After how many places of decimal, will the decimal expansion of `(141)/(120)` terminate ?

Options

A. 1
B. 2
C. 3
D. 4

Answer

To determine how many places of decimal the decimal expansion of \( \frac{141}{120} \) will terminate, we can follow these steps:

### Step 1: Factor the denominator
First, we need to factor the denominator \( 120 \) into its prime factors.

\[ 120 = 2^3 \times 3^1 \times 5^1 \]

### Step 2: Identify the prime factors
For a decimal expansion to terminate, the denominator (after simplification) must only have the prime factors \( 2 \) and/or \( 5 \).

In our case, \( 120 \) has a factor of \( 3 \) which is not allowed for a terminating decimal.

### Step 3: Simplify the fraction
Next, we simplify the fraction \( \frac{141}{120} \).

Since \( 141 \) and \( 120 \) have no common factors (as \( 141 \) is \( 3 \times 47 \) and \( 120 \) has a factor of \( 3 \)), we can simplify it:

\[ \frac{141}{120} = \frac{141 \div 3}{120 \div 3} = \frac{47}{40} \]

### Step 4: Factor the new denominator
Now, we factor the new denominator \( 40 \):

\[ 40 = 2^3 \times 5^1 \]

### Step 5: Check for terminating condition
The denominator \( 40 \) consists only of the prime factors \( 2 \) and \( 5 \). Therefore, the decimal expansion of \( \frac{47}{40} \) will terminate.

### Step 6: Determine the number of decimal places
To find out how many decimal places it will terminate after, we look at the highest power of \( 2 \) and \( 5 \) in the denominator:

- The highest power of \( 2 \) is \( 3 \) (from \( 2^3 \)).
- The highest power of \( 5 \) is \( 1 \) (from \( 5^1 \)).

The number of decimal places is determined by the maximum of these two powers:

\[ \text{Number of decimal places} = \max(3, 1) = 3 \]

### Conclusion
Thus, the decimal expansion of \( \frac{141}{120} \) will terminate after **3 decimal places**.

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