Question

Math

Posted 7 months ago

this can draw image omg!

`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?`

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Answer from Sia

Posted 7 months ago

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.

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