Working alone at a constant rate, Alan can paint a house in a hours.

Question

Working alone at a constant rate, Alan can paint a house in a hours. Working alone at a constant rate, Bob can point 1/4 of the same house in b hours. Working together, Alan and Bob can paint 1/3 of the house in c hours. What is the value of b in terms of a and c?

a) (3ac)/(a+c)
b) (4a-12c)/(3ac)
c) (3ac)/(4a-12c)
d) (ac)/(a+2c)
e) (ac)/(a+c)

a) (3ac)/(a+c)
b) (4a-12c)/(3ac)
c) (3ac)/(4a-12c)
d) (ac)/(a+2c)
e) (ac)/(a+c)

Spidy001 · Answer

work time

A 1 a
B 1/4 b

A+B 1/3 c

A's rate = 1/a

B's rate = 1/4b

when they are working together , their combined rate is 1/a + 1/(4b) = (1/3)/c

1/(4b) = 1/(3c) - 1/a

=> b = 3ac /(4a-12c)

Answer is C.

fluke · Answer

Alan's rate= 1/a
Bob's rate= 1/(4b)
Combined rate= 1/(3c)

\frac{1}{a}+\frac{1}{4b}=\frac{1}{3c}

Upon solving:
 b=\frac{3ac}{4a-12c}

Ans: "C"

shahideh · Answer

Alan's time to complete the task: a
Bob's time to complete the task: 4b
Together: 3c

\frac{1}{a}+\frac{1}{4b}=\frac{1}{3c}

b=\frac{3ac}{4a-12c}

Answer C

nycgirl212 · Answer

how do you isolate B on its own to figure out the other side of the equation?

EMPOWERgmatRichC · Answer

Hi All,

This question is a variation of a 'Work Formula' question (it involves 2 'entities' working on the same task together), so we can use the Work Formula to solve it.

Work = (A)(B)/(A+B) where A and B are the individual times that it takes the 2 entities to complete the task on their own.

We're told that:
1) Alan can paint a house in A hours
2) Bob can paint the same house in 4B hours
3) Together, Alan and Bob can paint the house in 3C hours

Using the Work Formula, we would have....

(A)(4B) / (A + 4B) = 3C

We're asked to solve for B in terms of A and C....

(A)(4B) = (3C)(A + 4B)
4AB = 3AC + 12BC
4AB - 12BC = 3AC
B(4A - 12C) = 3AC
B = (3AC)/(4A - 12C)

Final Answer:  C

GMAT assassins aren't born, they're made,
Rich

nycgirl212 · Answer

But isn't the equation:

\frac{1}{A} + \frac{1}{4B} =3C

EMPOWERgmatRichC · Answer

Hi nycgirl212,

The approach I used involved the Work Formula. If you want to use the 'unit' approach, then you should review the explanations offered by Spidy001, shahideh and fluke.

GMAT assassins aren't born, they're made,
Rich

JeffTargetTestPrep · Answer

We are given that Alan can paint a house in a hours and Bob can paint 1/4 of the same hours in b hours. Thus, we can say the following:

rate of Alan = 1/a

rate of Bob = (1/4)/b = 1/(4b)

Rate together = 1/a + 1/(4b).

However, since we are also given that working together Alan and Bob can paint 1/3 of the house in c hours, we can express their combined rate as (1/3)/c = 1/(3c). Since the combined rate has to be the sum of their individual rates, we can say 1/a + 1/(4b) = 1/(3c), and we can solve for b in terms of a and c.

Multiplying the entire equation by 12abc, we obtain:

12bc + 3ac = 4ab

3ac = 4ab - 12bc

3ac = b(4a - 12c)

3ac/(4a - 12c) = b

Answer: C

cavana · Answer

1) First of all we need to find the rates of Alan and Bob: A:  r*a=1; r=\frac{1}{a}, B:r*b=\frac{1}{4}, r=\frac{1}{4}/b=\frac{1}{4b} 
2) Working together Alan and Bob's rate is  \frac{1}{a}+\frac{1}{4b} 
3) Let's set up the equation using the information in the third sentence.  (\frac{1}{a}+\frac{1}{4b})*c=\frac{1}{3}, \frac{c}{a}+\frac{c}{4b}=\frac{1}{3} .........b= \frac{3ac}{4a-12c}

The correct answer is C

LeoN88 · Answer

3 Equations,
xa=J...........i
yb=J/4.......ii
(x+y)c=J/3.......... iii

Jis job and x and y are Alan and Bob's efficiency respectively.

x= (4yb)/a from i and ii

Substitute x in ii and iii to get
b= (3ca)/{4(a-3c)}...Option  C

ThatDudeKnows · Answer

Plugging In AND Hidden Plug In all in one question!

Plugging In: Any time I see variables that are repeated in the answer choices, I'm going to lean toward plugging in. Let's just make up values for a and b. 
Hidden Plug In: On work/rate questions, we can often make our lives easier by changing the definition of the job. Instead of painting a house, let's make the job solving 24 GMAT questions.

Alan can solve the 24 questions in a hours. Let's make a=6. So Alan can solve 4 questions per hour.
Bob can solve 1/4 the number of GMAT questions (so 6 questions) in b hours. Let's make b=3. So Bob can solve 2 questions per hour.
Together they can solve 4+2=6 questions per hour.
Together they can solve 1/3 the number of GMAT questions (so 8 questions) in c hours. How long does it take them to solve 8 questions at a rate of 6 questions per hour? 1.333. So c=1.333.
What is the value of b in terms of a and c? We have b=3. Let's go find that in the answer choices using a=6 and c=1.333.

(A) 24/7.333 Wrong.
(B) (24-16)/24 = 8/24 Wrong.
(C) 24/(24-16) = 24/8 = 3 Keep it.
(D) 8/8.667 Wrong.
(E) 8/7.333 Wrong.

Answer choice C.

ThatDudeKnowsPluggingIn
ThatDudeKnowsHiddenPlugIn

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

Working alone at a constant rate, Alan can paint a house in a hours.

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

Working alone at a constant rate, Alan can paint a house in a hours.

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards