The mean of four positive integers is 9. If the mode of the set of numbers is 12 and their… — JAMB UTME

The mean of four positive integers is $9$ . If the mode of the set of numbers is $12$ and their median is $10$ , what is the value of the smallest integer?

$4$ Answer
B
$2$
C
$6$
D
$5$

Answer

The smallest integer is 4.

Let the four positive integers in ascending order be

a \le b \le c \le d

. The sum of the four numbers is

4 \times 9 = 36

. For

12

to be the mode, it must appear at least twice, so

c = 12

and

d = 12

. The median of four numbers is

\frac{b + c}{2} = 10

, which gives

\frac{b + 12}{2} = 10

, so

b = 8

. Substituting these into the sum equation

a + 8 + 12 + 12 = 36

gives

a = 4

Step-by-Step Solution

Calculate the sum of all four integers from the given mean.

Sum

= 4 \times 9 = 36

The mean of

n

numbers is the sum divided by

n

Determine the two largest numbers using the mode.

Let the ordered numbers be

a \le b \le c \le d

. Since the mode is

12

and the set contains

4

numbers with a single mode,

c = 12

and

d = 12

For

12

to be the mode in a 4-element set without tie, it must appear at least twice.

Use the median to find the second integer

b

Median

= \frac{b + c}{2} = \frac{b + 12}{2} = 10 \implies b + 12 = 20 \implies b = 8

The median of an even number of ordered elements is the arithmetic mean of the two middle elements.

Solve for the smallest integer

a

a + b + c + d = 36 \implies a + 8 + 12 + 12 = 36 \implies a + 32 = 36 \implies a = 4

Subtracting the sum of the known three numbers from the total sum gives the smallest integer.

Key Concept

Using mean, median, and mode definitions simultaneously to deduce unknown values in an ungrouped dataset

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