Answer :
To find the percent of lemonade in the punch recipe, we can use the concept of proportions. Here's how you can solve it step by step:
1. Understand the Ratio: The recipe states that there are 6 cups of lemonade for every 24 cups of punch. This ratio can be represented as:
[tex]\[
\frac{\text{cups of lemonade}}{\text{cups of punch}} = \frac{6}{24}
\][/tex]
2. Convert Ratio to Percentage: To find out what percentage of the punch is lemonade, you need to express this ratio as a percentage. The percentage formula for a part-to-whole relationship is:
[tex]\[
\frac{\text{part}}{\text{whole}} \times 100 = \text{percent}
\][/tex]
Here, the "part" is the cups of lemonade (6) and the "whole" is the total cups of punch (24).
3. Set Up the Equation: Set up the equation based on the above formula to solve for [tex]\( x \)[/tex], the percent of lemonade in the punch:
[tex]\[
\frac{6}{24} = \frac{x}{100}
\][/tex]
4. Solve for [tex]\( x \)[/tex]: Solve this proportion to find [tex]\( x \)[/tex].
[tex]\[
6 \times 100 = 24 \times x
\][/tex]
[tex]\[
600 = 24x
\][/tex]
[tex]\[
x = \frac{600}{24} = 25
\][/tex]
5. Conclusion: Therefore, the percentage of lemonade in the punch is 25%, and the equation that represents this is:
A. [tex]\(\frac{6}{24} = \frac{x}{100}\)[/tex]
This logical and proportional reasoning guides us to the conclusion that the correct answer is option A.
1. Understand the Ratio: The recipe states that there are 6 cups of lemonade for every 24 cups of punch. This ratio can be represented as:
[tex]\[
\frac{\text{cups of lemonade}}{\text{cups of punch}} = \frac{6}{24}
\][/tex]
2. Convert Ratio to Percentage: To find out what percentage of the punch is lemonade, you need to express this ratio as a percentage. The percentage formula for a part-to-whole relationship is:
[tex]\[
\frac{\text{part}}{\text{whole}} \times 100 = \text{percent}
\][/tex]
Here, the "part" is the cups of lemonade (6) and the "whole" is the total cups of punch (24).
3. Set Up the Equation: Set up the equation based on the above formula to solve for [tex]\( x \)[/tex], the percent of lemonade in the punch:
[tex]\[
\frac{6}{24} = \frac{x}{100}
\][/tex]
4. Solve for [tex]\( x \)[/tex]: Solve this proportion to find [tex]\( x \)[/tex].
[tex]\[
6 \times 100 = 24 \times x
\][/tex]
[tex]\[
600 = 24x
\][/tex]
[tex]\[
x = \frac{600}{24} = 25
\][/tex]
5. Conclusion: Therefore, the percentage of lemonade in the punch is 25%, and the equation that represents this is:
A. [tex]\(\frac{6}{24} = \frac{x}{100}\)[/tex]
This logical and proportional reasoning guides us to the conclusion that the correct answer is option A.
