Question

The decimal expansion of $\frac{43}{2^4 \cdot 5^3}$ will terminate after how many places?

• A. 7
• B. 4
• C. 5
• D. 3

Hint:

For terminating decimals: \[ \text{Decimal places} = \max(\text{power of 2}, \text{power of 5}) \]

Answer 1

The Correct Option is B

Concept: If a fraction is of the form: \[ \frac{p}{2^m \cdot 5^n} \] then the number of decimal places after which it terminates is: \[ \max(m,n) \]

Step 1: Identify powers of 2 and 5.
\[ \frac{43}{2^4 \cdot 5^3} \] So: \[ m = 4,\quad n = 3 \]

Step 2: Find maximum exponent.
\[ \max(4,3) = 4 \] So decimal will terminate after 4 places.

Step 3: Verification.
Make denominator equal powers: \[ \frac{43}{2^4 \cdot 5^3} \times \frac{5}{5} = \frac{215}{2^4 \cdot 5^4} = \frac{215}{10^4} = 0.0215 \] There are 4 digits after decimal. Conclusion: \[ \boxed{4} \]