Question:
There are 5 cards in order from smallest to largest, 5 _ _ _ _. The range of the cards is 6, the median of the cards is 7, the mean of the cards is 8, work out the missing numbers
Answer:
5, 6, 7, 11, 11 or 5, 7, 7, 10, 11
Step-by-step explanation:
Given
[tex]Range = 6[/tex]
[tex]Median = 7[/tex]
[tex]Mean = 8[/tex]
Required
Complete the blanks
Range is calculated as:
[tex]Range = Highest - Lowest[/tex]
From the given numbers, we have:
[tex]Range = 6[/tex]
[tex]Lowest = 5[/tex]
So, the equation becomes
[tex]6 = Highest - 5[/tex]
[tex]Highest = 5 + 6 = 11[/tex]
So, we have:
5 _ _ _ 11
Next,
[tex]Median = 7[/tex] means that the middle number is 7;
So, we have:
5 _ 7 _ 11
Next,
[tex]Mean = 8[/tex]
Let the two remaining blanks be x and y: 5 x 7 y 11
Where
[tex]5 \le x \le 7[/tex] and [tex]7 \le y \le 11[/tex]
So, we have:
[tex]\frac{5 + x + 7 + y + 11}{5} = 8[/tex]
Collect like terms
[tex]\frac{5 + 7+11+ x + y }{5} = 8[/tex]
[tex]\frac{23+ x + y }{5} = 8[/tex]
Cross Multiply
[tex]23 + x + y = 8 * 5[/tex]
[tex]23 + x + y = 40[/tex]
[tex]x + y = 40 - 23[/tex]
[tex]x + y = 17[/tex]
Recall that: [tex]5 \le x \le 7[/tex] and [tex]7 \le y \le 11[/tex]
Using trial by error, we have:
[tex]x + y = 17[/tex]
[tex]6 + 11 = 17[/tex]
[tex]x = 6\ and \ y = 11[/tex]
[tex]7 + 10 = 17[/tex]
[tex]x = 7\ and \ y = 10\\[/tex]
Hence, the complete cards could be: 5, 6, 7, 11, 11 or 5, 7, 7, 10, 11
