Some friends decided to go on a picnic and expected expense is assumed as 768. Four friends did not come for which each paid 16 extra. How many friends went for picnic?

Options

A. 24
B. 16
C. 12
D. 8

Answer

To solve the problem step by step, let's denote the total number of friends as \( x \).

### Step 1: Set up the equation for expected expenses
The total expected expense for the picnic is given as 768. If all \( x \) friends had come, each would pay:
\[ \text{Cost per friend} = \frac{768}{x} \]

### Step 2: Calculate the cost per friend when 4 friends do not come
If 4 friends do not come, the number of friends who actually went to the picnic is \( x - 4 \). The cost per friend in this case would be:
\[ \text{Cost per friend with 4 absent} = \frac{768}{x - 4} \]

### Step 3: Set up the equation based on the extra amount paid
According to the problem, each of the friends who went had to pay 16 extra because 4 friends did not come. Therefore, we can set up the equation:
\[ \frac{768}{x - 4} - \frac{768}{x} = 16 \]

### Step 4: Solve the equation
To solve this equation, we first find a common denominator, which is \( x(x - 4) \):
\[ \frac{768x - 768(x - 4)}{x(x - 4)} = 16 \]

Simplifying the numerator:
\[ 768x - 768x + 3072 = 16x(x - 4) \]
\[ 3072 = 16x^2 - 64x \]

Rearranging the equation gives:
\[ 16x^2 - 64x - 3072 = 0 \]

### Step 5: Simplify the quadratic equation
Dividing the entire equation by 16:
\[ x^2 - 4x - 192 = 0 \]

### Step 6: Factor the quadratic equation
Now we need to factor the quadratic equation:
\[ (x - 16)(x + 12) = 0 \]

### Step 7: Solve for \( x \)
Setting each factor to zero gives us:
\[ x - 16 = 0 \quad \Rightarrow \quad x = 16 \]
\[ x + 12 = 0 \quad \Rightarrow \quad x = -12 \quad (\text{not valid since number of friends cannot be negative}) \]

Thus, the number of friends who went for the picnic is \( x = 16 \).

### Final Answer:
The number of friends who went for the picnic is **16**.

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