Albert and Bob are painting rooms at constant, but different

Question

Albert and Bob are painting rooms at constant, but different rates. Albert takes 1 hour longer than Bob to paint n rooms. Working side by side, they can paint a total of 3n/5 rooms in 4/3 hours. How many hours would it take Albert to paint 3n rooms by himself?

a) 7
b) 9
c) 11
d)13
e) 15

Please help, totally stumped. Looking for some simpler and quick approach.

A. 7
B. 9
C. 11
D.13
E. 15

KarishmaB · Accepted Answer

First of all, I don't like unnecessary variables. The data given is in terms of n rooms but the answer is independent of n. This means, no matter what value I put for n, the answer would be the same. So I will just put n = 1 and proceed.

Albert takes an hour longer than Bob to paint one room.
Time taken by Albert to paint 1 room = A hrs, Time taken by Bob = A-1 hrs

"they can paint a total of 3/5 rooms in 4/3 hours"
W = R*T
3/5 = R*(4/3)
R = 9/20 which is the combined rate of the two.

We obtained this by adding their rates together: 9/20 = 1/A + 1/(A-1)
Over here, please remember that numbers fall in place in GMAT questions. You should be able to figure that since the denominator is 20, A = 5 gives 5*4 in the denominator so it should work. Check before proceeding though.

So Albert paints a room in 5 hrs. For 3 rooms, he takes 3*5 = 15 hrs.

kys123 · Answer

rate of bob to paint n rooms is n/T(bob) 
rate of Albert to paint n rooms is n/T(albert). 
albert paints the room 1 hour slower than bob, so T (albert) = T(bob) -1
Together they paint the 3n/5 rooms in 4/3 hours. Rate is equal to work over time

Therefore
n/x + n/x-1 = (3n/5)/(4/3) = 
n/x + n/x-1 = 9n/20. Fastest way for me is to think how would make the denominator 20. 4*5 = 20 and it fits x and x-1 or you can solve the quadratic 
4n/20 + 5n/20 = 9n/20. Therefore you know it takes Albert 5 hours to paint n room, since Albert's rate is n/5.
5*3 = 3n
15 = 3n.

Answer is E

kartik222 · Answer

Hi, This question can be done very quickly if you understand what they asking for. I will present a very general way of solving ANY work,rate and time related question: Question will always ask for something out of rate or time or work. (obviously coz its a work time rate related question

) ex. in this question they want "time" for "3n" work done by Albert, right?? This means ------> rate * TIME = 3n =====> All you have to do it find out the RATE for albert and then put it here to find the TIME!!!!! (easy right??) Let see what is given here: some relation between Albert and bob's individual time and their combined time. In combined time: some work and time is given, we don't care what it is, all we will do is find their rate by writing WORK/TIME i.e. (3n/5) / 4/3 = 9n/20 = 9/20 From individual relationships just find Albert and Bob's respective rates: A: takes 'a' time to complete n rooms===> rate becomes : n/a = 1/a B: takes 1 hr less than A to complete n rooms =====> rate becomes : n/(a-1) = 1/a-1 now add individual rates and equate 'em to the combined rate: 1/a + 1/(a-1) = 9/20 From my experience I feel that from thus point its faster and much easier to put values than solve the quadratic: if u see that answer I feel 9 and 15 stand out as they are multiples of 3 and they represent 3n work, we can do 3n=15 => n = 5 and put "5" in place of "a" in the equation. it will be equal to 9/20 thats ur answer. BOTTOM LINE: first thing always look for what is need to in final answer and how can I get their, if yo start by what's given then u will start doing actions not even necessary to reach the solutions. I hope this helps, thanks, -Kartik

jlgdr · Answer

Did a similar approach but somewhat shorter A = B+1 Then 4/3 (1/B+1/(B+1)) = 3/5 Hence 1/B + 1/(B+1) = 9/20 So B = 4 Therefore A = 5 And so we will take 5*3 = 15 to paint three rooms Answer is E Just my 2c Hope it helps Cheers! J

TGC · Answer

Rate * Time = Work

Let B be the hours that Albert takes to paint 'n' rooms then

For (Albert) => n/B * B = n
For (BOB) => n/(B-1) ( (B-1) = n

Combined rate = n/B + n/(B-1)

