Question

There are nine water bottles in​ Devin's refrigerator. He adds three full boxes of water bottles to the refrigerator. Then he adds two more boxes that each has 1 fewer bottle than a full box. When he is​ done, there are 67 bottles in the refrigerator. Write and solve an equation to find the number of bottles in a full box.

Answer 1

Let's denote the number of bottles in a full box as \( x \).

1. Initially, Devin has 9 bottles in the refrigerator.
2. He adds 3 full boxes of water bottles, which would contain \( 3x \) bottles.
3. Next, he adds 2 more boxes, each containing \( x - 1 \) bottles. So, the total for these 2 boxes is \( 2(x - 1) \).

Now we can set up the equation based on the total number of bottles:

\[
9 + 3x + 2(x - 1) = 67
\]

Now, we'll simplify and solve the equation:

1. Distribute the \( 2 \) in \( 2(x - 1) \):
\[
2(x - 1) = 2x - 2
\]

2. Substitute back into the equation:
\[
9 + 3x + 2x - 2 = 67
\]

3. Combine like terms:
\[
9 - 2 + 3x + 2x = 67
\]
\[
7 + 5x = 67
\]

4. Subtract 7 from both sides:
\[
5x = 67 - 7
\]
\[
5x = 60
\]

5. Divide both sides by 5:
\[
x = \frac{60}{5}
\]
\[
x = 12
\]

Thus, the number of bottles in a full box is \( \boxed{12} \).

To verify:
- Full boxes added: \( 3 \times 12 = 36 \)
- Boxes with one fewer bottle: \( 2 \times (12 - 1) = 2 \times 11 = 22 \)
- Total bottles: \( 9 + 36 + 22 = 67 \), which matches the problem statement. Hence, the solution is correct.