Let's analyze the given problem step-by-step.
First, we are provided with pairs of values representing the number of boxes of batteries, [tex]\( b \)[/tex], and the number of batteries, [tex]\( n \)[/tex]:
[tex]\[
\begin{array}{|c|c|c|c|}
\hline
\text{number of boxes, } b & 11 & 15 & 21 \\
\hline
\text{number of batteries, } n & 44 & 60 & 84 \\
\hline
\end{array}
\][/tex]
We need to determine the ratio of the number of batteries to the number of boxes for each recorded pair.
### Step 1: Calculate the Ratio for Each Pair
The ratio can be found by dividing the number of batteries [tex]\( n \)[/tex] by the number of boxes [tex]\( b \)[/tex] for each pair:
1. For 11 boxes and 44 batteries:
[tex]\[
\frac{n}{b} = \frac{44}{11} = 4
\][/tex]
2. For 15 boxes and 60 batteries:
[tex]\[
\frac{n}{b} = \frac{60}{15} = 4
\][/tex]
3. For 21 boxes and 84 batteries:
[tex]\[
\frac{n}{b} = \frac{84}{21} = 4
\][/tex]
### Step 2: Verify the Consistency
All these ratios are the same: 4.0. This consistency is significant because it confirms that the number of batteries [tex]\( n \)[/tex] is indeed proportional to the number of boxes [tex]\( b \)[/tex], meaning the relationship [tex]\( n = kb \)[/tex] holds for some constant [tex]\( k \)[/tex].
### Step 3: Interpretation of the Ratio
The ratio of the number of batteries to the number of boxes is 4. This means that for every box of batteries Donna orders, there are 4 batteries. In other words, each box contains 4 batteries.
### Conclusion
The ratios for each recorded pair are as follows:
- [tex]\([4.0, 4.0, 4.0]\)[/tex]
The constant ratio is:
- [tex]\(4.0\)[/tex]
This ratio represents the fact that each box of batteries contains 4 batteries. Thus, if Donna knows the number of boxes she orders, she can determine the total number of batteries by multiplying the number of boxes by 4.
