Two taps A and B can fill a tank with water in 20 minutes and 15 minutes respectively. A third tap C can empty the full tank in 6 minutes. After the taps A and B fill the tank for 5 minutes, the tap C is opened to empty while A and B continue to fill it. In how many minutes after the start shall the tank get empty ?<\/p>\n
[amp_mcq option1=”$10\\frac{1}{3}$ minutes” option2=”$10\\frac{2}{5}$ minutes” option3=”$10\\frac{4}{5}$ minutes” option4=”$11\\frac{2}{3}$ minutes” correct=”option4″]<\/p>\n
Initially, taps A and B fill the tank for 5 minutes.
\nCombined filling rate of A and B = $\\frac{1}{20} + \\frac{1}{15}$.
\nLCM of 20 and 15 is 60.
\nCombined rate of A and B = $\\frac{3}{60} + \\frac{4}{60} = \\frac{7}{60}$ tank per minute.
\nVolume filled in the first 5 minutes = $5 \\times \\frac{7}{60} = \\frac{35}{60} = \\frac{7}{12}$ of the tank.<\/p>\n
After 5 minutes, tap C is opened, while A and B continue to fill. All three taps are now open.
\nThe net rate of change in tank volume is the sum of individual rates:
\nNet rate = Rate of A + Rate of B + Rate of C
\nNet rate = $\\frac{1}{20} + \\frac{1}{15} – \\frac{1}{6}$.
\nLCM of 20, 15, and 6 is 60.
\nNet rate = $\\frac{3}{60} + \\frac{4}{60} – \\frac{10}{60} = \\frac{3+4-10}{60} = \\frac{-3}{60} = -\\frac{1}{20}$ tank per minute.
\nThe net rate is negative, which means the tank is emptying at a rate of $\\frac{1}{20}$ tank per minute.<\/p>\n
At the moment C is opened, the tank is $\\frac{7}{12}$ full. Since the net rate is negative, the tank will empty from this level.
\nThe time taken to empty the tank from this point is the current volume divided by the emptying rate:
\nTime to empty = $\\frac{\\text{Current Volume}}{\\text{Net Emptying Rate}} = \\frac{7\/12}{1\/20}$ (using the absolute value of the negative rate).
\nTime to empty = $\\frac{7}{12} \\times 20 = \\frac{140}{12} = \\frac{35}{3}$ minutes.<\/p>\n
The question asks “In how many minutes after the start shall the tank get empty ?”. This phrasing suggests the total time from t=0.
\nTotal time from start = Time A and B filled alone + Time A, B, C worked together until empty.
\nTotal time = 5 minutes + $\\frac{35}{3}$ minutes = $\\frac{15}{3} + \\frac{35}{3} = \\frac{50}{3}$ minutes.
\n$\\frac{50}{3} = 16 \\frac{2}{3}$ minutes.<\/p>\n