Question

The decimal expansion of the rational number $\dfrac{17}{2^2 \times 5}$ will terminate after how many places of decimal?

• A. 1
• B. 2
• C. 3
• D. 4

Hint:

For a rational number to have a terminating decimal expansion, the denominator (in simplest form) must have only the prime factors 2 and 5.

Answer 1

The Correct Option is A

Step 1: Analyze the given expression.
We are given the rational number $\frac{17}{2^2 \times 5}$. This simplifies to: \[ \frac{17}{4 \times 5} = \frac{17}{20}. \]
Step 2: Check for terminating decimal condition.
A rational number has a terminating decimal expansion if the denominator, in its simplest form, has only the prime factors 2 and/or 5. Here, the denominator $20 = 2^2 \times 5$, which only contains the primes 2 and 5. Hence, the decimal expansion will terminate.

Step 3: Perform the division.
Now, divide $17$ by $20$: \[ \frac{17}{20} = 0.85 \] The decimal expansion terminates after 2 decimal places.
Step 4: Conclusion.
Thus, the decimal expansion of $\frac{17}{20}$ will terminate after 2 places of decimal.