A man ate 100 grapes in 5 days. Each day, he ate 6 more grapes than those he ate on the earlier day. How many grapes did he eat on the first day ?

Options

A. 8
B. 12
C. 54
D. 76

Answer

To solve the problem, we need to determine how many grapes the man ate on the first day, given that he ate a total of 100 grapes over 5 days, with an increase of 6 grapes each day.

### Step-by-Step Solution:

1. **Define the Variables**:
Let the number of grapes eaten on the first day be \( x \).

2. **Express Grapes Eaten Each Day**:
- On the first day, he eats \( x \) grapes.
- On the second day, he eats \( x + 6 \) grapes.
- On the third day, he eats \( x + 12 \) grapes.
- On the fourth day, he eats \( x + 18 \) grapes.
- On the fifth day, he eats \( x + 24 \) grapes.

3. **Set Up the Equation**:
The total number of grapes eaten over the 5 days is given as 100. Therefore, we can write the equation:
\[ x + (x + 6) + (x + 12) + (x + 18) + (x + 24) = 100 \]

4. **Simplify the Equation**:
Combine like terms:
\[ 5x + (6 + 12 + 18 + 24) = 100 \]
Calculate the sum of the constants:
\[ 6 + 12 + 18 + 24 = 60 \]
So the equation becomes:
\[ 5x + 60 = 100 \]

5. **Solve for \( x \)**:
Subtract 60 from both sides:
\[ 5x = 100 - 60 \]
\[ 5x = 40 \]
Now, divide both sides by 5:
\[ x = \frac{40}{5} = 8 \]

6. **Conclusion**:
The man ate **8 grapes on the first day**.