Mean, Median, Mode,…, Sum?

Question

Five test scores have a mean of $91$, a median of $93$, and a mode of $95$. The possible scores on the tests are from $0$ to $100$. a) What is the sum of the lowest two test scores? b) What are the possible values of the lowest two test scores?

• so i came up with the numbers $82,90,93,95,95$
• I know that $95$ will appear more than once because its the mode
• however, I do not know how to prove my answer, as I got these answers through the use of an average calculator
• what would be a way to prove my answer, what equation should I use? -thanks

Answer 1

We know the median is $93$ and there are $5$ numbers. So we can arrange the numbers in increasing order $a,b,93,c,d$ because $93$ is the median.

Now we know the mode is $95$, which means it should occur more than once. If it only appears one time, $93$ would have also been the mode. So we have $a,b,93,95,95$.

Now the mean is $\frac{a+b+93+95+95}{5} = 91$, so $a+b=172$. That is part a). Now $b=172-a$ and $b ≥a$ and $b<93$. $b$ cannot be $93$ because then $93$ would also be the mode. So $b$ can take values from $92$ to $86$ and $a$ will range from $80$ to $86$ correspondingly.

$$80,92,93,95,95$$

$$81,91,93,95,95$$

$$\cdots$$

$$\cdots$$

$$86,86,93,95,95$$