To solve the problem, we will break it down into Part A and Part B.
Part A
To create equations based on the number of candles, we first need to interpret the information given.
- One large box holds \(1 \frac{1}{2}\) dozen candles, which is equal to \(1.5\) dozen candles.
- One small box holds \( \frac{3}{4} \) dozen candles.
- The candlemaker can fill either 8 large boxes or 16 small boxes.
From the information given, we can set up two equations:
- If she fills 8 large boxes, the total number of candles is: \[ 8 \text{ boxes} \times 1.5 \text{ dozen/box} = 12 \text{ dozen candles} \]
- If she fills 16 small boxes, the total number of candles is: \[ 16 \text{ boxes} \times 0.75 \text{ dozen/box} = 12 \text{ dozen candles} \]
Thus, we can state that \(x\) (the total number of dozen candles) is equal to 12. We can form equations to express this relationship.
The two relevant equations correspond to the filled boxes:
-
For the large boxes: \[ \frac{x}{1.5} = 8 \quad \text{(which can be derived from the total candles being equal)} \]
-
For the small boxes: \[ \frac{x}{0.75} = 16 \quad \text{(which can also be derived from the total candles being equal)} \]
Thus, the correct answers for Part A are:
- B. \( \frac{x}{1.5} = 8 \)
- C. \( \frac{x}{0.75} = 16 \)
Part B
Now let's calculate how many large boxes can be filled with a batch of 18 dozen candles.
Since one large box holds \(1.5\) dozen candles, we need to divide the total number of candles by the amount each large box can hold:
\[ \text{Number of large boxes} = \frac{18 \text{ dozen}}{1.5 \text{ dozen/box}} = 12 \text{ boxes} \]
Thus, the answer for Part B is:
- A. 12
In summary:
- Part A: B and C
- Part B: A (12)