Without actual division, find which of the following rational number are teminating decimals.
`(i) (13)/(80) " " (ii) (7)/(4) " " (iii)(5)/(12) " " (iv) (31)/(375) " " (v) (16)/(125)`

Answer

To determine which of the given rational numbers are terminating decimals, we need to analyze the denominators of each fraction. A rational number in the form \( \frac{p}{q} \) is a terminating decimal if the denominator \( q \) (when expressed in its simplest form) can be written as \( 2^n \times 5^m \), where \( n \) and \( m \) are non-negative integers.

Let's analyze each of the given rational numbers step by step:

### Step 1: Analyze \( \frac{13}{80} \)
1. Factor the denominator:
\[ 80 = 2^4 \times 5^1 \]
2. Since \( 80 \) can be expressed as \( 2^n \times 5^m \) (where \( n = 4 \) and \( m = 1 \)), this fraction is a terminating decimal.

### Step 2: Analyze \( \frac{7}{4} \)
1. Factor the denominator:
\[ 4 = 2^2 \times 5^0 \]
2. Since \( 4 \) can be expressed as \( 2^n \times 5^m \) (where \( n = 2 \) and \( m = 0 \)), this fraction is also a terminating decimal.

### Step 3: Analyze \( \frac{5}{12} \)
1. Factor the denominator:
\[ 12 = 2^2 \times 3^1 \]
2. Since \( 12 \) has a factor of \( 3 \), it cannot be expressed solely as \( 2^n \times 5^m \). Therefore, this fraction is a non-terminating decimal.

### Step 4: Analyze \( \frac{31}{375} \)
1. Factor the denominator:
\[ 375 = 5^3 \times 3^1 \]
2. Since \( 375 \) has a factor of \( 3 \), it cannot be expressed solely as \( 2^n \times 5^m \). Therefore, this fraction is a non-terminating decimal.

### Step 5: Analyze \( \frac{16}{125} \)
1. Factor the denominator:
\[ 125 = 5^3 \times 2^0 \]
2. Since \( 125 \) can be expressed as \( 2^n \times 5^m \) (where \( n = 0 \) and \( m = 3 \)), this fraction is a terminating decimal.

### Summary of Results:
- **Terminating decimals**: \( \frac{13}{80}, \frac{7}{4}, \frac{16}{125} \)
- **Non-terminating decimals**: \( \frac{5}{12}, \frac{31}{375} \)