- Identify the original yield (\(10\) servings) and the target yield (\(75\) servings).
- Set up the scale factor formula: \(\frac{\text{Target Servings}}{\text{Original Servings}}\).
- Perform the division: \(\frac{75}{10} = 7.5\).
- The final scale factor for the recipe is \(\boxed{7.5}\).
Explanation
- Analyzing the Problem Requirements
To scale a recipe, we need to determine the ratio between the number of servings we want to make and the number of servings the original recipe yields. In this case, the original recipe for churros provides \(10\) servings, and our goal is to prepare enough for \(75\) guests. By comparing these two quantities, we can calculate the multiplier needed to adjust all ingredient amounts proportionally.
- Defining the Scale Factor Formula
The scale factor is determined by dividing the target number of servings by the original number of servings defined in the recipe. The relationship is expressed as:
\[\text{Scale Factor} = \frac{\text{Target Servings}}{\text{Original Servings}}\]
With our data, this becomes:
\[\text{Scale Factor} = \frac{75}{10}\]
- Calculating and Concluding the Scale Factor
To complete the calculation, we divide \(75\) by \(10\):
\[\text{Scale Factor} = 7.5\]
This result indicates that every ingredient in the original recipe must be multiplied by \(7.5\) to produce exactly \(75\) servings. For example, if the recipe calls for \(2\) cups of flour for \(10\) servings, we would need \(2 \times 7.5 = 15\) cups of flour for our target of \(75\) servings. The final scale factor required is \(\boxed{7.5}\).
Examples
Scale factors are essential in culinary arts and chemistry whenever you need to adjust a base formula to a new desired output. Whether you are scaling a cookie recipe for a large bake sale or adjusting a chemical concentration for a laboratory experiment, using a consistent scale factor ensures that the ratios of ingredients remain identical, maintaining the quality and outcome of your final product. Understanding how to calculate this factor helps ensure you have exactly the right amount of resources for any project.