A lemonade stand has both cups of lemonade and bottles of water for sale. The bottle of water costs three times as much as the lemonade. One day , the stand sold 25 cups of lemonade , 20 bottles of water for $42.50
a) Write an equation relating the price of one cup of lemonade L, to the price of one bottle of water, W
b) Write an equation relating to the total amount , in dollars, of drinks sold to the cost of one cup of lemonadeL and the cost of one bottle of w
c) what is the price of one cup of lemonade
The Correct Answer and Explanation is :
a) Equation relating the price of one cup of lemonade ( L ) to the price of one bottle of water ( W ):
From the problem statement, we know that the price of one bottle of water is three times the price of one cup of lemonade. Therefore, we can express this relationship mathematically as:
[
W = 3L
]
This equation tells us that the cost of a bottle of water ( W ) is three times the cost of a cup of lemonade ( L ).
b) Equation relating the total amount of drinks sold to the cost of one cup of lemonade ( L ) and the cost of one bottle of water ( W ):
The total amount of drinks sold consists of the sales of both lemonade and water. We can express the total revenue from lemonade and water separately, then add them up to get the total amount sold.
- The revenue from lemonade: The stand sold 25 cups of lemonade, and each cup costs ( L ). So, the revenue from lemonade is ( 25L ).
- The revenue from water: The stand sold 20 bottles of water, and each bottle costs ( W ). So, the revenue from water is ( 20W ).
The total revenue from both drinks is:
[
25L + 20W = 42.50
]
Now, substitute ( W = 3L ) from part (a) into this equation:
[
25L + 20(3L) = 42.50
]
Simplify:
[
25L + 60L = 42.50
]
[
85L = 42.50
]
c) Finding the price of one cup of lemonade:
To solve for ( L ), divide both sides of the equation by 85:
[
L = \frac{42.50}{85} = 0.50
]
Thus, the price of one cup of lemonade is $0.50.
Explanation:
The lemonade stand sells two items: lemonade and water. The relationship between their prices is that the bottle of water costs three times as much as the lemonade. When we know the quantities sold and the total amount earned, we can set up equations that represent this situation.
First, we express the price of water in terms of the price of lemonade using the given relationship. Then, by calculating the total amount earned from both lemonade and water sales, we can form an equation where both quantities (the number of lemonade cups and water bottles) and their prices are included. Solving this equation allows us to find the price of a cup of lemonade, which is $0.50.
This method involves algebraic manipulation and substitution, key tools in solving real-world problems like this one.
Let me generate an image to represent the breakdown of the solution.
Here is the visual breakdown of the lemonade stand problem, showing the quantities sold and the price of each item. The total sales and the solution to the equation are also clearly illustrated. Let me know if you need any further explanation!
