- Determine the ratio of carrots to broccoli from the table: \(\frac{C}{B} = \frac{1}{3}\), so \(B = 3C\).
- Define the total number of vegetables as \(V = C + B = 4C\).
- Minimize the total number of vegetables by selecting the smallest positive integer for the number of carrots, \(C = 1\).
- Calculate the result: \(V = 4 \times 1 = \boxed{4}\).
Explanation
- Determining the Constant Ratio
The problem asks for the minimum total number of vegetables (carrots and broccoli) required to make a salad, given that the ratio of carrots to broccoli is constant and only whole vegetables can be used. From the provided table, we can determine the ratio of carrots (\(C\)) to broccoli (\(B\)):
- For 3 carrots, there are 9 broccoli: \(\frac{C}{B} = \frac{3}{9} = \frac{1}{3}\).
- For 4 carrots, there are 12 broccoli: \(\frac{C}{B} = \frac{4}{12} = \frac{1}{3}\).
- For 6 carrots, there are 18 broccoli: \(\frac{C}{B} = \frac{6}{18} = \frac{1}{3}\).
- For 7 carrots, there are 21 broccoli: \(\frac{C}{B} = \frac{7}{21} = \frac{1}{3}\).
The ratio of carrots to broccoli is always \(1:3\), which means for every 1 carrot, there are 3 broccoli. Therefore, we can express the number of broccoli as \(B = 3C\).
- Calculating the Minimum Total Number of Vegetables
The total number of vegetables \(V\) in the salad is the sum of the number of carrots and the number of broccoli:
\[V = C + B\]
Substituting \(B = 3C\) into this equation, we get:
\[V = C + 3C = 4C\]
To find the fewest number of vegetables, we need to minimize \(V\). Since the salad requires whole vegetables and the ratio implies positive quantities, \(C\) must be the smallest possible positive integer, which is \(C = 1\).
- Final Calculation and Conclusion
By substituting \(C = 1\) into the equation for the total number of vegetables:
\[V = 4 \times 1 = 4\]
If \(C = 1\), then \(B = 3(1) = 3\), and the total number of vegetables is \(1 + 3 = 4\). Since the number of vegetables must be a positive integer and we have found the smallest positive integer value for \(C\), 4 is indeed the minimum number of vegetables.
Examples
This concept of ratios is essential for many daily tasks, such as cooking or mixing materials. For instance, if you are making a drink mix that requires 1 part of syrup to 3 parts of water, knowing the minimum ratio helps you scale your recipe accurately to make just enough for a single serving without waste. Understanding these proportions ensures that you can maintain the desired balance regardless of the quantity you are preparing.