To determine the fewest number of vegetables that can be used to maintain the constant ratio of carrots to broccoli, we first assess the given combinations of vegetables:
- 3 carrots and 9 broccoli (total = 12)
- 4 carrots and 12 broccoli (total = 16)
- 6 carrots and 18 broccoli (total = 24)
- 7 carrots and 21 broccoli (total = 28)
Next, we calculate the ratio of carrots to broccoli for each combination:
- For 3 carrots and 9 broccoli, the ratio is \( \frac{3}{9} = \frac{1}{3} \)
- For 4 carrots and 12 broccoli, the ratio is \( \frac{4}{12} = \frac{1}{3} \)
- For 6 carrots and 18 broccoli, the ratio is \( \frac{6}{18} = \frac{1}{3} \)
- For 7 carrots and 21 broccoli, the ratio is \( \frac{7}{21} = \frac{1}{3} \)
All combinations maintain the same ratio of \( \frac{1}{3} \), meaning for every 1 carrot, there are 3 broccolis.
To find the fewest number of whole vegetables combined, we can represent the number of carrots as \( c \) and the number of broccolis as \( b \), adhering to the ratio \( \frac{c}{b} = \frac{1}{3} \), or rewritten as \( b = 3c \).
The total number of vegetables (carrots + broccoli) becomes:
\[ c + b = c + 3c = 4c \]
To minimize \( c + b \), we must select the smallest whole number for \( c \), which is 1.
Substituting \( c = 1 \):
\[ b = 3(1) = 3 \]
Thus, the total number of vegetables is:
\[ c + b = 1 + 3 = 4 \]
Therefore, the fewest number of vegetables that can be used to make the salad is 4.
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