A portable recharging device can recharge 3 batteries at once, and it

Question

A portable recharging device can recharge 3 batteries at once, and it can charge the batteries while itself being charged from an outlet. The device can recharge a single dead battery in 3 hours, and it can be completely recharged from an outlet in 2 hours when it contains no batteries. A fully discharged device holding 3 dead batteries is plugged into an outlet. If the capacity of a battery is one-third the capacity of the device, and the device and batteries charge at constant rates, how much time will pass before the device is completely recharged?

a) 5 hours

b) 6 hours

c) 8 hours

d) 9 hours

e) 11 hours

a) 5 hours  
  b) 6 hours  
  c) 8 hours  
  d) 9 hours  
  e) 11 hours

I'm unable to understand the explanation provided. Does anyone have a better explanation?

The question asks for the time it takes the device to fully recharge when it holds 3 batteries. Determine the individual rates at which the device is recharged and discharged. Then, subtract the discharge rate from the recharge rate to determine the overall rate at which the device is charged when it holds 3 batteries. If the device is completely recharged from an outlet in 2 hours when it holds no batteries, then the device is recharged at a rate of  \frac{ 1 device}{ 2 hours }  .
Now, determine the discharge rate for a device holding 3 dead batteries. If the device can charge 1 battery in 3 hours, and it can charge 3 batteries at once, then the device can charge 3 batteries in 3 hours. Since the capacity of a single battery is one-third the capacity of the device, the capacity of three batteries is equal to the capacity of the device, and a device containing 3 batteries is discharged at a rate of  \frac{1 device}{3 hours} . Now, subtract the discharge rate from the recharge rate in order to determine the overall rate at which the device is charged or :

\frac{1 device}{2 hours} - \frac{1 device}{3 hours } = \frac{1 device}{6 hours}

The correct answer is choice B.

sujoykrdatta · Accepted Answer

Understand that: The power source charges ONLY the device
The charge in the device now charges the batteries
While the device is being charged by the power source, it keeps charging the batteries using its stored charge

Let the charge capacity of the device be 3 units => Charge capacity of a battery is 1/3 * 3 = 1 unit

The device charges 1 battery (1 unit) in 3 hrs 
=>  Charge transfer rate from device to battery per hour = 1/3 unit/hr

To charge itself, the device takes 2 hours (without any batteries) 
=>  Charge transfer rate to device from power source = 3/2 unit/hr

When 3 dead batteries are connected to a discharged device: 
# Charge transfer rate to device from power source = 3/2 unit/hr
# Charge transfer rate from device to  3 batteries  per hour = 3/3 = 1 unit/hr
=> Charge stored in device per hour = 3/2 - 1 = 0.5 hours

=> Time taken to fully charge device (capacity 3 units) = 3/0.5 =  6 hours

Answer B

ScottTargetTestPrep · Answer

We can let each battery be 1 unit. Since the device charges 1 battery (or 1 unit) in 3 hours, the charge transfer rate from the device to a battery is 1/3 unit/hr and hence that rate is 3/3 = 1 unit/hr when there are 3 batteries in the device.

We are also given that the capacity of a battery is one-third the capacity of the device, so the device is equivalent to 3 batteries, or 3 units. Since it takes 2 hours to charge the device without any batteries, the charge transfer rate from an outlet to the device is 3/2 unit/hr.

Since 3 dead batteries are connected to a discharged device which is connected to an outlet, the charge transfer rate 3/2 - 1 = 1/2 unit/hr. Since the device is equivalent to 3 units, it takes 3/(1/2) = 6 hours to fully charge the device.

Answer: B

arvind910619 · Answer

Imo B

Let the capacity of the battery be X 
Then the using A = r*t equation we have x = r1*3 as it takes 3 hours for the battery to get charged
r1 = x/3

Capacity of the charger 3*x as we are given in the question
So again using A=r*t equation we have 
3x = r2*2 as it takes 2 hours to charge the charger.

No we have to find out the amount of time that the charger takes to get charged when batteries are also plugged in.
 
The time will be the difference in between their respective rates.
Think of this as a pipe equation where the charger is filler the charge and batteries are draining the charge, so we have the equation

r2 - (r1+r1+r1) as we have three batteries.

