It takes Franklin 14 hours to make a 200-square-foot cement patio. It takes Scott 10 hours to make the same size patio. Which equation can be used to find [tex]$x$[/tex], the number of hours it would take Franklin and Scott to make the patio together?
A. [tex]$14 x+10 x=200$[/tex]
B. [tex]$\frac{1}{14} x-\frac{1}{10} x=-$[/tex]
C. [tex]$\frac{1}{14} x+\frac{1}{10} x=1$[/tex]
D. [tex]$14 x-10 x=200$[/tex]
Answer (2)
The equation that represents how long it would take Franklin and Scott to build the patio together is 14 1 x + 10 1 x = 1 . This equation combines their individual work rates to find the total time required. The correct option is C.
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Define x as the time it takes for Franklin and Scott to make the patio together.
Calculate Franklin's work rate: 14 200 sq ft/hour.
Calculate Scott's work rate: 10 200 sq ft/hour.
Set up the equation: 14 1 x + 10 1 x = 1 .
Explanation
Problem Analysis Let's analyze the problem. We are given the time it takes Franklin and Scott to make a 200-square-foot cement patio individually, and we want to find an equation that represents the time it would take them to make the same patio together.
Define Variables and Rates Let's define the variables:
Let x be the number of hours it takes Franklin and Scott to make the patio together.
Franklin takes 14 hours to make a 200 sq ft patio, so his work rate is 14 200 sq ft/hour.
Scott takes 10 hours to make a 200 sq ft patio, so his work rate is 10 200 sq ft/hour.
Combined Work Rate When they work together, their combined work rate is the sum of their individual work rates: 14 200 + 10 200 sq ft/hour. In x hours, they complete the 200 sq ft patio. So, we have the equation: x × ( 14 200 + 10 200 ) = 200
Simplify the Equation Now, we simplify the equation by dividing both sides by 200: 14 x + 10 x = 1 This can be written as: 14 1 x + 10 1 x = 1
Final Equation Therefore, the equation that can be used to find x , the number of hours it would take Franklin and Scott to make the patio together is: 14 1 x + 10 1 x = 1
Examples
Imagine you and a friend are painting a room. You know how long it takes each of you to paint the room alone. This problem helps you determine how long it will take if you work together. By calculating individual work rates and combining them, you can predict the total time needed to complete the task. This concept applies to many real-world scenarios, such as construction projects, manufacturing processes, and even cooking, where combining efforts can significantly reduce the overall time required.
