In a company of 100 employees, 75 have at Laptop, 80 have at least one

In total, there are 100 employees
75 laptop —> 25 no laptop
80 cell phone —> 20 no phone
55 Alexa —> 45 no Alexa.

All three devices:
—> the greatest possible number —55 = x

The lowest possible number — 55–45= 10 =y

—> x —y = 55–10 = 45

The answer is C

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If we find the maximum number of employees who has all the three- laptop, Mobile phone and Alexa- the count is 75

If we find the minimum number of employees who has all the three- laptop, Mobile phone and Alexa- the count is 10

Answer is A

we have
100= 75+80+55-2*( all three)
x= 55 i.e max
and min over lap is 10 each
∆ ; 55-10 ; 45
IMO C


In a company of 100 employees, 75 have at Laptop, 80 have at least one cell phone, and 55 have at least one Alexa. If x and y are respectively the greatest and lowest possible number of employees that have all three of these devices, what is the value of x – y ?
A. 65
B. 55
C. 45
D. 35
E. 25

In order to find maximum number of employees who have all three devices, We need to find out maximum common number which comes to 55 which would be value of x
Now to find out minimum we are assuming out of 80 employee having laptops, 25 are different employees having at least one cell phone, so here 55 employees are common and out of 55 employees having at least one Alexa 25 employees are different having at least one Alexa, so here 30 employees are common, so minimum common number comes to 30 which is y
So x-y comes to 55-30 which is 25, hence Option E

Ans:C
Maximum value of intersection of all the three sets( Laptop, cell phone, Alexa)= Minimum value of(any three set)=55
Minimum Value of three set = A+B+C-2*Total Value
Hence, Minimum Value of three set = 210-2*100=10
hence x-y= 55-10=45

The answer is C: 45.

EXPLANATION:
Let's assume:
A = number of people having one device.
B = number of people having two devices.
C = number of people having three devices.
D = number of people having zero devices.

So, total number of people A + B + C + D = 100.
And number of devices: A + 2B + 3C + D = 210.

For this question, we are assuming that D, or number of people having no devices, is zero (although it is not given in the question, I would show that we are not getting the answer if we assume there are people having no devices).

So, A + B + C = 100
A + 2B + 3C = 210.

The above equations must be equal, to satisfy the condition of same people having the same devices.

To maximize people having 3 devices, we minimize people having 2 and 1 device.
Hence, we assume A + B = 0, and likewise, A + 2B = 0.

So, 100 - C = 210 - 3C.
This means 2C = 110.
This means, C = 55.

This is the maximum value, or X.

Now, to get minimum value, or Y, let's assume no people have 3 devices. So C = 0.

This means A + B = 100, and A + 2B = 210.
Solving both, we get A = -10. Which is not possible (number of people can't be negative integer).

Now, let's assume number of people having three devices as 10 (as in the answer choice, difference is of 10).
So C = 10. This means 3C = 30.

So, A + B = 90, and A + 2B = 180.
Solving, we get A = 0, and B = 90. This is a valid solution, as it shows there are 90 people having 2 devices each (so 180 devices), and zero people having one device each, and 10 people having three devices each (30 devices). So 100 people having 210 devices is satisfied.

So, Y = 10.
Hence, X - Y = 55 - 10 = 45, the answer.

Why there are no people with zero devices?

If we assume D = 20 (as it is given 80 people have atleast cell phone, so the maximum number of people who would have zero devices is 20, as the same 80 people can possess all three devices):

So equation would be:
A + B + C = 80.
A + 2B + 3C = 210.

Going by the same thought process we applied earlier, first we maximize C, to get X. (A + B, and A + 2B assumed to be zero).
So we get 80 - C = 210 - 3C.
C = 65 = X.

Now, to find minimal value, we again try to get the valid solution, on C = 10, 20, 30, and so forth.

We would find that, only on C = 50, we would get the valid equation:
A + B = 80 - 50
A + 2B = 210 - 150

We would get A = 0, B = 30, and C = 50. Means, zero people with one device, 30 people with 2 devices, and minimum 50 people with 3 devices, and 20 people with 0 devices.

So, again, Y = 50.

