After how many place will decimal expansion of `294/1200` terminate ?

Options

A. 2
B. 3
C. 4
D. will not terminate

Answer

To determine how many places the decimal expansion of \( \frac{294}{1200} \) will terminate, we can follow these steps:

### Step 1: Simplify the Fraction
First, we need to simplify the fraction \( \frac{294}{1200} \).

**Calculation:**
- Find the greatest common divisor (GCD) of 294 and 1200.
- The prime factorization of 294 is \( 2 \times 3 \times 7 \times 7 \) (or \( 2 \times 3 \times 49 \)).
- The prime factorization of 1200 is \( 2^4 \times 3 \times 5^2 \).
- The GCD is \( 2 \times 3 = 6 \).

Now divide both the numerator and the denominator by their GCD:
\[ \frac{294 \div 6}{1200 \div 6} = \frac{49}{200} \]

### Step 2: Analyze the Denominator
Next, we need to check the denominator \( 200 \) to see if it can be expressed in the form \( 2^m \times 5^n \).

**Calculation:**
- The prime factorization of \( 200 \) is \( 2^3 \times 5^2 \).

### Step 3: Determine the Termination of Decimal
A fraction in its simplest form will have a terminating decimal expansion if the denominator (after simplification) has no prime factors other than \( 2 \) and \( 5 \).

Since \( 200 = 2^3 \times 5^2 \), it only contains the prime factors \( 2 \) and \( 5 \).

### Step 4: Count the Decimal Places
To find out how many decimal places the decimal expansion will terminate, we take the maximum of the powers of \( 2 \) and \( 5 \) in the denominator.

**Calculation:**
- The power of \( 2 \) is \( 3 \).
- The power of \( 5 \) is \( 2 \).

The maximum of these powers is \( 3 \).

### Conclusion
Thus, the decimal expansion of \( \frac{294}{1200} \) will terminate after **3 decimal places**.

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