Solving Water Tank Filling Problems Using Rates and Ratios
Understanding and applying the principles of rate and ratio is crucial in solving real-world problems involving the filling or emptying of tanks or containers. In this article, we will explore a classic problem that involves two taps, P and Q, to fill a tank. We'll break down the process step-by-step and use mathematical calculations to find the solution.
The Problem Statement
Two taps, P and Q, can fill a water tank in 10 minutes and 20 minutes respectively. Both taps are opened simultaneously and run for 2 minutes. Then, tap P is closed. How long will it take for tap Q alone to fill the remaining part of the tank?
Calculation of Filling Rates
To solve this problem, we need to determine the rates at which each tap fills the tank and use these rates to find out how much of the tank is filled and how much remains to be filled.
Filling Rate of Tap P:
The rate of tap P is the amount of the tank it can fill per minute. Mathematically, this is expressed as:
[ text{Rate of P} frac{1}{10} text{tank per minute} ]Filling Rate of Tap Q:
The rate of tap Q is:
[ text{Rate of Q} frac{1}{20} text{tank per minute} ]Combined Filling Rate
When both taps are open, the combined rate at which the tank is being filled is the sum of the individual rates of taps P and Q. We use a common denominator of 20 to add the fractions:
[ text{Combined rate} frac{1}{10} frac{1}{20} frac{2}{20} frac{1}{20} frac{3}{20} text{tank per minute} ]Amount of Tank Filled in 2 Minutes
The total amount of the tank that is filled in 2 minutes can be calculated by multiplying the combined rate by the time:
[ text{Amount filled in 2 minutes} frac{3}{20} times 2 frac{3}{10} text{of the tank} ]Remaining Amount of the Tank
Now, we need to calculate how much of the tank is left to be filled after 2 minutes:
[ text{Remaining amount} 1 - frac{3}{10} frac{7}{10} text{of the tank} ]Time to Fill Remaining Amount with Tap Q
Tap Q alone is now filling the remaining (frac{7}{10}) of the tank. The time required is the remaining amount divided by the rate of tap Q:
[ t frac{frac{7}{10}}{frac{1}{20}} frac{7}{10} times 20 14 text{minutes} ]Therefore, after tap P is closed, it will take 14 minutes for tap Q to fill the remaining part of the tank.
Additional Example Problem
Let's solve another problem with a similar setup:
[ 1 - frac{1}{12} - frac{1}{15}x 0 ]Solving for x:
[ x frac{3 times 15 - 15}{4} - 3 11frac{1}{4} - 3 8frac{1}{4} text{minutes} ]After working 2 minutes, the tank is filled to (frac{3}{10}) of its capacity, leaving (frac{7}{10}). The time required for tap Q to fill the remaining (frac{7}{10}) is 14 minutes.
Conclusion
By utilizing the principles of rate and ratio, we can efficiently solve problems related to filling and emptying tanks. This problem demonstrates the step-by-step process of combining rates and determining remaining time, which is a valuable skill in math and practical problem-solving scenarios.