rahul.s wrote: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
The answers are nice concrete numbers, so let's try backsolving (i.e. working backwards from the choices).
Before we automatically dive in with B or D (where we usually start for backsolving), let's do a bit of critical thinking: each answer represents the length of time it takes to paint 3n rooms, so answers divisible by 3 are going to be much simpler to work with. We also have the term "3n/5" in the question, so if 3n is a multple of 5 the numbers will work out much nicer. Which answere is a multiple of 3 and 5? Only 15 (making it a great guess if we're short on time), so let's start there.
If it takes 15 hours to pain 3n rooms, then it takes albert 5 hours to paint n rooms working alone.
We know that it takes Albert 1 hour longer to paint n rooms than it takes Bob, so it takes Bob 4 hours to paint n rooms.
So, working together, it would take (5*4)/(5+4) = 20/9 hours to paint n rooms.
(The basic work formula for 2 workers is:
combined time = a*b/(a+b).)
Now we can set up the ratio:
n/(20/9) = (3n/5)/(4/3)
cross multipying:
9n/20 = 9n/20... bingo! Everything worked out properly, so E is in fact the correct answer.
Now, as we can see, with complicated terms backsolving definitely takes a lot more work than with simple ones, but it did allow us to avoid quadratics and complicated algebra. If we hadn't nailed it on the first shot, this could have turned into a 3-5 minute question, which we definitely want to avoid on Test Day.
Accordingly, this is a great question for strategic guess and, if you got to "multple of 3 and 5 will be much simpler" as I did early on and guessed 15, you were much happier!
* * *
Let's look at how we could have strategically chosen E with 100% certainty:
Together, they can paint 3n/5 rooms in 4/3 of an hour; let's solve for the time for n rooms:
(3/5)n = 4/3
n = (4/3)(5/3) = 20/9
So, it takes them 20/9 hours to paint n rooms together.
To paint 3n rooms, it would take 3*(20/9) = 60/9 hours together.
If they worked at the same rate, then we would double 60/9 to get their individual rates (since at the same rate they each do half the work).
2 * 60/9 = 120/9 = 13 1/3 hours.
Now, we know that it takes Albert longer than it takes Bob; therefore, Albert's "alone" time must be more than 13 1/3 hours.
Only E is greater than 13 1/3... done!
