A restaurant freezes a cherry and lime juice mixture to create slushes. Cherry juice costs \$5 per quart, and lime juice costs \$3 per quart. Each day, the restaurant spends a total of \$36 on 8 quarts of juice. The restaurant manager organizes the information in the table below.
| | Amount (quarts) | Cost | Total |
| :------------- | :-------------- | :--- | :------- |
| Cherry Juice | q | 5 | 5q |
| Lime Juice | $8-q$ | 3 | $3(8-q)$ |
| Mixture | 8 | | 36 |
Which equation can be used to determine the amount of cherry juice in each mixture?
A. $5 q+3(8-q)=36$
B. $5 q+3(8-q)=8$
C. $q+(8-q)=8$
Answer (1)
Define 'q' as the amount of cherry juice.
Express the cost of cherry juice as 5 q and lime juice as 3 ( 8 − q ) .
Set up the equation for the total cost: 5 q + 3 ( 8 − q ) = 36 .
The equation to determine the amount of cherry juice is 5 q + 3 ( 8 − q ) = 36 .
Explanation
Problem Analysis Let's analyze the problem. We are given the cost of cherry juice and lime juice per quart, the total amount of juice (8 quarts), and the total cost ($36). We need to find the equation that represents the total cost in terms of the amount of cherry juice, denoted by 'q'.
Equation Setup The cost of cherry juice is the price per quart times the number of quarts, which is 5 q . The amount of lime juice is 8 − q quarts, so the cost of lime juice is 3 ( 8 − q ) . The total cost is the sum of the cost of cherry juice and the cost of lime juice, which should equal $36. Therefore, the equation is: 5 q + 3 ( 8 − q ) = 36
Verification Now, let's verify this equation with the given options. The first option is 5 q + 3 ( 8 − q ) = 36 , which matches our derived equation. The second option is 5 q + 3 ( 8 − q ) = 8 , which is incorrect because the total cost is $36, not $8. The third option is q + ( 8 − q ) = 8 , which represents the total amount of juice but doesn't account for the cost.
Final Answer Therefore, the correct equation is 5 q + 3 ( 8 − q ) = 36 .
Examples
Imagine you're planning a party and need to mix fruit punch. You know the cost of each juice and the total amount you want to spend. This problem helps you determine how much of each juice to buy to stay within your budget. For example, if orange juice costs $2 per liter and apple juice costs $1 per liter, and you want to spend $10 in total for 6 liters of juice, you can use a similar equation to find out how many liters of each juice to buy. This kind of problem is also useful in business for optimizing costs and resource allocation.
