To solve this problem, we need to analyze the given ratios of carrots to broccoli and find the least number of vegetables that can be used while keeping these ratios constant.
Let's examine the ratios:
1. For 3 carrots to 9 broccoli:
[tex]\[
\frac{3}{9} = \frac{1}{3}
\][/tex]
2. For 4 carrots to 12 broccoli:
[tex]\[
\frac{4}{12} = \frac{1}{3}
\][/tex]
3. For 6 carrots to 18 broccoli:
[tex]\[
\frac{6}{18} = \frac{1}{3}
\][/tex]
4. For 7 carrots to 21 broccoli:
[tex]\[
\frac{7}{21} = \frac{1}{3}
\][/tex]
From the ratios above, we can see that each combination simplifies to a ratio of [tex]\(1:3\)[/tex]. This means for every 1 carrot, we must have 3 broccoli to maintain the given ratio.
To find the fewest number of vegetables that can be used, let's combine the minimum number of each type of vegetable while maintaining the correct ratio. The smallest ratio pair that meets the requirement is 1 carrot and 3 broccoli.
Adding these together, we get the total number of vegetables:
[tex]\[
1 \text{ carrot} + 3 \text{ broccoli} = 4 \text{ vegetables}
\][/tex]
Thus, the fewest number of vegetables that can be used to make the salad while keeping the ratio constant is:
[tex]\[
\boxed{4}
\][/tex]
