Simultaneously 81441

The tank is filled through three openings, A, B, and C. The simultaneously open inlets A and B are filled in 1 hour, inlets A and C in 45 minutes, and inlets B and C in 1.5 hours. How long would it take to fill each inlet separately?

Correct answer:

a =  1.2 h
b =  6 h
c =  2 h

Step-by-step explanation:

$$\begin{gathered} t_1=1\text{h} \\ {t}_2=0:45=0hr45min=0+\frac{45}{60}=0.75hr=0.75 \\ {t}_3=1.5\text{h} \\ 1/a+1/b=1/t_1 \\ 1/a+1/c=1/t_2 \\ 1/b+1/c=1/t_3 \\ A+B=1/t_1 \\ A+C=1/t_2 \\ B+C=1/t_3 \\ A+B=1/1 \\ A+C=1/0.75 \\ B+C=1/1.5 \\ A+B=1 \\ A+C=1.333333 \\ B+C=0.666667 \\ Row2-Row1\to Row2 \\ A+B=1 \\ -B+C=0.333 \\ B+C=0.667 \\ Row3+Row2\to Row3 \\ A+B=1 \\ -B+C=0.333 \\ 2C=1 \\ C=\frac{1}{2}=0.5 \\ B=\frac{0.33333333-C}{-1}=\frac{0.33333333-0.5}{-1}=0.16666667 \\ A=1-B=1-0.16666667=0.83333333 \\ A=\frac{5}{6}\doteq 0.833333 \\ B=\frac{1}{6}\doteq 0.166667 \\ C=\frac{1}{2}=0.5 \\ a=1/A=1/0.8333=\frac{6}{5}=1.2=1.2\text{h} \\ b=1/B=1/0.1667=6 \\ c=1/C=1/0.5=2 \end{gathered}$$

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