Fred combines peaches and cherries to make a fruit salad. The ratio of cups of peaches to cups of cherries is 2 to 3. How many cups of peaches will Fred need to make 60 cups of fruit salad?
The problem describes a fruit salad mixture of peaches (\(P\)) and cherries (\(C\)). We are given two pieces of information:
Our goal is to find the value of \(P\).
We have a system of two equations:
(1) \(\frac{P}{C} = \frac{2}{3}\)
(2) \(P + C = 60\)
From equation (1), we can express \(C\) in terms of \(P\):
\(3P = 2C \implies C = \frac{3}{2}P\)
Now, substitute this expression for \(C\) into equation (2):
\(P + \left(\frac{3}{2}P\right) = 60\)
\(\frac{2}{2}P + \frac{3}{2}P = 60\)
\(\frac{5}{2}P = 60\)
To isolate \(P\), multiply both sides of the equation by \(\frac{2}{5}\):
\(P = 60 \times \frac{2}{5}\)
\(P = \frac{120}{5}\)
\(P = 24\)
Fred needs \(24\) cups of peaches. We can also find the number of cherries: \(C = \frac{3}{2} \times 24 = 36\). Checking the total, \(24 + 36 = 60\), which matches our requirement. Therefore, the number of cups of peaches is \(\boxed{24}\).
Understanding how to use ratios to determine specific quantities is very useful in everyday activities like cooking. For instance, if you are following a recipe that calls for a specific ratio of ingredients (like flour and sugar in baking), you can use the same algebraic method to scale the recipe up or down to make the perfect amount, ensuring your culinary creation always has the right balance of flavors.