- Convert the milk quantity from a mixed number to an improper fraction: \(1 \frac{1}{4} = \frac{5}{4}\).
- Set up a proportion between milk and servings: \(\frac{5/4}{7} = \frac{6}{S}\).
- Solve for \(S\) to find the maximum number of servings: \(\frac{5}{4}S = 42\), so \(S = \frac{168}{5} = 33 \frac{3}{5}\).
- The final answer is \(\boxed{33 \frac{3}{5}}\).
Explanation
- Understanding the Data and Problem
The problem asks for the maximum number of servings Ming can make with 6 cups of milk, given that the original recipe requires \(1 \frac{1}{4}\) cups of milk for 7 servings. The amount of flour is irrelevant as Ming has an unlimited supply. First, we need to convert the mixed number \(1 \frac{1}{4}\) to an improper fraction: \(1 \frac{1}{4} = \frac{1 \times 4 + 1}{4} = \frac{5}{4}\) cups. This means the ratio of milk to servings is \(\frac{5/4}{7}\).
- Setting Up the Proportion
Let \(S\) be the number of servings Ming can make with 6 cups of milk. We can set up a proportion using the ratio of milk to servings:
\[\frac{\text{Milk}_1}{\text{Servings}_1} = \frac{\text{Milk}_2}{\text{Servings}_2}\]
Substituting the known values, we have:
\[\frac{5/4}{7} = \frac{6}{S}\]
To simplify the left side, we multiply the numerator and denominator by 4:
\[\frac{5}{28} = \frac{6}{S}\]
This ratio \(\frac{5}{28}\) represents the amount of milk needed for one serving.
- Solving for the Number of Servings
Now, we solve for \(S\):
\[5 \times S = 28 \times 6\]
\[5 \times S = 168\]
\[S = \frac{168}{5}\]
Converting the improper fraction \(\frac{168}{5}\) into a decimal or a mixed number:
\[S = 33.6 = 33 \frac{3}{5}\]
Thus, Ming can make \(33 \frac{3}{5}\) servings.
- Conclusion
Based on the calculations, Ming can make \(33 \frac{3}{5}\) servings with 6 cups of milk. Therefore, the total number of servings is \(\boxed{33 \frac{3}{5}}\).
Examples
This type of calculation is very common in everyday cooking and resource management. If you are preparing a large meal for an event and only have a limited amount of a key ingredient, you can use these proportions to adjust your recipe scale to ensure you don't run out. It allows for precise planning, whether you are scaling up for a party or scaling down for an individual portion, ensuring that every serving maintains the correct ratio of ingredients.