Albert and Bob are painting rooms at constant, but different

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

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

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)

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

Hi Karishma,

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

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.

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

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

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

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) + [n/(b + 1)](4/3) = 3n/5

4n/3b + 4n/[3(b + 1)] = 3n/5

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

4n[5(b + 1)] + 4n(5b) = 3n[3b(b + 1)]

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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