Suppose that Dog-o-Rama provides day care for 30 dogs and regular groo

48 and 17

This one got my head spinning as none of the combinations were working. Then i thought to try this with inequality. I assume this solution is correct, but not sure.

Given:
day_care = 30 dogs
regular_grooming = 24 dogs

either_day_care_or_grooming = day_care + regular_grooming - both_day_care_and_grooming

either_day_care_or_grooming = 54 - both_day_care_and_grooming

at this point i assume everyone would have come. but no matter what value you pick for 2 unknown variables, equality will not be measured. let's try inequality.

with some logical thinking, we can assume, both_day_care_and_grooming < min(30,24) or can we write this as,

both_day_care_and_grooming < 24
here options in table may help us. we can narrow down value for this variable to 17.
=> both_day_care_and_grooming = 17

Now, coming back to our equation
either_day_care_or_grooming = 54 - 17 = 37

37 not there in table, i can feel you. Let's apply some logic again
either_day_care_or_grooming > max(37,24)
=> either_day_care_or_grooming must be greater than 37, AND either_day_care_or_grooming must be less than or equal to 54(calculated above)
only one option in table, 48.

This was tough for me. Please correct the solution if i am wrong.

the maximum 24+30=54 can receive either services. The maximum value we can choose from the option is 48.The maximum number of dogs that can receive both the services is 24. 17 is the maximum available from the option. So (d,a) is the answer

Let a be the number of dogs who received only day care.
Let b be the number of dogs who received only regular grooming.
Let c be the number of dogs who received both day care and regular grooming.
Here a, b and c will be non negative integers.

Dog-o-Rama provides day care for 30 dogs and regular grooming for 24 dogs.
This can be translated as a + b = 30 let's call this equation 1, and b + c = 24, let's call this equation 2.

Column 1: Select the row that displays the largest number of dogs at Dog-o-Rama that are receiving either day care or regular grooming.

We need to maximize the value of a + c in order to answer column 1.

Adding equation 1 and 2 we get a + 2b + c = 54.
Therefore, a + c = 54 - 2b. Let's call this equation 3.

To maximize equation 3 we need to reduce the value of 2b. i.e. reduce the value of b.
The minimum value b can take is 0. Therefore, the maximum value a + c can take is 54. Hence we can eliminate two options 56 and 60.

We also, see that 54 is not one of the options, hence we also need some other constraint to get the answer.
If we close look at equation 3, we see that 54 is an even number and 2b is also an even number. Recalling Even / Odd concepts Even - Even = Even.
Therefore, out of the 4 remaining options we need to eliminate odd numbers. Post POE we see that the only option remains which is 48.

Hence, the largest number of dogs at Dog-o-Rama that are receiving either day care or regular grooming from the given option choices is 48.

Therefore, there are 48 dogs at Dog-o-Rama that are receiving either day care or regular grooming and 3 dogs at Dog-o-Rama that are receiving both day care and regular grooming.

Column 2: Select the row that displays the largest number of dogs at Dog-o-Rama that are receiving both day care and regular grooming.

We need to maximize the value of b.

From equation 3 we get b = (54 - a - c)/2.
This can be simplified as b = 27 - (a+c)/2.

To maximize the value of b we need to reduce the value of a + c.
The minimum value a + c can take is 0. Hence the maximum value b can take is 27. Which is one of the options.
However, looking closely at equation 2, we see that b + c = 24 and both b and c are non negative. Hence the maximum value b can take is 24 which is not an option. However, there is only one option which is lesser than 24 i.e. row 1 (17).

Hence the largest number of dogs at Dog-o-Rama that are receiving both day care and regular grooming from the given option choices is 17.

Therefore, there are 17 dogs at Dog-o-Rama that are receiving both day care and regular grooming and 20 dogs at Dog-o-Rama that are receiving either day care or regular grooming.