P, Q and R are channels which discharge solutions A, B, C respectively in a tank. When the tank is empty and all the three channels are opened what is the proportion of the solution C in the tank after 3 minutes, if the channels P, Q and R can fill the tank from empty to full in 30 minutes and 20 minutes and 10 minutes respectively when they are opened one at a time.
6/11
This problem involves understanding the concept of work rates applied to filling a tank using multiple channels. We need to determine the proportion of one specific solution (Solution C) in the tank after a certain time when all channels are working together.
Understanding the Tank Filling Problem
We have three channels, P, Q, and R, filling a tank. Each channel discharges a different solution (A, B, and C respectively). We know the time each channel takes to fill the tank individually. We want to find the proportion of solution C after 3 minutes when all channels are open simultaneously.
Calculating Individual Filling Rates
The rate at which a channel fills the tank is the reciprocal of the time it takes to fill the tank completely. If a channel fills the tank in $T$ minutes, its rate is $\frac{1}{T}$ tank per minute.
- Channel P fills the tank in 30 minutes. Its rate is $\text{Rate}_P = \frac{1}{30}$ tank/minute. This channel adds solution A.
- Channel Q fills the tank in 20 minutes. Its rate is $\text{Rate}_Q = \frac{1}{20}$ tank/minute. This channel adds solution B.
- Channel R fills the tank in 10 minutes. Its rate is $\text{Rate}_R = \frac{1}{10}$ tank/minute. This channel adds solution C.
Calculating the Combined Filling Rate
When all three channels are open, their rates add up to give the combined filling rate of the tank.
Combined Rate $ = \text{Rate}_P + \text{Rate}_Q + \text{Rate}_R $
Combined Rate $ = \frac{1}{30} + \frac{1}{20} + \frac{1}{10} $
To add these fractions, we find a common denominator, which is the Least Common Multiple (LCM) of 30, 20, and 10. The LCM is 60.
Combined Rate $ = \frac{1 \times 2}{30 \times 2} + \frac{1 \times 3}{20 \times 3} + \frac{1 \times 6}{10 \times 6} $
Combined Rate $ = \frac{2}{60} + \frac{3}{60} + \frac{6}{60} $
Combined Rate $ = \frac{2 + 3 + 6}{60} = \frac{11}{60}$ tank/minute.
Volume Filled by Each Channel in 3 Minutes
In 3 minutes, the volume of solution added by each channel is its rate multiplied by the time (3 minutes).
- Volume of solution A (from P) in 3 mins $ = \text{Rate}_P \times 3 = \frac{1}{30} \times 3 = \frac{3}{30} = \frac{1}{10}$ of the tank.
- Volume of solution B (from Q) in 3 mins $ = \text{Rate}_Q \times 3 = \frac{1}{20} \times 3 = \frac{3}{20}$ of the tank.
- Volume of solution C (from R) in 3 mins $ = \text{Rate}_R \times 3 = \frac{1}{10} \times 3 = \frac{3}{10}$ of the tank.
Total Volume Filled in 3 Minutes
The total volume filled in 3 minutes is the sum of the volumes added by each channel, or the combined rate multiplied by the time.
Total Volume $ = (\text{Combined Rate}) \times 3 $
Total Volume $ = \frac{11}{60} \times 3 = \frac{33}{60} = \frac{11}{20}$ of the tank.
Alternatively, sum of individual volumes:
Total Volume $ = \frac{1}{10} + \frac{3}{20} + \frac{3}{10} = \frac{2}{20} + \frac{3}{20} + \frac{6}{20} = \frac{2+3+6}{20} = \frac{11}{20}$ of the tank.
Finding the Proportion of Solution C
The proportion of solution C in the tank after 3 minutes is the ratio of the volume of solution C to the total volume of solution in the tank after 3 minutes.
Proportion of C $ = \frac{\text{Volume of Solution C in 3 mins}}{\text{Total Volume in 3 mins}} $
Proportion of C $ = \frac{\frac{3}{10}}{\frac{11}{20}} $
To divide by a fraction, we multiply by its reciprocal:
Proportion of C $ = \frac{3}{10} \times \frac{20}{11} $
Proportion of C $ = \frac{3 \times 20}{10 \times 11} = \frac{60}{110} $
Simplifying the fraction by dividing the numerator and denominator by their greatest common divisor, 10:
Proportion of C $ = \frac{60 \div 10}{110 \div 10} = \frac{6}{11} $
Therefore, the proportion of solution C in the tank after 3 minutes is $\frac{6}{11}$.
| Channel | Time to Fill Tank (mins) | Rate (tank/min) | Volume in 3 mins (fraction of tank) |
|---|---|---|---|
| P (Solution A) | 30 | $\frac{1}{30}$ | $\frac{1}{30} \times 3 = \frac{1}{10}$ |
| Q (Solution B) | 20 | $\frac{1}{20}$ | $\frac{1}{20} \times 3 = \frac{3}{20}$ |
| R (Solution C) | 10 | $\frac{1}{10}$ | $\frac{1}{10} \times 3 = \frac{3}{10}$ |
Total Volume in 3 mins $ = \frac{1}{10} + \frac{3}{20} + \frac{3}{10} = \frac{2}{20} + \frac{3}{20} + \frac{6}{20} = \frac{11}{20} $
Proportion of Solution C $ = \frac{\text{Volume of C}}{\text{Total Volume}} = \frac{3/10}{11/20} = \frac{3}{10} \times \frac{20}{11} = \frac{60}{110} = \frac{6}{11} $
Revision Table: Key Concepts in Tank Filling Problems
| Concept | Description | Formula/Approach |
|---|---|---|
| Rate of Work | The amount of work done per unit of time. For tank filling, it's the fraction of the tank filled per minute. | Rate $ = \frac{1}{\text{Time to complete the work}} $ |
| Combined Rate | The sum of individual rates when multiple entities work together. | Combined Rate $ = \text{Rate}_1 + \text{Rate}_2 + \dots $ |
| Work Done | The total amount completed. For tank filling, it's the fraction or total capacity of the tank filled. | Work Done $ = \text{Rate} \times \text{Time} $ |
| Proportion of a Component | The ratio of the volume of one component to the total volume of the mixture. | Proportion $ = \frac{\text{Volume of Component}}{\text{Total Volume}} $ |
Additional Information: Solving Time and Work Problems
Problems involving pipes, channels, or individuals filling or emptying tanks are common applications of time and work concepts. The key is to convert the time taken into a 'rate' or 'efficiency'.
- If a pipe fills a tank in $T$ hours, it fills $\frac{1}{T}$ of the tank in 1 hour. This is its filling rate.
- If a pipe empties a tank in $T'$ hours, it empties $\frac{1}{T'}$ of the tank in 1 hour. This is its emptying rate (often considered negative when combined with filling rates).
- When multiple pipes work together, their rates are added (for filling) or subtracted (if emptying).
- The total work done is usually considered as '1 unit' (representing the full tank).
- Time taken when working together = $\frac{\text{Total Work}}{\text{Combined Rate}}$.
- In problems involving mixtures (like different solutions), calculate the volume contributed by each source and the total volume to find proportions.
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