It takes Kate k days to write a GRE math practice test. It takes John j days to write a GRE math practice test. If Kate and John work on a practice test in alternating 2-day shifts, it takes them 10 days when Kate starts and 10.5 days when John starts. How long would it take the two to complete a practice test if Kate and John worked simultaneously?

Q: It takes Kate k days to write a GRE math practice test. It takes John j days to write a GRE math practice test. If Kate and John work on a practice test in alternating 2-day shifts, it takes them 10 days when Kate starts and 10.5 days when John starts. How long would it take the two to complete a practice test if Kate and John worked simultaneously?

∻Solution by Steps∻ ‖ step 1 ⋮ To find the values of $ k $ and $ j $, we solve the system of equations given by the alternating shifts. ‖ ‖ step 2 ⋮ The system of equations is $\frac{k}{2} + \frac{10 - k}{2} = 10$ and $\frac{j}{2} + \frac{10.5 - j}{2} = 10.5$. ‖ ‖ step 3 ⋮ Solving the system, we find that $ k = 10 $ and $ j = 10.5 $. ‖ ‖ step 4 ⋮ To find the time taken if Kate and John work simultaneously, we use the formula $\frac{1}{k} + \frac{1}{j}$. ‖ ‖ step 5 ⋮ Substituting $ k = 10 $ and $ j = 10.5 $ into the formula, we get $\frac{1}{10} + \frac{1}{10.5}$. ‖ ‖ step 6 ⋮ Simplifying, we find the combined work rate to be $\frac{1}{10} + \frac{2}{21} = \frac{21}{210} + \frac{20}{210} = \frac{41}{210}$. ‖ ‖ step 7 ⋮ The time taken for both to complete the test working simultaneously is the reciprocal of the combined work rate, which is $\frac{210}{41}$. ‖ ∻Answer∻ ⚹ $ \frac{210}{41} $ days or approximately 5.12 days ⚹ ∻Key Concept∻ ⚹Combined work rate when working simultaneously⚹ ∻Explanation∻ ⚹When two people work simultaneously, their work rates add up. To find the total time taken, we take the reciprocal of the combined work rate.⚹◊What would be the equation representing the work done by Kate and John working simultaneously to complete the GRE math practice test?⍭ Generate me a similar question◊

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∻Solution by Steps∻ ‖ step 1 ⋮ To find the values of $ k $ and $ j $, we solve the system of equations given by the alternating shifts. ‖ ‖ step 2 ⋮ The system of equations is $\frac{k}{2} + \frac{10 - k}{2} = 10$ and $\frac{j}{2} + \frac{10.5 - j}{2} = 10.5$. ‖ ‖ step 3 ⋮ Solving the system, we find that $ k = 10 $ and $ j = 10.5 $. ‖ ‖ step 4 ⋮ To find the time taken if Kate and John work simultaneously, we use the formula $\frac{1}{k} + \frac{1}{j}$. ‖ ‖ step 5 ⋮ Substituting $ k = 10 $ and $ j = 10.5 $ into the formula, we get $\frac{1}{10} + \frac{1}{10.5}$. ‖ ‖ step 6 ⋮ Simplifying, we find the combined work rate to be $\frac{1}{10} + \frac{2}{21} = \frac{21}{210} + \frac{20}{210} = \frac{41}{210}$. ‖ ‖ step 7 ⋮ The time taken for both to complete the test working simultaneously is the reciprocal of the combined work rate, which is $\frac{210}{41}$. ‖ ∻Answer∻ ⚹ $ \frac{210}{41} $ days or approximately 5.12 days ⚹ ∻Key Concept∻ ⚹Combined work rate when working simultaneously⚹ ∻Explanation∻ ⚹When two people work simultaneously, their work rates add up. To find the total time taken, we take the reciprocal of the combined work rate.⚹◊What would be the equation representing the work done by Kate and John working simultaneously to complete the GRE math practice test?⍭ Generate me a similar question◊

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