Lindsay can paint 1/x of a certain room in 20 minutes. What fraction

Great approach by bunuel.

It is also an option to plug in numbers.
If x=3 then Lindsay paints 1/3 in 20 minutes = 3/3 in 1 hour, i.e. she paints the entire room herself. Therefore x=3 should make Joseph = 0.
C and E achieves this goal
C: 3-3/3*3 = 0/9 = 0
E: 3-3/3 = 0/3 = 0

x=6 means Lindsay paints 1/6 room per 20 minuters = 3/6 = 1/2 room in 1 hour. Therefore x=6 should make Joseph paint 1/2 in 1 hour, i.e. 1/6 in 20 minutes (same work rate as Lindsay actually).
C: (6-3)/(3*6) = 3/18 = 1/6 correct
E: (6-3)/6 = 3/6 NOT correct

Answer is C.

What a great question! I got so caught up solving for the portion that Joseph completed in the hour that I completely forgot to multiply by 1/3 at the end to actually answer the question. In an effort to complete the question fast, I picked E. If I read the question one more time after doing the math, I would have gotten it correct. Lesson learned! Thank you for sharing.

Let's assume that total unit of work is 1 unit
Since Lindsay paints 1/x of certain paint in 20 minutes; therefore, the efficiency of Lindsay is 1/20x
and together they complete the 1 unit of work in 60 minutes, therefore their combined efficiency is 1/60

eff(Lindsay)+eff(joseph)=1/60
i/20x + eff(joseph) = 1/60
eff(joseph) = 1/60 - 1/20x =(x-3)/60x

so joseph will complete the work in 60x/(x-3) minutes

so in 20 minutes he will complete (x-3)/60x * 20 units of work = (x-3)/3x

We are given that Lindsay can paint 1/x of a room in 20 minutes; thus, she can paint 3/x of a room in 60 minutes (or in 1 hour). Thus, her hourly rate is 3/x room/hr. We are also given that when she works with Joseph, they can paint the entire room in 1 hour. If we let total work = 1 and j = the number of hours it takes Joseph to paint the room by himself, then Joseph’s rate = 1/j room/hr. We can create the following equation and isolate j:

work of Lindsay + work of Joseph = 1

(3/x)(1) + (1/j)(1) = 1

3/x + 1/j = 1

Multiplying the entire equation by xj, we obtain:

3j + x = xj

x = xj - 3j

x = j(x - 3)

x/(x - 3) = j

Since j = x/(x - 3) and 1/j = Joseph’s rate, then Joseph’s rate, in terms of x, is (x - 3)/x.

Since 20 minutes = 1/3 of an hour, and since work = rate x time, Joseph can complete:

[(x - 3)/x](1/3) = (x - 3)/(3x) of the job in 20 minutes.

Alternate Solution:

Since Lindsay and Joseph, working together, can paint the whole room in 1 hour, then in 20 minutes, they can paint 1/3 of the room. If we let r be the fraction of the room that Joseph can paint in 20 minutes, then it must be true that:

1/x + r = 1/3

r = 1/3 - 1/x

Using a common denominator of (3x), we obtain:

r = (x - 3)/(3x)

Answer: C

Hi Bunuel,

Can you explain how you got 1-(3/x) I didn't get the part how you took 1.
Did you consider the total work done as 1?
I took the Total work done as X and thus got Work done by Joseph as : \(X-\frac{3}{X}\)

Yes, the whole job is 1 unit.

The question says that Lindsay can paint \(\frac{3}{x}\) of a room in 1 hour. If 3/x is confusing there, consider x to be say 6 and it will become easier to understand: Lindsay can paint \(\frac{3}{6}=\frac{1}{2}\) of a room in 1 hour. Together they paint the whole room in 1 hour then Joseph can paint \(1-\frac{1}{2}=\frac{1}{2}\) of a room in 1 hour. In 20 minute or in \(\frac{1}{3}\) of an hour Joseph can paint \(\frac{1}{6}\) of a room.

Now, plug x = 6 into the options to see which one gives you 1/6.

Yes, the whole job is 1 unit.

The question says that Lindsay can paint \(\frac{3}{x}\) of a room in 1 hour. If 3/x is confusing there, consider x to be say 6 and it will become easier to understand: Lindsay can paint \(\frac{3}{6}=\frac{1}{2}\) of a room in 1 hour. Together they paint the whole room in 1 hour then Joseph can paint \(1-\frac{1}{2}=\frac{1}{2}\) of a room in 1 hour. In 20 minute or in \(\frac{1}{3}\) of an hour Joseph can paint \(\frac{1}{6}\) of a room.

Now, plug x = 6 into the options to see which one gives you 1/6.[/quote]


Thanks a lot Bunuel! I understood your method now. You make things very simple.

Assume the work to be something that is easily divisible by 20 and x => 100x

Lindsay can paint 1/x of the room in 20 minutes so it means Rate of work = 100x/20x = 5/minute (Call it RL)
Now, Lindsay and Joseph together can do the work in 60 minutes so we have => (5+RJ)*60 = 100x
From this we get RJ = (5x-15)/3 per minute

In 20 minutes Rj will do 20*(5x-15)/3

Fraction of work => (20(5x-15))/3*100x => (x-3)/3x