Again Rate * Time = Work

Then {n/B + n/(B-1)} * 4/3 = 3n/5

2B-1/{B(B-1)}=9/20

Substituting values B=5 then answer takes 3B since 3n rooms then (E)

jlgdr · Answer

Here's another way. May be shorter

A = B+1
B = A-1

Now then we have

1/A + 1/ (A-1) = (3/5) / (3/4) = 9n/20

We need to find 3n

Let's see

To get 9/20, we need something like 1/4 + 1/5 = 9/20

So A must be 5

If A takes 5 to paint 'n', it takes 15 to paint '3n'

Answer: E

ronr34 · Answer

Hi Karishma,

What do you mean "numbers fall in place in GMAT"?

KarishmaB · Answer

It means that GMAT doesn't expect you to do any funny decimal calculations. You won't find yourself multiplying 2.3 by 6.7. If the rate of work comes out to be 1/11, I would expect that the time for which you would have to work to complete it would be 22 hrs/33 hrs... etc

Since GMAT doesn't give you a calculator, it doesn't expect you to waste time doing these calculations in the test when you can easily use a calculator in real life. You don't need to have the skill to perform arduous calculations in a short span of time. You need to understand concepts and that's what they test. The numbers they use to test those concepts will be clean and easy to work with. Recognizing the concept will be the tough part.

gracie · Answer

let t=Albert's time to paint n rooms
Albert's rate=n/t
Bob's rate=n/(t-1)
Albert and Bob's rate=(3n/5)/(4/3)=9n/20
dividing out n, 1/t+1/(t-1)=9/20
t=5 hours
3t=15 hours

HolaMaven · Answer

Since options are taking about hrs taken by Albert to paint 3n rooms, we can use options which are divisible by 3, thus working on option B and E,
 Option B 
If for 3n rooms, Albert takes 9 hrs, so for n rooms he takes 3 hrs. Thus Bob will take 2 hrs to paint n rooms
Now if n = 60
Rate of Albert =  \frac{60}{3}  = 20 rooms per hour
Rate of Bob =  \frac{60}{2}  = 30 rooms per hour
So combined rate = 20+30 = 50 rooms per hour
So together they complete 50*  \frac{4}{3} = \frac{200}{3}  rooms which is not equal to  \frac{3*60}{5}

Option E: 
If for 3n rooms, Albert takes 15 hrs, so for n rooms he takes 5 hrs. Thus Bob will take 4 hrs to paint n rooms
Now if n = 60
Rate of Albert =  \frac{60}{5}  = 12 rooms per hour
Rate of Bob =  \frac{60}{4}  = 15 rooms per hour
So combined rate = 12+15 = 27 rooms per hour
So together they complete 27*  \frac{4}{3}  = 36 rooms which is equal to  \frac{3*60}{5} 
 Thus option E

gracie · Answer

let n=1
A's rate=1/t+1
B's rate=1/t
A+B rate=(3/5)/(4/3)=9/20
because the ratio of A's rate to B's rate=t/t+1,
assume A's rate to B's rate=(4/20)/(5/20) 
20/4=5 hours
3*5=15 hours
E

ScottTargetTestPrep · Answer

We can let b = the number of hours Bob takes to paint n rooms; thus, Bob’s rate = n/b and Albert’s rate = n/(b + 1). Knowing that they complete 3n/5 rooms in 4/3 hours, we can create the following equation:

(n/b)(4/3) +  (4/3) = 3n/5

4n/3b + 4n/  = 3n/5

Multiplying the entire equation by 3b x 5 x (b + 1) = 15b(b + 1), we have:

4n  + 4n(5b) = 3n

Dividing by n, we have:

20(b + 1) + 20b = 9b(b + 1)

20b + 20 + 20b = 9b^2 + 9b

9b^2 - 31b - 20 = 0

(9b + 5)(b - 4) = 0

b = -5/9 or b = 4

Since b can’t be negative, b = 4. Thus, it takes Albert 4 + 1 = 5 hours to paint n rooms and 5 x 3 = 15 hours to paint 3n rooms.

Answer: E

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Albert and Bob are painting rooms at constant, but different

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