Problem 10

Number 10

Given an ordered data set $4, 4, a, b, \text{ and } c$ . The average of these five numbers is $8$ . The largest data value is known to be $3$ times the median value. Which of the following relationships is correct between quantities $P$ and $Q$ based on the information provided?

$P$	$Q$
Median	$4$

$P > Q$

$P < Q$

$P = Q$

$PQ = 32$

Cannot be determined

Explanation

The given ordered data set is $4, 4, a, b, c$ . Since the data is already ordered and there are an odd number of values ( $5$ values), the median is the $3$ rd value, which is $a$ .

The largest data value is $c$ , and it is $3$ times the median, so:

c = 3a

The average of the five numbers is $8$ :

\frac{4 + 4 + a + b + c}{5} = 8

8 + a + b + c = 40

a + b + c = 32

Substitute $c = 3a$ into the equation:

a + b + 3a = 32

b + 4a = 32

b = 32 - 4a

Since the data is ordered, $4 \le a \le b \le c$ holds true. We will find the bounds for the value of $a$ .

Upper Bound of a

From the inequality $a \le b$ :

a \le 32 - 4a

5a \le 32

a \le 6.4

Lower Bound of a

From the inequality $b \le c$ :

32 - 4a \le 3a

32 \le 7a

a \ge \frac{32}{7}

a \ge 4.57

Note also that $a \ge 4$ (from the data order). Since $4.57 > 4$ , the stronger condition is $a \ge 4.57$ .

Conclusion

The range of values for $a$ is $4.57 \le a \le 6.4$ .

The value of $P$ is the median ( $a$ ) and the value of $Q$ is $4$ . Since the minimum value of $a$ is $4.57$ , which is greater than $4$ , it is certain that $P > Q$ .

Therefore, the correct relationship is $P > Q$ .

Command Palette

Number 10

Explanation

Upper Bound of a

Lower Bound of a

Conclusion