Question

Out of 500 employees in an office, 48% preferred Coffee, 54% liked Tea, 320 employees used to Smoke. Out of the total 28% drink both Coffee and Tea, 32% drink Tea and also Smokes and 30% drink Coffee and Smokes. Only 30 did none of these. How many employees like all three?

Answer 1

Solution:

Total Employees , n(T) = 500

Preference is Coffee , n(C) = 48% which means $\frac{48}{100} * 500$ = 240 employees

Preference is Tea , n(T) = 54% which means $\frac{54}{100} * 500$ = 270 employees

Preference to Smoke , n(S) = 320 employees

Preference to both Coffee and Tea , n(C∩T) = 28% which means $\frac{28}{100} * 500$ = 140 employees

Preference to both Tea and Smoke , n(T∩S) = 32% which means $\frac{32}{100} * 500$ = 160 employees

Preference to both Coffee and Smoke , n(C∩S) = 30% which means $\frac{30}{100} * 500$ = 150 employees

No preference = 30 employees

Employees those who have at least one preference or more than that ,

n(CᴜTᴜS) = 500 - 30 = 470

Employees who like all three , n(C∩T∩S) = ?

n(CᴜTᴜS) = n(C) + n(T) + n(S) – n(C∩T) – n(T∩S) – n(C∩S) + n(C∩T∩S)

470 = 240 + 270 + 320  - 140 - 160 - 150 + n(C∩T∩S)

470 = 830 - 450 + n(C∩T∩S)

470 = 380 + n(C∩T∩S)

n(C∩T∩S) = 470 - 380

n(C∩T∩S) = 90

Employees who like all three (Tea, Coffee, Smoke) are 90 employees.

Answer 2

The number of employees that like all three are 90

Given:

Total number of employees = n(T) = 500

Number of employees that prefer coffee only = n(C) = 48% = 48/100 × 500 = 240

Number of employees that prefer tea only = n(T) = 54% = 54/100 × 500 = 270

Number of employees that prefer smoke only = n(S) = 320

Number of employees that prefer both coffee and tea = n(C ∩ T) = 28% = 28/100 × 500 = 140

Number of employees that prefer both tea and smoke = n(T ∩ S) = 32% = 32/100 × 500 = 160

Number of employees that prefer both coffee and smoke = n(C ∩ S) = 30% = 30/100 × 500 = 150

Number of employees that prefer none = 30

Step-by-step explanation:

The number of employees that prefer all the is intersection of all which is given by the formula below:

n(C ∩ T ∩ S) = n(C) + n(T) + n(S) - n(C ∩ T) - n(T ∩ S) - n(C ∩ S) - n(C ∪T∪S)

Now,

n(C ∪ T ∪ S) = Total number of employees - Number of employees that like none

∴ n(C ∪ T ∪ S) = 500 - 30 = 470

On substituting the values, we get,

n(C ∩ T ∩ S) = 240 + 270 + 320  - 140 - 160 - 150 - 470

∴ n(C ∩ T ∩ S) = 90