3x/2 - 3x/3 = x/2

Now it takes 2 hours for x capacity to charge 
then for 3x capacity it will take 2*3x/x hours = 6 hours.

A very difficult question indeed

h31337u · Answer

A portable recharging device can recharge 3 batteries at once, and it can charge the batteries while itself being charged from an outlet. The device can recharge a single dead battery in 3 hours, and it can be completely recharged from an outlet in 2 hours when it contains no batteries. A fully discharged device holding 3 dead batteries is plugged into an outlet. If the capacity of a battery is one-third the capacity of the device, and the device and batteries charge at constant rates, how much time will pass before the device is completely recharged?

a) 5 hours

b) 6 hours

c) 8 hours

d) 9 hours

e) 11 hours

a) 5 hours
b) 6 hours
c) 8 hours
d) 9 hours
e) 11 hours

_______________________________

Think of this problem as a filling a tank problem. Then it will all make sense and be easy to solve.

A outlet equals a pipe to fill a tank.
A battery equals a pipe to empty a tank.

To fill a tank, it will take 2 hours. So 1/2 per hour.
To empty a tank, it takes 3 hours (3 dead devices), so 1/3 per hour.

With that, per hour you have 1/2 - 1/3 = 1/6. 
You want to refill the full tank, so you need to know how long it will take to fill 12/6.

After 3 hours, you fill 3/6 out of 12/6, therefore only 9/6 left.
From here, you only need to fill the device, which is 3/6.
 3/6 x 3(hours) = 9/6.

So you need 3 hours + 3 hours = 6 hours.

Therefore the answer is B.

Waldfee124 · Answer

ScottTargetTestPrep  Aren't the 3 batteries fully charged after 3 hours so that the full charge rate of 3/2 unit/hr goes into the device which is by that time already half charged so that only one more hour is needed = 4hours in total?

I think the question is intentionally ambiguous. "The device can recharge a single dead battery in 3 hours" Why can you assume that this means that 3 batteries will also be charged in 3 hours? Where does it say that the charge rate was limited by the battery and not the device itself? 
Also it says that itself can be charged while charging batteries, but the recharge time of 2 hours is given only for the state "when it contains no batteries"
This is a horrible question for me, difficult by being ambiguous

Regor60 · Answer

A key point is the statement that the capacity of the device is equal to 3 batteries, so the 1/3 battery per hour also works for all 3.

Posted from my mobile device

IanStewart · Answer

Everything you're saying is absolutely correct. None of the solutions in this thread are correct (including the "OA"), unless you make a very strange assumption. It's easy to see why the solutions above are wrong just by changing the question: if the device takes 2 hours to charge, and the device takes 2 hours (instead of 3) to charge three batteries, using the method in the solutions in this thread, it would take an infinite amount of time to charge the device. So you'd have a paradoxical situation where if you make the charger better at charging batteries, you make it impossible to ever charge the charger itself. That can't be right.

The answer is only 6 hours here if you assume the charger keeps charging the batteries after the batteries are all fully charged, which doesn't make much sense. If three batteries are equivalent to one device, and we need to charge three batteries and one device, we just need to charge the equivalent of two devices. That will take 4 hours, unless something charges so slowly we need more time than that. So four hours is the answer. If that's not convincing, the question can be solved in parts. In the first 3 hours, the device charges the three batteries fully, and in each hour 1/2 of the device gets charged, but 1/3 of a device is fed into the batteries, so the extra charge going to the device is 1/2 - 1/3 = 1/6 of a device per hour. So after 3 hours the device is 3/6 = 1/2 charged and the batteries are fully charged. Assuming the device stops charging the fully charged batteries, it will take one more hour to fill the last 1/2 of the device, for 4 hours in total.

But as you say, the wording is so opaque there's no easy way to guess what the question means. I'm not sure why we assume here that the charger is able to charge all three batteries in three hours, as you correctly point out; I don't think it's unreasonable to guess that the charger charges batteries in sequence, in which case it will take 9 hours to charge everything (the charger itself will get more than enough extra charge in 9 hours). So I agree completely with your assessment of the question.

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A portable recharging device can recharge 3 batteries at once, and it

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