SOLUTION: There are 150 people at an International Medical Conference. 40 are Africans, 70 are women and 110 are doctors. 12 of the women are Africans, 46 of the doctors are women and 31

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Question 1177476: There are 150 people at an International
Medical Conference. 40 are Africans, 70
are women and 110 are doctors. 12 of the
women are Africans, 46 of the doctors are
women and 31 of the Africans are doctors.
If 5 of the African men are not doctors:
a how many of the African women are
doctors
b how many of the men are neither African
nor doctors?
Found 2 solutions by CPhill, ikleyn:
Answer by CPhill(1959)   (Show Source): You can put this solution on YOUR website!
**a) How many of the African women are doctors?**
1. **Start with the African doctors:** We know 31 Africans are doctors.
2. **Subtract the African men who are not doctors:** 31 - 5 = 26 African doctors are men.
3. **Find the number of African women who are doctors:** Since 12 women are African, and 31 Africans are doctors, then 31 - 26 = **5 African women are doctors**.
**b) How many of the men are neither African nor doctors?**
1. **Find the total number of men:** There are 150 people, and 70 are women, so there are 150 - 70 = 80 men.
2. **Find the number of African men:** There are 40 Africans, and 12 are women, so there are 40 - 12 = 28 African men.
3. **Find the number of male doctors:** There are 110 doctors, and 46 are women, so there are 110 - 46 = 64 male doctors.
4. **Find the number of men who are either African or doctors:** We can't simply add the number of African men and male doctors, because some men are both. We use the principle of inclusion-exclusion:
Number of men who are African or doctors = Number of African men + Number of male doctors - Number of African male doctors
We know 5 African men are not doctors, so 28 - 5 = 23 African men are doctors. Hence, the number of men who are African or doctors is 28 + 64 - 23 = 69.
5. **Find the number of men who are neither African nor doctors:** There are 80 men total, and 69 are either African or doctors, so 80 - 69 = **11 men are neither African nor doctors**.
Answer by ikleyn(52781)   (Show Source): You can put this solution on YOUR website!
.
There are 150 people at an International Medical Conference.
40 are Africans, 70 are women and 110 are doctors.
12 of the women are Africans, 46 of the doctors are women and 31 of the Africans are doctors.
If 5 of the African men are not doctors:
(a) how many of the African women are doctors ?
(b) how many of the men are neither African nor doctors?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~


        In the post by  @CPhill,  his solution to part  (a)  is  OUT  of  LOGIC,  so it is  INCORRECT.
        Below is my solution for part  (a).


(a)   How many of the  African women are doctors ? 

              The solution is in 3 logical steps.


     (1)  We are given that there are 40 Africans, and 12 of women are Africans. 
          Hence, the number of African men is 40-12 = 28.


     (2)  Next, we are given that 5 of African men are not doctors.
          Hence, 28-5 = 23 is the number of African men who are doctors.


     (3)  Next, we are given that 31 of Africans are doctors.
          Hence, the number of African women who are doctors is 31-23 = 8.


ANSWER.  The number of African women who are doctors is 8.

Part  (a)  is solved.



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