Calculating Combined Pipe Filling Time
This problem involves finding the time it takes for multiple pipes operating simultaneously to fill a tank, given their individual filling times.
Individual Pipe Rates
First, determine the rate at which each pipe fills the tank. The rate is the fraction of the tank filled per hour.
- Rate of Pipe A: $\frac{1}{19}$ tank per hour.
- Rate of Pipe B: $\frac{1}{21}$ tank per hour.
- Rate of Pipe C: $\frac{1}{8}$ tank per hour.
Combined Rate Calculation
To find the combined rate when all pipes operate together, add their individual rates.
Combined Rate = Rate A + Rate B + Rate C
Combined Rate = $\frac{1}{19} + \frac{1}{21} + \frac{1}{8}$
Find a common denominator, which is the least common multiple (LCM) of 19, 21, and 8. LCM(19, 21, 8) = $19 \times 21 \times 8 = 3192$.
Combined Rate = $\frac{1 \times (21 \times 8)}{19 \times 21 \times 8} + \frac{1 \times (19 \times 8)}{19 \times 21 \times 8} + \frac{1 \times (19 \times 21)}{19 \times 21 \times 8}$
Combined Rate = $\frac{168}{3192} + \frac{152}{3192} + \frac{399}{3192}$
Combined Rate = $\frac{168 + 152 + 399}{3192} = \frac{719}{3192}$ tank per hour.
Total Time Calculation
The time taken to fill the tank together is the reciprocal of the combined rate.
Time = $\frac{1}{\text{Combined Rate}}$
Time = $\frac{1}{\frac{719}{3192}} = \frac{3192}{719}$ hours.
Convert to Mixed Fraction
Convert the improper fraction $\frac{3192}{719}$ into a mixed number.
Divide 3192 by 719:
$3192 \div 719 = 4$ with a remainder of $3192 - (4 \times 719) = 3192 - 2876 = 316$.
Therefore, the time taken is $4 \frac{316}{719}$ hours.
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