After how many places of decimal will the number `(343)/(1400)` terminate?

Options

A. Two
B. Three
C. Four
D. Five

Answer

To determine how many decimal places the number \( \frac{343}{1400} \) will terminate after, we need to analyze the denominator after simplifying the fraction.

### Step-by-Step Solution:

1. **Prime Factorization of the Denominator:**
We start with the denominator, which is 1400. We will find its prime factors.
- Divide 1400 by 2:
\[ 1400 \div 2 = 700 \]
- Divide 700 by 2:
\[ 700 \div 2 = 350 \]
- Divide 350 by 2:
\[ 350 \div 2 = 175 \]
- Divide 175 by 5:
\[ 175 \div 5 = 35 \]
- Divide 35 by 5:
\[ 35 \div 5 = 7 \]
- Finally, divide 7 by 7:
\[ 7 \div 7 = 1 \]
Thus, the prime factorization of 1400 is:
\[ 1400 = 2^3 \times 5^2 \times 7^1 \]

2. **Prime Factorization of the Numerator:**
Now, we factor the numerator, which is 343.
- We can express 343 as:
\[ 343 = 7^3 \]

3. **Simplifying the Fraction:**
Now, we can write the fraction \( \frac{343}{1400} \) using its prime factors:
\[ \frac{7^3}{2^3 \times 5^2 \times 7^1} \]
We can cancel one \( 7 \) from the numerator and the denominator:
\[ = \frac{7^{3-1}}{2^3 \times 5^2} = \frac{7^2}{2^3 \times 5^2} = \frac{49}{2^3 \times 5^2} \]

4. **Form of the Denominator:**
The simplified denominator is \( 2^3 \times 5^2 \). For a decimal to terminate, the denominator (after simplification) must only have the prime factors 2 and 5.

5. **Finding the Number of Decimal Places:**
The number of decimal places is determined by the highest power of 10 that can be formed from the factors of 2 and 5.
- The highest power of 2 in the denominator is 3 (from \( 2^3 \)).
- The highest power of 5 in the denominator is 2 (from \( 5^2 \)).
- The number of decimal places is given by the maximum of these two powers:
\[ \text{Decimal Places} = \max(3, 2) = 3 \]

### Conclusion:
The number \( \frac{343}{1400} \) will terminate after **3 decimal places**.