Helena uses a mixture of compost and topsoil in her garden. She purchased a total of 10 cubic yards of compost and topsoil for $180. If compost costs $25 per cubic yard and topsoil costs $15 per cubic yard, how many cubic yards of topsoil did she purchase?
Answer (2)
Define variables: Let c be the amount of compost and t be the amount of topsoil.
Set up equations: c + t = 10 and 25 c + 15 t = 180 .
Solve for t using substitution: c = 10 − t , so 25 ( 10 − t ) + 15 t = 180 .
Simplify and find t : 250 − 25 t + 15 t = 180 ⇒ − 10 t = − 70 ⇒ t = 7 . Therefore, the final answer is 7 .
Explanation
Problem Analysis and Setup Let's analyze the problem. We know that Helena purchased a total of 10 cubic yards of compost and topsoil for $180 . We also know the cost per cubic yard for each material. Our goal is to find out how many cubic yards of topsoil she purchased. Let's use variables to represent the unknowns.
Setting up the Equations Let c be the number of cubic yards of compost and t be the number of cubic yards of topsoil. We can set up two equations based on the given information:
The total amount of compost and topsoil is 10 cubic yards: c + t = 10
The total cost of compost and topsoil is $180 : 25 c + 15 t = 180
Solving the System of Equations Now we have a system of two equations with two variables. We can solve this system using substitution or elimination. Let's use substitution. From the first equation, we can express c in terms of t : c = 10 − t .
Substitution and Simplification Substitute this expression for c into the second equation:
25 ( 10 − t ) + 15 t = 180
Expand and simplify:
250 − 25 t + 15 t = 180
Combine like terms:
250 − 10 t = 180
Isolating t Now, isolate t :
− 10 t = 180 − 250
− 10 t = − 70
Divide by -10:
t = − 10 − 70
t = 7
Final Answer So, Helena purchased 7 cubic yards of topsoil.
Examples
This type of problem is useful in many real-world scenarios, such as when you need to mix two different ingredients to achieve a specific cost and quantity. For example, a baker might need to mix two different types of flour to create a specific blend, or a chemist might need to mix two different solutions to create a specific concentration. Understanding how to set up and solve systems of equations can help you solve these types of problems efficiently.
Helena purchased 7 cubic yards of topsoil. This was found by setting up and solving a system of equations based on the total volume and cost of compost and topsoil she bought. The calculations showed that she bought 3 cubic yards of compost along with the topsoil.
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