A painting crew painted 80 houses. They painted the first y houses at

Let w1 and w2 be number of weeks.
So
First Week= x * w1 =y

Second Week = 5/4x * w2 = 80-y (Remaining houses)

Question is to find w1 + w2

i.e y/x + 80-y / 5/4x

= y+320/5x
= B

IMO answer is B
Total weeks= y/x +(80−y)/1.25x ...
=(y+320)/(5x)

first part:
y houses at a x houses per week = y/x

second part:
rest of the houses at a 1.25x houses per week = 80-y / 1.25x

calculation:
y/x + 80-y/1.25x (multiple by 4 to make it easy)
4y/4x + 320 - 4y/5x
20y + 1280 - 16y / 20x
4y + 1280 / 20x (divide by 4 again)
y + 320 / 5x

Kudos if you like it

y/x + (80-y)/1.25 --> Work together

Solve above equation gives

(y/4 + 80)/(5/4)x=y+320/5x. Ans is B

Painted first y houses @x houses per week
To paint 1 house, time taken=1/x weeks
Therefore, to paint y houses, time taken=y/x weeks
Similarly, time taken to paint 80-y houses=80-y/1.25x
Total time taken=y/x+80-y/1.2x=4(0.25y+80)/5x=y+320/5x
Answer B

MANHATTAN GMAT OFFICIAL SOLUTION:

This is a combined work problem, so we will use the work formula: rate × time = work. The work and rates are given, but we need to calculate time, so we manipulate the formula: time = work/rate. This is also a Variable In the answer Choices (VIC) problem, so it is efficient to pick numbers and test the answer choices.

We are told that there are 80 houses, that y houses are painted at a rate of x houses per week, and that the rate increases to 1.25x houses per week for the remaining 80 – y houses. We will pick values such that x and 1.25x are integers (i.e., x is a multiple of 4) and y and 80 – y are divisible by x and 1.25x, respectively.

The total painting time is:
20 houses painted at a rate of houses/week = 5 weeks
60 houses painted at a rate of 5 houses/week = 12 weeks
Total time for 80 houses = 5 + 12 = 17 weeks

The correct answer is B.

Ans B
Plug some numbers
Let y = 40 houses and x = 20
then they painted 20 houses per week
as painters were added to the group , the rate of the group increased to 1.25 x => 1.25 * 20 = 25 houses per week

total time they took = (40/20) + (40/25) = 3.6 weeks

Putting the values of x and y in equation B

(y +320)/(5x) = (40+320)/5*20 = 360/100 = 3.6

Plug x=4 and y=40. Then It took 10 weeks to paint the first 40 houses.
Now the new working rate is 1.5*4=5 houes per week. So it takes 8 weeks to paint the remaining 40 houses.
Total 10+8=18 weeks.

Now plug the values in the answer choices and see only B gives 18.

Let Y=60 houses at X=4 houses per week. So it takes 15 weeks
Now balance 20 house at 1.25X is 5 house per week. So it takes 4 weeks.
On putting Y= 60 and X=4
Total weeks= 15+4= 19 weeks.
Option B satisfies.

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Bunuel

why we are not considering the number of painters who are increased later to calculate the answer?

I believe that's because after the new workers arrived the rate of work increased and the new rate of work was given as 1.25x. And so collectively their rate of work is 1.25x. Therefore we dont have to factor in the number of workers that were added because that would logically imply that the new rate is per individual worker.

I hope this makes sense