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

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

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

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:

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

But isn't the equation:

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

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

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

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

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

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

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