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 will simplify the fraction \( \frac{136}{1400} \).

1. Find the greatest common divisor (GCD) of 136 and 1400.
- The GCD of 136 and 1400 is 4.

2. Divide both the numerator and the denominator by their GCD:
\[ \frac{136 \div 4}{1400 \div 4} = \frac{34}{350} \]

### Step 2: Further Simplify
Next, we can simplify \( \frac{34}{350} \).

1. The GCD of 34 and 350 is 2.

2. Divide both the numerator and the denominator by their GCD:
\[ \frac{34 \div 2}{350 \div 2} = \frac{17}{175} \]

### Step 3: Convert to Decimal
Now, we will convert \( \frac{17}{175} \) to its decimal form by performing the division.

1. Divide 17 by 175 using long division:
- 175 goes into 17, 0 times. So we write 0. and add a decimal point and a zero to 17, making it 170.
- 175 goes into 170, 0 times. Add another zero making it 1700.
- 175 goes into 1700, 9 times (since \( 175 \times 9 = 1575 \)).
- Subtract \( 1700 - 1575 = 125 \).
- Bring down another 0 to make it 1250.
- 175 goes into 1250, 7 times (since \( 175 \times 7 = 1225 \)).
- Subtract \( 1250 - 1225 = 25 \).
- Bring down another 0 to make it 250.
- 175 goes into 250, 1 time (since \( 175 \times 1 = 175 \)).
- Subtract \( 250 - 175 = 75 \).
- Bring down another 0 to make it 750.
- 175 goes into 750, 4 times (since \( 175 \times 4 = 700 \)).
- Subtract \( 750 - 700 = 50 \).
- Bring down another 0 to make it 500.
- 175 goes into 500, 2 times (since \( 175 \times 2 = 350 \)).
- Subtract \( 500 - 350 = 150 \).
- Bring down another 0 to make it 1500.
- 175 goes into 1500, 8 times (since \( 175 \times 8 = 1400 \)).
- Subtract \( 1500 - 1400 = 100 \).
- Bring down another 0 to make it 1000.
- 175 goes into 1000, 5 times (since \( 175 \times 5 = 875 \)).
- Subtract \( 1000 - 875 = 125 \).
- This process will continue, leading to a repeating decimal.

The result of the division gives us:
\[ 0.097142857142857... \]
This decimal is non-terminating and non-repeating.

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

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