A student was asked to simplify the following : `((3)/(1.6)+(5)/(3.2))/((5)/(4.8)+(1)/(9.6))`
His answer was `3(1)/(5)`. Find the per cent error in the answer

Options

A. `10%`
B. `20%`
C. `20/3%`
D. `30%`

Answer

To solve the problem of finding the percent error in the student's answer, we will follow these steps:

### Step 1: Simplify the Expression
We start with the expression:
\[ \frac{\frac{3}{1.6} + \frac{5}{3.2}}{\frac{5}{4.8} + \frac{1}{9.6}} \]

### Step 2: Convert Decimal to Fraction
Convert the decimal numbers to fractions:
- \(1.6 = \frac{16}{10} = \frac{8}{5}\)
- \(3.2 = \frac{32}{10} = \frac{16}{5}\)
- \(4.8 = \frac{48}{10} = \frac{24}{5}\)
- \(9.6 = \frac{96}{10} = \frac{48}{5}\)

Now the expression becomes:
\[ \frac{\frac{3}{\frac{8}{5}} + \frac{5}{\frac{16}{5}}}{\frac{5}{\frac{24}{5}} + \frac{1}{\frac{48}{5}}} \]

### Step 3: Simplify Each Fraction
Now, simplify each fraction:
- \(\frac{3}{\frac{8}{5}} = 3 \times \frac{5}{8} = \frac{15}{8}\)
- \(\frac{5}{\frac{16}{5}} = 5 \times \frac{5}{16} = \frac{25}{16}\)
- \(\frac{5}{\frac{24}{5}} = 5 \times \frac{5}{24} = \frac{25}{24}\)
- \(\frac{1}{\frac{48}{5}} = 1 \times \frac{5}{48} = \frac{5}{48}\)

Now the expression becomes:
\[ \frac{\frac{15}{8} + \frac{25}{16}}{\frac{25}{24} + \frac{5}{48}} \]

### Step 4: Find a Common Denominator
Find a common denominator for the numerators and denominators:
- For \(\frac{15}{8} + \frac{25}{16}\), the common denominator is 16:
\[ \frac{15 \times 2}{16} + \frac{25}{16} = \frac{30 + 25}{16} = \frac{55}{16} \]

- For \(\frac{25}{24} + \frac{5}{48}\), the common denominator is 48:
\[ \frac{25 \times 2}{48} + \frac{5}{48} = \frac{50 + 5}{48} = \frac{55}{48} \]

### Step 5: Rewrite the Expression
Now we can rewrite the expression:
\[ \frac{\frac{55}{16}}{\frac{55}{48}} = \frac{55}{16} \times \frac{48}{55} = \frac{48}{16} = 3 \]

### Step 6: Compare with Student's Answer
The student's answer was \(3 \frac{1}{5} = \frac{16}{5}\).

### Step 7: Calculate the Percent Error
To find the percent error, we use the formula:
\[ \text{Percent Error} = \frac{\text{Reported Value} - \text{Actual Value}}{\text{Actual Value}} \times 100 \]
Substituting the values:
\[ \text{Percent Error} = \frac{\frac{16}{5} - 3}{3} \times 100 \]

Convert 3 to a fraction:
\[ 3 = \frac{15}{5} \]
So,
\[ \text{Percent Error} = \frac{\frac{16}{5} - \frac{15}{5}}{3} \times 100 = \frac{\frac{1}{5}}{3} \times 100 = \frac{1}{15} \times 100 = \frac{100}{15} \approx 6.67\% \]

### Final Answer
The percent error in the student's answer is approximately \(6.67\%\).

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