Verified Answer :
- Define the total amount equation as \(s + f = 8\).
- Define the adjusted sugar equation as \(s + \frac{1}{4} = \frac{1}{2}f\).
- Substitute \(f = 8 - s\) into the second equation to solve for \(s\).
- The calculated amount of sugar is \(\boxed{2.5}\) cups.
Explanation
- Problem Analysis and Variable Definition
We are given that the total amount of sugar (\(s\)) and flour (\(f\)) in a recipe is 8 cups, expressed as:
\[s + f = 8.\]
We are also given that if the amount of sugar were increased by \(\frac{1}{4}\) cup, it would be equal to half the amount of flour:
\[s + \frac{1}{4} = \frac{1}{2}f.\]
Our objective is to find the amount of sugar \(s\) used in the recipe.
- Setting up the System of Equations
From the equation \(s + f = 8\), we can express the amount of flour \(f\) in terms of the sugar \(s\):
\[f = 8 - s.\]
Next, substitute this expression for \(f\) into the second equation \(s + \frac{1}{4} = \frac{1}{2}f\):
\[s + \frac{1}{4} = \frac{1}{2}(8 - s).\]
- Solving for the Amount of Sugar
Now, solve for \(s\) by expanding the right side of the equation:
\[s + \frac{1}{4} = 4 - \frac{1}{2}s.\]
Add \(\frac{1}{2}s\) to both sides of the equation:
\[s + \frac{1}{2}s + \frac{1}{4} = 4.\]
Combine the terms with \(s\):
\[\frac{3}{2}s + \frac{1}{4} = 4.\]
Subtract \(\frac{1}{4}\) from both sides:
\[\frac{3}{2}s = 4 - \frac{1}{4} = \frac{16}{4} - \frac{1}{4} = \frac{15}{4}.\]
Finally, solve for \(s\) by multiplying both sides by \(\frac{2}{3}\):
\[s = \frac{15}{4} \times \frac{2}{3} = \frac{30}{12} = 2.5.\]
Thus, the recipe requires 2.5 cups of sugar. Checking the result: if \(s = 2.5\), then \(f = 8 - 2.5 = 5.5\). The condition \(s + 0.25 = 2.5 + 0.25 = 2.75\) and \(\frac{1}{2}f = 0.5 \times 5.5 = 2.75\). Both sides are equal, confirming the result.
Examples
This type of problem helps in practical culinary scenarios where recipes need adjustment based on changing ingredient proportions. By representing recipe ingredients as a system of linear equations, you can accurately calculate the required changes to ensure that the balance of ingredients remains correct, helping you to modify recipes for smaller or larger portions without compromising the result. Understanding how to solve these systems is a valuable skill in both professional baking and daily cooking tasks.