Question

a group of 75 math students were asked whether they like algebra and whether they like geometry. a total of 45 students like algebra, 53 like geometry, and 6 do not like either subject. what are the correct values of a, b, c, d, and e? algebra vs. geometry likes geometry does not like geometry total likes algebra a b 45 does not like algebra c 6 d total 53 e 75 a = 16, b = 29, c = 22, d = 30, e = 24 a = 29, b = 16, c = 30, d = 22, e = 24 a = 16, b = 29, c = 24, d = 22, e = 30 a = 29, b = 16, c = 24, d = 30, e = 22

Explanation:

Step1: Find the number of students who like at least one subject

The total number of students is 75 and 6 do not like either subject. So the number of students who like at least one subject is $75 - 6=69$.

Step2: Use the principle of inclusion - exclusion

We know that $n(A\cup B)=n(A)+n(B)-n(A\cap B)$, where $n(A)$ is the number of students who like algebra, $n(B)$ is the number of students who like geometry and $n(A\cup B)$ is the number of students who like at least one subject. Given $n(A) = 45$, $n(B)=53$ and $n(A\cup B)=69$. Then $n(A\cap B)=n(A)+n(B)-n(A\cup B)=45 + 53-69=29$. So $a = 29$.

Step3: Find the value of $b$

Since $n(A)=45$ and $a = 29$, then $b=n(A)-a=45 - 29 = 16$.

Step4: Find the value of $c$

Since $n(B)=53$ and $a = 29$, then $c=n(B)-a=53 - 29=24$.

Step5: Find the value of $d$

The number of students who do not like algebra is $d=75 - 45=30$.

Step6: Find the value of $e$

The number of students who do not like geometry is $e=75 - 53=22$.

Answer:

a = 29, b = 16, c = 24, d = 30, e = 22, so the answer is a = 29, b = 16, c = 24, d = 30, e = 22 (the fourth option).