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Bot GPT 4-o mini To solve the equation \( \frac{2f}{5} - \frac{8}{5} + 2f = 8 \), we can follow these steps using the properties of equality:
1. **Combine like terms**: Start with the terms that involve \( f \).
The term \( 2f \) can be written as \( \frac{2f}{1} \) and converted to have a common denominator with \( \frac{2f}{5} \). To do this, we can convert \( 2f \) into fifths:
\[
2f = \frac{10f}{5}
\]
Now, we can combine the terms involving \( f \):
\[
\frac{2f}{5} + \frac{10f}{5} = \frac{12f}{5}
\]
Rewriting the original equation, we have:
\[
\frac{12f}{5} - \frac{8}{5} = 8
\]
2. **Isolate the term with \( f \)**: To isolate \( \frac{12f}{5} \), we need to add \( \frac{8}{5} \) to both sides of the equation:
\[
\frac{12f}{5} = 8 + \frac{8}{5}
\]
To add \( 8 \) and \( \frac{8}{5} \), we will convert \( 8 \) into fifths:
\[
8 = \frac{40}{5}
\]
Thus,
\[
8 + \frac{8}{5} = \frac{40}{5} + \frac{8}{5} = \frac{48}{5}
\]
3. **Now the equation is**:
\[
\frac{12f}{5} = \frac{48}{5}
\]
4. **Eliminate the denominator**: Multiply both sides of the equation by \( 5 \) to eliminate the fraction:
\[
12f = 48
\]
5. **Solve for \( f \)**: Divide both sides by \( 12 \):
\[
f = \frac{48}{12} = 4
\]
Thus, the solution to the equation is:
\[
\boxed{4}
\]