To determine how long it will take for Lisa and Tom to fill the order together, we can use the concept of work rates.
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Determine the work rates for each clerk:
- Lisa’s work rate: Since she can complete the job in 14 hours, her work rate is \( \frac{1}{14} \) of the job per hour.
- Tom’s work rate: Since he can complete the job in 17 hours, his work rate is \( \frac{1}{17} \) of the job per hour.
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Combine their work rates: \[ \text{Combined work rate} = \frac{1}{14} + \frac{1}{17} \]
To add these fractions, find a common denominator. The least common multiple of 14 and 17 is \( 238 \).
- Convert \( \frac{1}{14} \) to have a denominator of 238: \[ \frac{1}{14} = \frac{17}{238} \]
- Convert \( \frac{1}{17} \) to have a denominator of 238: \[ \frac{1}{17} = \frac{14}{238} \]
Now we can add the two work rates: \[ \text{Combined work rate} = \frac{17}{238} + \frac{14}{238} = \frac{31}{238} \]
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Calculate the time they take to finish the job together: The time \( T \) taken to complete the job is the reciprocal of their combined work rate: \[ T = \frac{1}{\text{Combined work rate}} = \frac{1}{\frac{31}{238}} = \frac{238}{31} \]
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Simplify the answer: To simplify \( \frac{238}{31} \), perform the division: \[ 238 \div 31 = 7.677 \ldots \] This shows that the fraction is not an integer, but it's still in its simplest form.
Therefore, the time it takes for Lisa and Tom to fill the order together is:
\[ \frac{238}{31} \text{ hours} \]
If preferred in mixed form: \[ 7 \frac{25}{31} \text{ hours} \]
So the final answer is \( \frac{238}{31} \) hours, which is the simplest form as a fraction.