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A certain water tank has 3 intake valves, valve A, valve B, and valve 27 Jan 2025, 01:30
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A certain water tank has 3 intake valves, valve A, valve B, and valve C, each of which fills the tank at a distinct constant rate. Valve A alone can fill the tank in 4 hours and valve B alone can fill the tank in 6 hours. If all 3 valves working together can fill the tank in 2 hours, what is the amount of time valves B and C working together take to fill the tank?

A. 3
B. 4
C. 6
D. 8
E. 12


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B
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A certain water tank has 3 intake valves, valve A, valve B, and valve 27 Jan 2025, 03:02
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Rate of valves A, B, C be \(\frac{1}{a}\), \(\frac{1}{b}\), \(\frac{1}{c}\)

Time taken to fill tank when all 3 valves working together, \(t=2\)

\(a=4\) & \(b=6\)

\(\frac{1}{a}+ \frac{1}{b} + \frac{1}{c}=\frac{1}{t} \)

Amount of time it takes for valve B and C working together take to fill the tank, \(x\) can be found by their combined rates

\(\frac{1}{b} + \frac{1}{c}= \frac{1}{x}\)

\(\frac{1}{a} + \frac{1}{x}= \frac{1}{t}\)

\(\frac{1}{4} + \frac{1}{x}= \frac{1}{2}\)

\(\frac{1}{x} = \frac{1}{2}-\frac{1}{4}= \frac{1}{4} \)

\(x=4\)

Answer: B
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A certain water tank has 3 intake valves, valve A, valve B, and valve 27 Jan 2025, 03:05
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Bunuel
A certain water tank has 3 intake valves, valve A, valve B, and valve C, each of which fills the tank at a distinct constant rate. Valve A alone can fill the tank in 4 hours and valve B alone can fill the tank in 6 hours. If all 3 valves working together can fill the tank in 2 hours, what is the amount of time valves B and C working together take to fill the tank?

A. 3
B. 4
C. 6
D. 8
E. 12
­
lets assume rates for valves A, B and C as a, b and c respectively.
Volume of tank = Rate * Time taken by valve
Volume of tank = a*4 = b*6
LCM of 4 and 6 is 12.
lets assume volume of tank as 12 units
now calculating rates a = 12/4 = 3 units/hr and b = 12/6 = 2 units/hr
Now 3 of them working together and fills the tank in 2 hrs.
2 * (a + b + c) = 12
2 * (3 + 2 + c) = 12
solving we will get c = 1
Now time when B and C works together = Volume of tank / Combine rate of B and C
= 12 / (b+c) = 12 / (2 + 1) = 12/3 = 4 Hrs
Answer is B.
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A certain water tank has 3 intake valves, valve A, valve B, and valve 27 Jan 2025, 08:28
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Method: Using efficiency

The time taken by \(A = 4, B= 6, (A+B+C) = 2\)

To find the efficiency, we needed to find the Total Work(TW).


Now we will find LCM to calculate total work, i.e, \( TW = LCM(A, B, (A+B+C)) = LCM(4, 6, 2) = 12\)

Therefore,


Efficiency of \(A = \dfrac{12}{4} = 3\)

Efficiency of \(B = \dfrac{12}{6} = 2\)

Efficiency of \((A+B+C) = \dfrac{12}{2} = 6\)


Using the above efficiency, we can calculate the efficiency of \(C\), i.e., \((A+B+C) = 6 \Longrightarrow C = 6 - (A+B) \Longrightarrow C = 6 -(3+2) = 1\)

Time to fill the tank using tap \((B+C) = \dfrac{12}{2+1} = \dfrac{12}{3} = 4 \)

ANS B
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A certain water tank has 3 intake valves, valve A, valve B, and valve 27 Jan 2025, 08:41
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A certain water tank has 3 intake valves, valve A, valve B, and valve C, each of which fills the tank at a distinct constant rate. Valve A alone can fill the tank in 4 hours and valve B alone can fill the tank in 6 hours. If all 3 valves working together can fill the tank in 2 hours, what is the amount of time valves B and C working together take to fill the tank?
We use the formula:

Rate of Work * Time =Total Work Done

Let the total work done be 1.
Rate of work of A, B and C be Wa, Wb, Wc
Hence given,
Wa*4=1 Hence Wa = 1/4 ..... (i)
Wb*6=1 Hence Wb = 1/6 ..... (ii)

Also, (Wa+Wb+Wc)*2 = 1
Substituting values from (i) and (ii), we get
Wc =1/12

Now we are asked What is the time required for B and C to finish the work together,

Therefore we are asked t where:
(Wb+Wc)*t=1
Substituting values of Wb and Wc ,
We get t= 4 hours.

IMO Ans B
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A certain water tank has 3 intake valves, valve A, valve B, and valve 27 Jan 2025, 09:00
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Assume rates for valves A, B, and C as a, b, and c, respectively.

1. Volume of tank = Rate × Time taken by valve.
- Given: a × 4 = b × 6
- LCM of 4 and 6 is 12. Assume tank volume = 12 units.

2. Calculate rates:
- a = 12/4 = 3 units/hr
- b = 12/6 = 2 units/hr

3. All three valves working together fill the tank in 2 hours:
- 2 × (a + b + c) = 12
- 2 × (3 + 2 + c) = 12
- Solve for c: c = 1 unit/hr

4. Time when valves B and C work together:
- Combine rate of B and C = b + c = 2 + 1 = 3 units/hr
- Time = Volume / Rate = 12 / 3 = 4 hours

Answer: B
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A certain water tank has 3 intake valves, valve A, valve B, and valve 27 Jan 2025, 10:24
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Valve A fills in 4 hours, Valve B fills in 6 hours, Valve C=?
1/A+1/B+1/C=1/2
C=12 hours

1/B+1/C= 1/6+1/12= 3/12 =1/4

Total time taken by valve B+C together to fill the tank= 4 hours

Answer: B