Arithmetic

Mean, Median & Mode

Medium

The possible values of $x$ in the set $\{1, 5, 13\}$ for which the mean of eight observations $5, 8, 3x + 2, 15, 27, 29, 36, 5x - 2$ equals their median are

only 1

5 and 13

only 13

only 5

✅ Correct Option: 2

Eight observations: $5, 8, 3x+2, 15, 27, 29, 36, 5x-2$ .

Sum $= 5 + 8 + (3x+2) + 15 + 27 + 29 + 36 + (5x-2) = 8x + 120$

Mean $= \dfrac{8x + 120}{8} = x + 15$

Median of $8$ sorted values is the average of the $4$ th and $5$ th values. Check each candidate.

Try $x = 1$ : values become $5, 8, 5, 15, 27, 29, 36, 3$ . Sorted: $3, 5, 5, 8, 15, 27, 29, 36$ .

Median $= \dfrac{8 + 15}{2} = 11.5$ , but mean $= 16$ . Not equal.

Try $x = 5$ : values become $5, 8, 17, 15, 27, 29, 36, 23$ . Sorted: $5, 8, 15, 17, 23, 27, 29, 36$ .

Median $= \dfrac{17 + 23}{2} = 20$ , and mean $= 20$ . Equal.

Try $x = 13$ : values become $5, 8, 41, 15, 27, 29, 36, 63$ . Sorted: $5, 8, 15, 27, 29, 36, 41, 63$ .

Median $= \dfrac{27 + 29}{2} = 28$ , and mean $= 28$ . Equal.

Both $x = 5$ and $x = 13$ work.

Answer $= 5$ and $13$

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IPMAT Indore Past Year Questions (Topic-Wise)

The possible values of $x$ in the set $\{1, 5, 13\}$ for which the mean of eight observations $5, 8, 3x + 2, 15, 27, 29, 36, 5x - 2$ equals their median are