But, X - Y = 65 - 50 = 15, which is not given in the answer choice.

Hence, we assume, zero people have no devices.

Clearly, the maximum no. of employees which can have all 3 devices is 55, which is limited by the number of Alexas.
Therefore, x = 55

For minimum number of employees,
We will find the sum of all the employees which might not have any of the product.

100 - 75 = 25 do not have laptop
100 - 80 = 20 do not have cell phone
100 - 55 = 45 do not have Alexa

Therefore, 25 + 20 + 45 = 90 employees might not have all of the three products.

Therefore, y = 100 - 90 = 10
Now, x - y = 55 - 10 = 45

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the maximum is limited by the smallest value which is 55. all values are above this number. so y= 55
if we see carefully, the difference among the items,

employee with No Laptop =100-75 = 25
employee with No Cell phone =100-80 = 20
employee with No Alexa 100-55 = 45

20+25+45 = 90 people dont have all three items.
That mean 10 may have three items as none is not specified and considered as zero.
so min x =10
the difference is 55-10= 45
answer is C

Let the employees be numbered from 1 to 100.

For greatest number of employees that have all three devices, let the distribution be as follows:
Laptops to be given to employees numbered from 1 to 75
Cell phones to be given to employees numbered from 1 to 80
Alexa to be given to employees numbered from 1 to 55

So, employees numbered 1 to 55 have all three devices.
⇒ x=55

For least number of employees that have all three devices, let the distribution be as follows:

Alexa to be given to employees numbered from 1 to 55
Laptops to be given to employees numbered from 56 to 100
Now, we are left with 30 laptops which we have to distribute among employees numbered from 1 to 55 (who already have Alexa)
Let employees numbered from 1 to 30 be given the remaining 30 laptops.
So, employees numbered from 1 to 30 have two devices each and from 31 to 100 have one device each.

Let cell phones be given to employees numbered from 31 to 100.
Now, we are left with 10 cell phones which we have to distribute among employees numbered 1 to 30 (who already have Alexa and laptop)
Let employees numbered from 1 to 10 be given cell phones.

So, employees numbered 1 to 10 have all three devices.
⇒ y=10

Therefore, x - y = 55 - 10 = 45

In a company of 100 employees, 75 have at Laptop, 80 have at least one cell phone, and 55 have at least one Alexa. If x and y are respectively the greatest and lowest possible number of employees that have all three of these devices, what is the value of x – y ?
A. 65
B. 55
C. 45 --> correct
D. 35
E. 25

SOLUTION:
let's say each bar/dash(-) =5
----------------:80
----------------:75
----------------: 55
so maximum overlap = x = 55 (left 11 blue overlap bars/dashes)

--------------------: 80
--------------------: 55
--------------------: 75
so minimum overlap = y = 10 (middle 2 blue overlap bars/dashes)

So x-y=55-10=45

In a company of 100 employees, 75 have at Laptop, 80 have at least one cell phone, and 55 have at least one Alexa. If x and y are respectively the greatest and lowest possible number of employees that have all three of these devices, what is the value of x – y ?
A. 65
B. 55
C. 45 --> correct
D. 35
E. 25

SOLUTION:
let's say each bar/dash(-) =5
----------------:80
----------------:75
----------------: 55
so maximum overlap = x = 55 (left 11 blue overlap bars/dashes)

--------------------: 80
--------------------: 55
--------------------: 75
so minimum overlap = y = 10 (middle 2 blue overlap bars/dashes)

So x-y=55-10=45

Max for Min(75,55,80)=55 so tot 100 one 0
Min for 75+80+55=210-100=110 but none (110) cannot be more than total (100) so none=110-100=10 thus None max is 100 therefore common Min=10
Max-min common= 55-10=45

One of those 700 level question that I did mentally!

Greatest possible number
The minimum of the three numbers, that is 55.
Assume that these 55 have Laptop and cell phone too.

Least possible number
Number line: Fill up till 100 and then move to the first person again. The last or third overlap will give you the minimum answer.
……..75(L)……………....|…25(A).. =100
…..30(A)..….|……………..70(C)…. =100
…10(C).
Thus, the last 10(C) have A and L too.


Answer = 55-10 = 45