Averages
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Q.

The average of five different positive numbers is 25. $x$ is the decrease in the average when the smallest number among them is replaced by 0. What can be said about $x$?

 A.

$x$ is less than 5

 B.

$x$ is greater than 5

 C.

$x$ is equal to 5

 D.

Nothing can be said

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Solution:
Option(A) is correct

Let $a, b, c, d$, and $e$ be the five positive numbers in the decreasing order of size such that $e$ is the

smallest number. We are given that the average of the five numbers is 25. Hence, we have the

equation

\(\dfrac{a+b+c+d+e}{5}=25\)

$a+b+c+d+e=125$----------- (1) by multiplying by 5

The smallest number in a set is at least less than the average of the numbers in the set if at least

one number is different. 

For example, the average of 1, 2, and 3 is 2, and the smallest number in the set 1 is less than the

average 2. Hence, we have the inequality

$0<e<25$
$0>–e>–25$ by multiplying both sides of the inequality by –1 and flipping the directions of

the inequalities.

Adding this inequality to equation (1) yields

$0+125>(a+b+c+d+e)+(–e)>125–25$

$125>(a+b+c+d)>100$

$125>(a+b+c+d+0)>100$ by adding by 0

\(25>\dfrac{a+b+c+d+0}{5}=>20\)by dividing the inequality by 5

$25>$ The average of numbers $a, b, c, d$ and $0>20$

Hence, $x$ equals 

(Average of the numbers $a, b, c, d$, and $e$)–(Average of the numbers $a, b, c$, and $d$)

$=25-$(A number between 20 and 25)

$⇒$ A number less than 5

Hence, x is less than 5.


(4) Comment(s)


Mamoona
 ()

or a alternative solution could be

we know that 25 is the mean or the middle value

hence we can suppose that the values are 23, 24, 25, 26 and 27

sum of these equal 125 and they average 25

if we were to remove and replace the lowest value i.e. 23 with 0; the sum will be

125 - 23 + 0 = 102

now by taking the average of the new no.s

102/5 = 20.4

x = 25-20.4 = 4.6

thus x > 5



PRATYUSH ANAND
 ()

Really nice one Cool



Sulekh
 ()

If we remove a number less than the average the average should increase and not decrease.

In the above question the average decreases by $x$ does it mean $x$ is negative. Huh


Niket
 ()

but that is if the number of quantities also decrease but in this case , number of quantities remain same and one quantity is put equal to zero hence the average is decreasing... Smile