Question

A student was asked to simplify the expression: \[ \frac{0.1216 \times 0.105 \times 0.0002}{0.625 \times 0.08512 \times 0.039 \times 0.16} \] His answer was \(\frac{1}{65}\). What is the difference between his answer and the correct answer?

• A. \(\frac{1}{65}\)
• B. \(\frac{1}{130}\)
• C. \(\frac{1}{26}\)
• D. \(\frac{1}{13}\)

Hint:

Expressing decimals as fractions and canceling common factors early makes problems easier and avoids calculation errors.

Answer 1

The Correct Option is B

Step 1: Write numbers as fractions for easier simplification.
\(0.1216 = \frac{1216}{10000} = \frac{76}{625}\),
\(0.105 = \frac{105}{1000} = \frac{21}{200}\),
\(0.0002 = \frac{2}{10000} = \frac{1}{5000}\).
\(0.625 = \frac{625}{1000} = \frac{5}{8}\),
\(0.08512 = \frac{8512}{100000} = \frac{532}{6250}\),
\(0.039 = \frac{39}{1000}\),
\(0.16 = \frac{16}{100} = \frac{4}{25}\).
Step 2: The given expression becomes: \[ \frac{\frac{76}{625} \times \frac{21}{200} \times \frac{1}{5000}} {\frac{5}{8} \times \frac{532}{6250} \times \frac{39}{1000} \times \frac{4}{25}}. \] Step 3: Simplify step-by-step.
The numerator: \[ \frac{76 \times 21 \times 1}{625 \times 200 \times 5000} = \frac{1596}{625 \times 200 \times 5000}. \] The denominator: \[ \frac{5 \times 532 \times 39 \times 4}{8 \times 6250 \times 1000 \times 25}. \] Step 4: Invert and multiply to simplify fully.
After systematic cancellations, the expression simplifies exactly to: \[ \frac{1}{130}. \] Step 5: Find the difference between the student’s answer and the correct answer.
Student’s answer = \(\frac{1}{65} = \frac{2}{130}\). Correct answer = \(\frac{1}{130}\). Difference: \[ \frac{2}{130} - \frac{1}{130} = \frac{1}{130}. \] \[ \boxed{\frac{1}{130}} \]