In a certain city only two cellular (mobile) phone services are available. One is Radiance and other is DSML. It is known that `25%` of the city population uses the services of Radiance and `20%` uses the services of DSML. while `8%` uses both Radiance and DSML. It is also known that `30%` of those who uses Radiance services but not services of DSML have prepaid cards and `40%` of those who uses DSML .but not Radiance have prepaid cards while`50%` of those who uses the services of both Radiance and DSML have prepaid cards. What percentage of the population have prepaid cards?

Options

A. `12.5%`
B. `10%`
C. `13.9%`
D. `20%`

Answer

To determine the percentage of the population that has prepaid cards, we will follow a step-by-step approach using the information provided in the question.

### Step 1: Define the Variables
Let:
- \( P(A) \) = Percentage of the population using Radiance = 25% = \( \frac{25}{100} = \frac{1}{4} \)
- \( P(B) \) = Percentage of the population using DSML = 20% = \( \frac{20}{100} = \frac{1}{5} \)
- \( P(A \cap B) \) = Percentage of the population using both services = 8% = \( \frac{8}{100} = \frac{2}{25} \)

### Step 2: Calculate Users of Only One Service
1. **Users of Radiance only**:
\[ P(A \cap B') = P(A) - P(A \cap B) = \frac{1}{4} - \frac{2}{25} \]
To compute this, we need a common denominator (100):
\[ P(A) = \frac{25}{100}, \quad P(A \cap B) = \frac{8}{100} \]
\[ P(A \cap B') = \frac{25}{100} - \frac{8}{100} = \frac{17}{100} = 17\% \]

2. **Users of DSML only**:
\[ P(B \cap A') = P(B) - P(A \cap B) = \frac{1}{5} - \frac{2}{25} \]
Again, using a common denominator (100):
\[ P(B) = \frac{20}{100}, \quad P(A \cap B) = \frac{8}{100} \]
\[ P(B \cap A') = \frac{20}{100} - \frac{8}{100} = \frac{12}{100} = 12\% \]

### Step 3: Calculate Prepaid Card Holders
Let \( P(C) \) be the percentage of users having prepaid cards.

1. **Prepaid card holders using Radiance only**:
- 30% of users of Radiance only:
\[ P(C | A \cap B') = 30\% = \frac{30}{100} = \frac{3}{10} \]
\[ \text{Prepaid holders} = P(C | A \cap B') \times P(A \cap B') = \frac{3}{10} \times \frac{17}{100} = \frac{51}{1000} \]

2. **Prepaid card holders using DSML only**:
- 40% of users of DSML only:
\[ P(C | B \cap A') = 40\% = \frac{40}{100} = \frac{2}{5} \]
\[ \text{Prepaid holders} = P(C | B \cap A') \times P(B \cap A') = \frac{2}{5} \times \frac{12}{100} = \frac{24}{500} = \frac{48}{1000} \]

3. **Prepaid card holders using both services**:
- 50% of users using both:
\[ P(C | A \cap B) = 50\% = \frac{50}{100} = \frac{1}{2} \]
\[ \text{Prepaid holders} = P(C | A \cap B) \times P(A \cap B) = \frac{1}{2} \times \frac{2}{25} = \frac{1}{25} = \frac{40}{1000} \]

### Step 4: Total Prepaid Card Holders
Now, we sum all the prepaid card holders:
\[ P(C) = \frac{51}{1000} + \frac{48}{1000} + \frac{40}{1000} = \frac{139}{1000} \]

### Step 5: Convert to Percentage
To express this as a percentage:
\[ P(C) = \frac{139}{1000} \times 100 = 13.9\% \]

### Final Answer
The percentage of the population that has prepaid cards is **13.9%**.