John can clear a lot in $1.5$ hours. His partner can do the same job in $8.5$ hours. How long will it take them to clear the lot working together?

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Q. John can clear a lot in

1.5

hours. His partner can do the same job in

8.5

hours. How long will it take them to clear the lot working together?

Given Time by John: Given: $\newline$ Time taken by John to clear a lot = $1.5$ hours. $\newline$ Find John's clearing rate per hour. $\newline$ The rates are given by the reciprocal of their respective times. $\newline$ John's\_rate = $\frac{1}{1.5}$
Calculate John's Rate: Calculate John's clearing rate per hour. $\newline$ John's $_{rate}$ = $\frac{1}{1.5}$ = $\frac{2}{3}$ $\newline$ John can clear $\frac{2}{3}$ of a lot per hour.
Given Time by Partner: Given: $\newline$ Time taken by John's partner to clear a lot = $8.5$ hours. $\newline$ Find his partner's clearing rate per hour. $\newline$ The rates are given by the reciprocal of their respective times. $\newline$ Partner's\_rate = $\frac{1}{8.5}$
Calculate Partner's Rate: Calculate John's partner's clearing rate per hour. $\newline$ $\text{Partner's\_rate} = \frac{1}{8.5}$ $\newline$ To make the calculation easier, convert $8.5$ to an improper fraction. $\newline$ $8.5 = \frac{17}{2}$ $\newline$ $\text{Partner's\_rate} = \frac{1}{\left(\frac{17}{2}\right)} = \frac{2}{17}$ $\newline$ John's partner can clear $\frac{2}{17}$ of a lot per hour.
Find Combined Rate: Find the combined clearing rate of John and his partner. $\newline$ $John's\_rate + Partner's\_rate = \frac{2}{3} + \frac{2}{17}$ $\newline$ To add these fractions, find a common denominator. $\newline$ The common denominator of $3$ and $17$ is $51$ . $\newline$ $\left(\frac{2}{3}\right)\left(\frac{17}{17}\right) + \left(\frac{2}{17}\right)\left(\frac{3}{3}\right) = \frac{34}{51} + \frac{6}{51}$ $\newline$ $Combined\_rate = \frac{40}{51}$ $\newline$ John and his partner can clear $\frac{40}{51}$ of a lot per hour together.
Find Time Taken Together: Find the number of hours it will take them to clear the lot working together. $\newline$ $T = \frac{1}{\text{Combined\_rate}}$ $\newline$ $T = \frac{1}{(\frac{40}{51})}$ $\newline$ To divide by a fraction, multiply by its reciprocal. $\newline$ $T = 1 \times (\frac{51}{40})$ $\newline$ $T = \frac{51}{40}$ $\newline$ Time taken together is $\frac{51}{40}$ hours.