A set of 7 numbers: 8, 12, 14, 15, 17, 18, 21 has a mean of 15. When two additional number… — JAMB UTME

A set of $7$ numbers: $8, 12, 14, 15, 17, 18, 21$ has a mean of $15$ . When two additional numbers, $x$ and $y$ (where $x < y$ ), are included in the dataset, the mean of all $9$ numbers becomes $17$ . Given that the mode of the $9$ numbers is $18$ , what is the median of the combined set of $9$ numbers?

A
$15$
$17$ Answer
C
$18$
D
$24$

Answer

The median of the combined set of

9

numbers is

17

The sum of the original 7 numbers is 105. For 9 numbers with a mean of 17, the sum is 153, meaning the two new numbers add up to 48. Since 18 is the mode, it must appear more than once, so one of the new numbers is 18 and the other is 30. Arranging the 9 numbers in order yields 8, 12, 14, 15, 17, 18, 18, 21, 30. The middle (5th) term is 17.

Step-by-Step Solution

Calculate the sum of the original 7 numbers and the total sum required for 9 numbers

Sum of 7 numbers =

8 + 12 + 14 + 15 + 17 + 18 + 21 = 105

. Sum of 9 numbers =

9 \times 17 = 153

The mean formula

\text{Mean} = \frac{\sum x}{n}

gives total sum =

\text{Mean} \times n

Determine the values of the two added numbers

x

and

y

x + y = 153 - 105 = 48

. Since

18

is the mode,

x = 18

and

y = 30

The original set has all distinct numbers. For

18

to be the mode,

18

must repeat, so one of the added values must be

18

Order all 9 numbers and find the median

Ordered set:

8, 12, 14, 15, 17, 18, 18, 21, 30

. Median (

5^{\text{th}}

term) =

17

For an odd number of items

n = 9

, the median is the

\frac{n+1}{2} = 5^{\text{th}}

term in ascending order.

Key Concept

Combining mean, mode, and median properties for ungrouped data

Estimated Time:1m 30s

Question

Answer

Step-by-Step Solution

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