{ country songs}}{x \text{ rock songs}} \]
Cross-multiplying gives: \[ 25x = 25 \times 6 \]
Solving for \(x\): \[ 25x = 150 \] \[ x = \frac{150}{25} = 6 \]
Therefore, the number of rock songs on Fernando's road trip playlist is \( \boxed{6} \).
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