Find the average test score for each of three groups, given overall averages

Question

I've been working on this question for the last hour and cannot determine a solution.

Q: 100 children, 200 teens and 300 adults write a test and an average is computed for all three groups. The average of these three averages is 85%. The overall average of the 600 people is 86%. Also, the average for the 300 children and teens is 4 marks higher than the average for adults. Determine the average for all three groups.

From this question I have derived the following relationships. Note $A_{c}$ is average for children, $A_{t}$ is average for teens, $A_{a}$ is average for adults and $S_{c}$ is sum of children grades, $S_{t}$ is sum of teen grades, and $S_{a}$ is sum of adult grades $$A_{c}=\dfrac{1}{100}\times S_{c} \;\;\;\;\;\;A_{t}=\dfrac{1}{200}\times S_{t}\;\;\;\;\;\; A_{a}=\dfrac{1}{300}\times S_{a}$$ $$\dfrac{1}{600}(S_{a}+S_{c}+S_{t})=0.86$$

My first attempt: I could use the equation $\dfrac{1}{3}(A_{c}+A_{t}+A_{a})$ where $A_{c}=A_{t}=A_{a}+0.04$. However, this assumes that the average of children and teens is the same which I don't think I'm justified in making. My problem is using the information in the second last sentence properly. From that sentence, I derived the following relationship but it has not helped determine the solution $$\left(\dfrac{A_{c}+A_{t}}{2}\right)=A_{a}+4$$ Any ideas and suggestions would be appreciated.

Answer 1

Let $A_c, A_t, A_a$ be the average for children, teens and adults respectively. The condition that "The average of these three averages is 85%" is given by the equation $$ \dfrac{A_c+A_t+A_a}{3}=0.85.$$ The condition "The overall average of the 600 people is 86%" can be written as $$ \dfrac{100A_c+200A_t+300A_a}{600}=0.86.$$ The last one is given by $$ \dfrac{100A_c+200A_t}{300}-A_a=0.04$$ Now you can solve the system given by the three equations above.

Answer 2

$A_c = A_t = A_a + 0.04$ is not correct. That would only apply if the children and teens have the same average. Also, $\left(\frac{A_{c}+A_{t}}{2}\right)=A_{a}+4$ is not correct either. That is the average of the average teen and average child score, which is not the same as the average of 200 teens and 100 children combined.

The correct way to use the last sentence would be $\frac{S_c + S_t}{300} = A_a + 0.04$. Since $A_a$ can be directly converted to $S_a$, you can solve using your first equation. Be careful because according to the original problem, all averages should be between $0$ and $1$.

Answer 3

the set of equations to solve $$ \dfrac{1}{3}\left(\dfrac{S_c}{100}+\dfrac{S_t}{200} + \dfrac{S_a}{300}\right) = 0.85,\\ \dfrac{S_c +S_t + S_a}{600} = 0.86,\\ \dfrac{1}{2}\left(\dfrac{S_c}{100}+\dfrac{S_t}{200}\right) = \dfrac{S_a}{300} + 4 $$