How do I average two numbers that are already averaged?

Question

$50$ juniors and seniors are tested. $35$ of them average $80%$. $15$ of them average $70%$. What is the average of the class of $50$?

We tried $(70+80)/2$ but that was $75$ and the real answer is $78$ but we don't know how to get there.

Obviously more kids had $80%$ than $70%$ so it makes sense that the number is higher than $75$ but I have gone back over our homeschooling materials and can't even figure out what to look under to find out how to do this.

Can someone show us the calculation and also tell us what to look up so we can brush up on this before college testing? Is it averaging averages??

Answer 1

Hint: How many total points were scored by the $35$? How many by the $15$? Add up the total points and divide by the number of students. You want to weight the average by the number of students in each group.

Answer 2

Use:

The students which average $80\%$ have an influence of $\dfrac{35}{35+15}$ on the average result and the ones who averaged $70\%$ have an influece of $\dfrac{15}{35+15}$

And: $$\text{total average} = \text{average 1}\cdot\text{influence 1}+\text{average 2}\cdot\text{influence 2}$$

Answer 3

If $35$ of them have an average of $80$, then their sum is $35\times 80=2800$.

Similarly the sum for the $15$ who have an average of $70$ is $15\times70=1050$.

The average for all of them therefore must be $$ \frac{\text{sum}}{\text{number of indiviuals}} = \frac{2800 + 1050}{35+15} = \cdots\cdots. $$

$(70+80)/2$ is wrong because it treats the contributions of the smaller group and the larger group equally despite the difference in their sizes.

Answer 4

The total score = overall_average x n

= average_of_some_students x number_of_students_making_that_average + average_of_rest_of_students x number_of_rest_of_students.

So

total score = overall_average x 50 = 80 x 35 + 70 x 15

so

$$\text{overall average} = \dfrac{80 \times 35 + 70 \times 15}{50}$$

In general:

$$\text{average} = \frac{\text{total}}{n} = \frac{\text{group Total}_1 + \text{group Total}_2 + \cdots + \text{group Total}_k}{n}=\frac{\text{avg}_1\cdot n_1 + \text{avg}_2\cdot n_2 + \cdots + \text{avg}_k\cdot n_k}n$$

Answer 5

Use the equation $$A_{overall}=\frac{A_1\times N_1 + A_2\times N_2}{N_1 + N_2}$$

Where:

$A_1$ Is the average of group one;

$A_2$ is the average of group two;

$N_1$ is the number of students in group one;

$N_2$ is the number of students in group two