To find the value of \( r \), which represents the fraction of the lawn Jing can complete in one hour, we can start by determining the work rates of Melissa and Jing separately.
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Melissa's Work Rate: Since Melissa can mow the lawn alone in 8 hours, her work rate is: \[ \text{Rate of Melissa} = \frac{1 \text{ lawn}}{8 \text{ hours}} = \frac{1}{8} \text{ lawns per hour} \]
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Combined Work Rate of Melissa and Jing: Together, Melissa and Jing can mow the lawn in 5 hours, so their combined work rate is: \[ \text{Combined Rate} = \frac{1 \text{ lawn}}{5 \text{ hours}} = \frac{1}{5} \text{ lawns per hour} \]
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Jing's Work Rate: Let \( r \) be the rate at which Jing can mow the lawn alone (in lawns per hour). The equation for their combined work rate can be set up as follows: \[ \text{Rate of Melissa} + \text{Rate of Jing} = \text{Combined Rate} \] Substituting the known values: \[ \frac{1}{8} + r = \frac{1}{5} \]
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Solve for \( r \): To solve for \( r \), first subtract \(\frac{1}{8}\) from both sides: \[ r = \frac{1}{5} - \frac{1}{8} \]
To perform this subtraction, we need a common denominator, which is 40: \[ \frac{1}{5} = \frac{8}{40}, \quad \frac{1}{8} = \frac{5}{40} \] So: \[ r = \frac{8}{40} - \frac{5}{40} = \frac{3}{40} \]
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Calculate \( r \): Now we calculate the decimal equivalent of \( r \): \[ r = \frac{3}{40} = 0.075 \]
Thus, the value of \( r \), the part of the lawn that Jing could complete in 1 hour, is \( 0.075 \).
The answer is: \[ \boxed{0.075} \]
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