In a friends group every person takes either tea or coffee or both. `76%` takes tea, `40%` takes coffee and `192` persons taken both tea and coffee then , find the total of persons in friends group ?

Options

A. `1500`
B. `1600`
C. `1200`
D. `1400`

Answer

To solve the problem step by step, we will use the information given about the percentages of people taking tea, coffee, and both.

### Step 1: Define the total number of persons
Let the total number of persons in the friends group be represented as \( N \).

### Step 2: Express the percentages in terms of \( N \)
According to the problem:
- The number of persons taking tea = \( 76\% \) of \( N = 0.76N \)
- The number of persons taking coffee = \( 40\% \) of \( N = 0.40N \)
- The number of persons taking both tea and coffee = \( 192 \)

### Step 3: Use the principle of inclusion-exclusion
The principle of inclusion-exclusion states that:
\[ \text{Total} = (\text{Tea}) + (\text{Coffee}) - (\text{Both}) \]
Substituting the values we have:
\[ N = (0.76N) + (0.40N) - 192 \]

### Step 4: Combine like terms
Combine the terms on the right side:
\[ N = 0.76N + 0.40N - 192 \]
\[ N = 1.16N - 192 \]

### Step 5: Isolate \( N \)
To isolate \( N \), we can rearrange the equation:
\[ N - 1.16N = -192 \]
\[ -0.16N = -192 \]
Dividing both sides by -0.16 gives:
\[ N = \frac{192}{0.16} \]

### Step 6: Calculate \( N \)
Now, calculate \( N \):
\[ N = 192 \div 0.16 = 1200 \]

### Conclusion
The total number of persons in the friends group is \( 1200 \).

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