Four identical pumps working together fill 18 identical storage tanks in hours. For a later job, 50% more pumps are used, but the volumetric flow rate of each pump is reduced by 20%. Also, each replacement tank has a capacity 25% less than an original tank. At these adjusted rates, which choice gives an expression for the number of replacement tanks the pumps fill in minutes?
For problems with several proportional changes, apply a multiplier for each one. Quantities such as the number of pumps and elapsed time affect output directly, while the capacity of each tank affects the number filled inversely. Keep all rates in consistent time units before simplifying.
Hints
Combine the pump changes
Consider separately how 50% more pumps and a 20% reduction in each pump's rate affect the total volumetric flow rate.
Treat tank size inversely
If each tank requires less volume, the same volumetric flow can fill more tanks. Divide by the replacement tank's capacity rather than multiplying by it.
Convert the time unit
Express minutes as a fraction of an hour before multiplying by the hourly filling rate.
Desmos Guide
Choose convenient variable values
Enter and on separate lines. These values can be used to compare the expressions numerically.
Calculate the adjusted number of tanks
Enter . This combines the change in the number of pumps, the change in each pump's rate, the smaller tank capacity, and the time conversion.
Compare the choices
Enter each answer-option expression using the assigned values of and . The value of is , and the expression that also evaluates to is . Therefore, the answer is .
Step-by-step Explanation
Represent the original flow rate
Let be the capacity of one original tank. The original pumps move a total volume of in hours, so their original volumetric flow rate is per hour.
Adjust the total flow rate
Using 50% more pumps multiplies the total flow rate by , while reducing each pump's rate by 20% multiplies it by . Therefore, the adjusted volumetric flow rate is
Account for tank capacity and time
Each replacement tank has capacity , so divide the adjusted volumetric flow rate by to find the number of replacement tanks filled per hour. Since minutes equals hours, the required number is
Simplify the expression
Cancel and simplify:
Therefore, the answer is .