GO Classes Test Series 2024 | Mock GATE | Test 12 | Question: 63

Answer 1

Let c represents cat, d represents dog, t represents turtle.

The largest possible number of house to have all three c,d,t would be to maximize the overlap which happens when houses having turtle will be a subset, i.e all three animals are present in those houses which have tuetles.

Therefore, x = 1182.

To calculate y, the minimal number of houses to have all three animals would be to minimize the overlap.

We can find the minimal number of house by assuming there are only two types of houses; one which has 2 types of animals (any one of ct / cd / dt) and another type of house in which all animals are present and then form linear equation and solve it;

ct + dt + cd + all = 2017 ; (cd / dt / ct represents the house which have only two types of animals in their house and all represents the number of house which has all three types of animals)  -eq 0

ct + dt + all = 1182 ( Number of houses with turtles)        -eq 1

cd + ct + all = 1651 (# houses with cats)                          -eq 2

cd + dt + all = 1820 (#houses with dogs)                         -eq 3

Solving the above the linear equations;

subtracting eq 1 from eq 0;

cd = 835

subtracting eq 3 from eq 0;

ct = 197

And from eq 2;

835 + 197 + all = 1651

all = 619 , which is equal to y

Hence, x-y = 1182 – 619 = 563

**************************************************

Also refer to article shared by Srken

https://math.stackexchange.com/questions/2389581/venn-diagram-question-involving-maximum-and-minimum-values

Comments

You didn’t take into consideration the Houses with only Dogs for eq-0

---

There will be no house with only dogs or even with only cats/ turtle, we have taken that assumption to find the minimal overlap

Answer 2

Here, X is 1182 and Y is 619. hence the final answer is X – Y = 563.

It is clear that 1182 is the max number of houses that can have dog, cat and turtle because this is the maximum overlap possible between the given 3 types of houses.

now, ill try to explain why 619 is the min number of houses that can have dogs and cats and turtles.

Given: total houses = 2017 ,

house with dogs(d) = 1820; house with no dogs(nd) = 2017-1820 = 197

house with cats( c) = 1651; house with no cats(nc) = 2017- 1651 = 366

house with turtles(t) = 1182; house with no turtles (nt) = 2017- 1182 = 835.

lets split house with dogs ( d ) into house with dogs & turtles (d & t) and house with dogs & no turtles (d & nt)

house with dogs & no turtles( d & nt) means those houses wont contain turtles, so ill take it to be 835 (nt).

now, lets split house with dogs & turtles (d & t) into house with dogs & turtles & cats (d & t & c) and house with dogs & turtles & no cats (d & t & nc)

house with dogs & turtles & no cats ( d & t & nc) means those houses wont contain cats, so ill take it 366 (nc).

intuitively what we are doing here is removing maximum chunks from what we have to get a final value as low as possible.

i have started to split from houses with dogs. you can also start splitting from house with cats or house with turtles. you will end up getting the same 619 as the answer.