Gordon and Alexis are assembling fruit baskets for a local retirement home. In the time it takes Gordon to assemble 80 baskets, Alexis assembles 90. In each hour, Alexis can assemble 2 more baskets than Gordon can. If [tex]$G$[/tex] and [tex]$A$[/tex] are the number of baskets that Gordon and Alexis can assemble in an hour, then the equation that describes the relationship is [tex]$\frac{80}{G}=\frac{90}{A}$[/tex]. How many baskets can Gordon assemble in one hour?
Answer (2)
Express Alexis's basket assembly rate in terms of Gordon's: A = G + 2 .
Substitute this into the given equation: G 80 = A 90 ⇒ G 80 = G + 2 90 .
Solve the equation for G : 80 ( G + 2 ) = 90 G ⇒ 80 G + 160 = 90 G ⇒ 10 G = 160 .
Find Gordon's basket assembly rate: G = 10 160 = 16 . The number of baskets Gordon can assemble in one hour is 16 .
Explanation
Understanding the Problem Let's analyze the problem. We are given that Gordon and Alexis assemble fruit baskets. The time it takes Gordon to assemble 80 baskets is the same as the time it takes Alexis to assemble 90 baskets. Also, Alexis can assemble 2 more baskets per hour than Gordon. We are given the equation G 80 = A 90 , where G is the number of baskets Gordon assembles in one hour, and A is the number of baskets Alexis assembles in one hour. Our goal is to find the value of G .
Expressing A in terms of G We know that Alexis assembles 2 more baskets per hour than Gordon, so we can write A = G + 2 . Now we can substitute this expression for A into the given equation: G 80 = G + 2 90
Solving for G To solve for G , we can cross-multiply: 80 ( G + 2 ) = 90 G Expanding the left side, we get: 80 G + 160 = 90 G
Isolating G Now, we can subtract 80 G from both sides of the equation: 160 = 90 G − 80 G 160 = 10 G
Finding the Value of G Finally, we can divide both sides by 10 to solve for G : G = 10 160 G = 16 So, Gordon can assemble 16 baskets in one hour.
Examples
Understanding how to calculate work rates, like the basket assembly rates in this problem, is useful in many real-world scenarios. For example, if you're managing a team on a project, knowing each person's work rate helps you estimate how long the project will take. If one person can complete 16 tasks per hour, and another can do 18, you can plan tasks accordingly to maximize efficiency and meet deadlines. This kind of calculation is also useful in manufacturing, logistics, and even in planning household chores!
Gordon can assemble 16 baskets in one hour. We determined this by expressing Alexis's assembly rate in terms of Gordon's and setting up an equation based on their rates. After solving, we found that G = 16 .
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