What is the decimal representation of `136/1400`?

Options

A. Non-terminating and non-repeating
B. Terminating and repeating
C. Non-terminating and repeating
D. Teminating and non-repeating

Answer

To find the decimal representation of \( \frac{136}{1400} \), we can follow these steps:

### Step 1: Simplify the Fraction
First, we can simplify the fraction \( \frac{136}{1400} \) by finding the greatest common divisor (GCD) of the numerator and the denominator.

1. **Finding GCD of 136 and 1400**:
- The prime factorization of 136 is \( 2^3 \times 17 \).
- The prime factorization of 1400 is \( 2^3 \times 5^2 \times 7 \).
- The common factor is \( 2^3 \).

2. **Dividing both the numerator and denominator by their GCD**:
\[ \frac{136 \div 8}{1400 \div 8} = \frac{17}{175} \]

### Step 2: Convert to Decimal
Next, we convert \( \frac{17}{175} \) into decimal form by performing the division.

1. **Perform long division of 17 by 175**:
- 175 goes into 17 zero times. So, we write 0. and add a decimal point and a zero.
- 175 goes into 170 (which is 17.0) zero times again. So, we add another zero.
- Now, we are considering 1700. 175 goes into 1700 nine times (since \( 175 \times 9 = 1575 \)).
- Subtract \( 1700 - 1575 = 125 \).
- Bring down another zero to make it 1250. 175 goes into 1250 seven times (since \( 175 \times 7 = 1225 \)).
- Subtract \( 1250 - 1225 = 25 \).
- Bring down another zero to make it 250. 175 goes into 250 one time (since \( 175 \times 1 = 175 \)).
- Subtract \( 250 - 175 = 75 \).
- Bring down another zero to make it 750. 175 goes into 750 four times (since \( 175 \times 4 = 700 \)).
- Subtract \( 750 - 700 = 50 \).
- Bring down another zero to make it 500. 175 goes into 500 two times (since \( 175 \times 2 = 350 \)).
- Subtract \( 500 - 350 = 150 \).
- Bring down another zero to make it 1500. 175 goes into 1500 eight times (since \( 175 \times 8 = 1400 \)).
- Subtract \( 1500 - 1400 = 100 \).
- Bring down another zero to make it 1000. 175 goes into 1000 five times (since \( 175 \times 5 = 875 \)).
- Subtract \( 1000 - 875 = 125 \).

At this point, we notice that we are back to 125, which means the decimal will start repeating.

So, the decimal representation of \( \frac{136}{1400} \) is:
\[ 0.097142857142857... \]

### Final Answer
Thus, the decimal representation of \( \frac{136}{1400} \) is \( 0.097142857142857... \), which is non-terminating and repeating.