Blood tests of 100 people showed that 65 had the \( A \) antigen and 36 had the \( B \) antigen. Another 25 had neither antigen and so are of type \( O \). How many people are of type \( A B \), having both the \( A \) and \( B \) antigens? Draw and label a Venn diagram that shows the number of people with blood types A, B, AB, and O. How many people are of type AB, having both the A and B antigens? H

To find the number of people with blood type AB, we can use the principle of inclusion-exclusion. We know: - Total people = 100 - People with A antigen = 65 - People with B antigen = 36 - People with neither antigen (type O) = 25 From the total number of people, if 25 are of type O, then 100 - 25 = 75 people have either A or B or both antigens. Let \( x \) be the number of people with both A and B (type AB). We can express the counts as follows: - Total with A only = A - AB = 65 - x - Total with B only = B - AB = 36 - x Now we set up the equation for the count of people with A or B: \[ (65 - x) + (36 - x) + x = 75 \] Simplifying, we get: \[ 65 + 36 - x = 75 \] \[ 101 - x = 75 \] \[ x = 101 - 75 = 26 \] So, there are 26 people of type AB. Now, to summarize using a Venn diagram: - Type A = 65 - 26 = 39 - Type B = 36 - 26 = 10 - Type AB = 26 - Type O = 25 This gives us a clear count as follows: - Type A: 39 - Type B: 10 - Type AB: 26 - Type O: 25 So, the population distribution looks like this: - **39 people** are of **blood type A** only. - **10 people** are of **blood type B** only. - **26 people** are of **blood type AB** (both antigens). - **25 people** are of **blood type O** (neither antigen). Feel free to draw the Venn diagram, representing portions for type A, type B, and overlapping for type AB, while the external portion represents type O. Enjoy playing with the numbers!