Without actual division, show that each of the rational numbers given below is expressible as a terminating decimal :
`123/200`

Answer

To determine whether the rational number \( \frac{123}{200} \) is expressible as a terminating decimal without performing actual division, we can analyze the denominator.

### Step-by-Step Solution:

1. **Identify the Rational Number**: We start with the rational number \( \frac{123}{200} \).

2. **Factor the Denominator**: We need to factor the denominator, which is 200.
\[ 200 = 2^3 \times 5^2 \]
This means that 200 can be expressed as the product of powers of 2 and 5.

3. **Check the Form of the Denominator**: For a rational number to be expressed as a terminating decimal, the denominator (after simplification) must be of the form \( 2^m \times 5^n \), where \( m \) and \( n \) are non-negative integers.

4. **Analyze the Factors**: Since we have factored 200 into \( 2^3 \times 5^2 \), we see that it is indeed in the required form.

5. **Conclusion**: Since the denominator \( 200 \) can be expressed as \( 2^3 \times 5^2 \), we conclude that \( \frac{123}{200} \) is expressible as a terminating decimal.

### Final Result:
Thus, \( \frac{123}{200} \) is a terminating decimal.

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