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Dane is using two differently sized water pumps to clean up flooded water. The larger pump can remove the water alone in 240 min. The smaller pump can remove the water alone in 400 min. How long would it take the pumps to remove the water working together? ◻ minutes

Dane is using two differently sized water pumps to clean up flooded water. The larger pump can remove the water alone in 240 min240\,\text{min}. The smaller pump can remove the water alone in 400 min400\,\text{min}. \newlineHow long would it take the pumps to remove the water working together?\newline□ minutes\square\,\text{minutes}

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Q. Dane is using two differently sized water pumps to clean up flooded water. The larger pump can remove the water alone in 240 min240\,\text{min}. The smaller pump can remove the water alone in 400 min400\,\text{min}. \newlineHow long would it take the pumps to remove the water working together?\newline□ minutes\square\,\text{minutes}
  1. Find Pump Rates: Let's find the rate at which each pump works. The larger pump's rate is 1240\frac{1}{240} of the water per minute, and the smaller pump's rate is 1400\frac{1}{400} of the water per minute.
  2. Calculate Combined Rate: Now, we add the rates of the two pumps to get their combined rate. So, 1240+1400=(400+240)(240×400)\frac{1}{240} + \frac{1}{400} = \frac{(400 + 240)}{(240 \times 400)}.
  3. Simplify Fraction: Calculate the combined rate: (400+240)/(240×400)=640/96000(400 + 240) / (240 \times 400) = 640 / 96000.
  4. Find Combined Rate: Simplify the fraction: 64096000=1150\frac{640}{96000} = \frac{1}{150}. So, the combined rate of the two pumps is 1150\frac{1}{150} of the water per minute.
  5. Calculate Time: To find the time it takes for both pumps to remove the water together, we take the reciprocal of the combined rate. The time is 1/(1/150)1 / (1/150) minutes.
  6. Final Calculation: Calculate the time: 1/(1/150)=1501 / (1/150) = 150 minutes. So, it would take the pumps 150150 minutes to remove the water working together.

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