There are five numbers. Some information of the numbers are as shown. Range = 6 mean = 7 median = 8 mode = 9 Find the numbers. Ans: ___, ___, ___, ___, ___ The average mass of 5 boys is 68 kg and that of 8 girls is 55 kg. To find the average mass of all the children, Caleb takes the average of 68 kg and 55 kg. Explain why Caleb is wrong and find the correct answer. Answer

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1. Use the median and mode to fix the middle and repeated largest values.

Arrange the five numbers in ascending order.

Since the median is $8$, the middle number is $8$:

$\_,\ \_,\ 8,\ \_,\ \_$

Since the mode is $9$, $9$ must occur more often than any other number. As $9>8$, the two $9$s must be after $8$:

$\_,\ \_,\ 8,\ 9,\ 9$

2. Use the range and mean to find the two missing numbers.

The largest number is $9$. Since the range is $6$,

$\text{smallest number}=9-6=3$

So the numbers are

$3,\ \_,\ 8,\ 9,\ 9$

The mean is $7$, so the total is

$5\times 7=35$

Let the missing number be $x$.

$3+x+8+9+9=35$

$x+29=35$

$x=6$

Hence the five numbers are

$3,\ 6,\ 8,\ 9,\ 9$

1. Explain why taking the simple average of the two given averages is incorrect.

Caleb is wrong because there are $5$ boys but $8$ girls. The two averages, $68$ kg and $55$ kg, are based on different numbers of children, so they must be weighted by the group sizes.

2. Find the total mass of all children and divide by the total number of children.

Total mass of boys $=5\times68=340$ kg.

Total mass of girls $=8\times55=440$ kg.

Combined average mass
$=\frac{340+440}{5+8}$
$=\frac{780}{13}=60\text{ kg}.$