Without doing any actual division, find which of the rational numbers have terminating decimal representation :
`(43)/(50)`

Answer

To determine whether the rational number \( \frac{43}{50} \) has a terminating decimal representation, we can follow these steps:

### Step 1: Identify the Denominator
The given rational number is \( \frac{43}{50} \). Here, the denominator is \( 50 \).

### Step 2: Factor the Denominator
Next, we need to factor the denominator \( 50 \) into its prime factors:
- Divide \( 50 \) by \( 2 \):
\[ 50 \div 2 = 25 \]
- Now, divide \( 25 \) by \( 5 \):
\[ 25 \div 5 = 5 \]
- Finally, divide \( 5 \) by \( 5 \):
\[ 5 \div 5 = 1 \]
Thus, the prime factorization of \( 50 \) is:
\[ 50 = 2^1 \times 5^2 \]

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

From our factorization:
\[ 50 = 2^1 \times 5^2 \]
This matches the required form, where \( m = 1 \) and \( n = 2 \).

### Step 4: Conclusion
Since the denominator \( 50 \) can be expressed in the form \( 2^m \times 5^n \), we conclude that the rational number \( \frac{43}{50} \) has a terminating decimal representation.

### Final Answer
Thus, \( \frac{43}{50} \) has a terminating decimal representation.

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