Question

Leonardo is solving the equation 4 (x minus one-fifth) = 2 and two-thirds. His work is shown. Where is his error?

4 (x minus one-fifth) = 2 and two-thirds. Step 1, 4 x minus four-fifths = 2 and two-thirds. Step 2, 4 x = StartFraction 8 over 3 EndFraction + four-fifths. Step 3, 4 x StartFraction 40 over 15 EndFraction + StartFraction 16 over 15 EndFraction. Step 4, 4 x = StartFraction 56 over 15 EndFraction times one-fourth. X = StartFraction 14 over 15 EndFraction.

step 1
step 2
step 3
step 4

Answer 1

Let's analyze Leonardo's work step by step to identify where the error occurred.

1. **Starting Equation**:
\[
4 \left( x - \frac{1}{5} \right) = 2 \frac{2}{3}
\]

2. **Convert Mixed Number to Improper Fraction**:
\[
2 \frac{2}{3} = \frac{8}{3}
\]
So the equation is:
\[
4 \left( x - \frac{1}{5} \right) = \frac{8}{3}
\]

3. **Step 1**: Distributing the 4:
\[
4x - 4 \cdot \frac{1}{5} = \frac{8}{3}
\]
This simplifies to:
\[
4x - \frac{4}{5} = \frac{8}{3}
\]
- **This step is correct.**

4. **Step 2**: Adding \(\frac{4}{5}\) to both sides:
\[
4x = \frac{8}{3} + \frac{4}{5}
\]

To add \(\frac{8}{3}\) and \(\frac{4}{5}\), we need a common denominator. The least common multiple of 3 and 5 is 15.

Convert each fraction:
\[
\frac{8}{3} = \frac{8 \cdot 5}{3 \cdot 5} = \frac{40}{15}
\]
\[
\frac{4}{5} = \frac{4 \cdot 3}{5 \cdot 3} = \frac{12}{15}
\]
So,
\[
4x = \frac{40}{15} + \frac{12}{15} = \frac{52}{15}
\]
- **This is where Leonardo makes an error**. He wrote \(4x = \frac{56}{15}\) instead of \(\frac{52}{15}\).

5. **Step 3**: Dividing by 4 to solve for \(x\):
Correct calculation based on the correct equation:
\[
x = \frac{52}{15} \cdot \frac{1}{4} = \frac{52}{60} = \frac{13}{15}
\]
Leonardo incorrectly calculated \(x = \frac{14}{15}\).

The error is in **Step 2**, where he incorrectly added \(\frac{8}{3} + \frac{4}{5}\) and ended up with \(\frac{56}{15}\) instead of the correct sum, which is \(\frac{52}{15}\).