Question

The mean of three positive numbers is 10 more than the smallest of the numbers and 15 less than the largest of the three. If the median of the three numbers is 5, then the mean of the squares of the numbers is:

Answer 1

Given,

The mean of three positive numbers is 10 more than the smallest of the numbers and 15 less than the largest of the three. The median of the three numbers is 5.

To find,

The mean of the squares of the numbers.

Solution,

The mean of the squares of the numbers is 216.66

To answer this question, we will follow the following steps:

Let the three numbers be a, b and c where a is the smallest number and c is the largest number.

Now,

According to the question, we have,

Median of three numbers = 5

This means,

The value of b = 5

(as the median of odd observations

$= {( \frac{n + 1}{2} )}^{th} \: observation$

where n = 3)

Now,

Mean of three numbers

$= 10 + a$

Also,

Mean of three numbers

$= \frac{a + b + c}{3}$

So,

$\frac{a + b + c}{3} = 10 + a$

On solving the above, we get

$a + b + c = 30 + 3a$

$b + c - 2a = 30 \: \:$

$c - 2a = 25 \: (as \: b \: = 5) \: \: (i)$

Similarly,

We have,

$\frac{a + b + c}{3} = c - 15$

On solving the above, we get

$a + b + c = 3c - 45$

$2c - a - b = 45 \: \:$

$2c - a = 50 \: (as \: b \: = 5) \: (ii)$

On multiplying (i) by 2, we get

$2c - 4a = 50 \: \: (iii)$

Now,

On subtracting (iii) from (ii), we get

$a = 0$

On putting the value of a = 0 in (iii), we get

$c = 25$

So,

The mean of squares of numbers

$= \frac{ {a}^{2} + {b}^{2} + {c}^{2} }{3}$

$= \frac{0 + 25 + 625}{3}$

$= \frac{650}{3}$

$= 216.66$

Henceforth, the mean of squares of numbers is 216.66

Answer 2

The mean of the squares of the   positive numbers is 216.66.

Step-by-step explanation:

Given:

Three positive numbers is 10 more than the smallest of the three numbers.

Three positive numbers  is 15 less than the largest of the three numbers.

The median of the three positive numbers is 5.

To Find:

The mean of the squares of the positive  numbers.

Formula Use:

The mean is the   average value = Sum of all observations /Number of observations

The median is the middle value.

Solution:

Let   smallest positive number is  x,  middle positive number    is  y.   largest positive   number is  z.

Let mean of three positive numbers are p.

$p=\frac{(x+y+z)}{3}$  ---------------          equation no.01

As given-   the three positive numbers is 10 more than the smallest of the three numbers

$x=p-10$           ------------------------ equation no. 02

As given-   the three positive numbers  is 15 less than the largest of the three numbers.

$z=p+15$                 ----------------------- equation no.03

As given- the median of the three positive  numbers is 5.

$y=5$      ------------------------ equation 04

Putting the values of x,y,z and equation no.01.

   $p=\frac{[(p-10)+5+(p+15) ]}{3}$

   $3p=2p+10$

    $p=10$  

Putting the value of p in equation no.02 and equation no.03.

$x=p-10=10-10\\x=10$          

$z=p+15=10+15\\z=25$            

The mean of the squares of the  positive numbers

$=\frac{x^{2} +y^{2} +z^{2} }{3}$

putting the values of x, y and z, we get.

$=\frac{0^{2}+5^{2} +25^{2} }{3}$

$=\frac{0+25+625}{3} \\=\frac{650}{3} \\=216.66$

                                                                               

                                                                               

Thus, The mean of the squares of the positive numbers=216.66