Question

If mean of ten consecutive odd numbers is 120
then the mean of first five odd numbers among
them is
(a) 113
(b) 115
(c) 114
(d) 116



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Answer 1

Given:

• Mean of ten consecutive odd numbers is 120.

To Find:

• Mean of first five odd numbers.

Answer:

Let us take first odd number be x ,

Then second odd number = (x+2) , third = (x+4) , fourth = (x+6) ..........tenth = (x+18).

Now we know formula of mean as;

$\large{\boxed{\bf{\leadsto Mean=\dfrac{Sum\:of\: Numbers}{Total\: number\:of\:numbers}}}}$

⇒120 = x+(x+2)+(x+4)+(x+6)+..........+(x+16)/10

⇒120×10 = 10 × x + 2+4+6+.....+18

⇒1200 = 10x + 2(1+2+3+4+.....+9)

⇒1200 = 10x +2[9(9+1)]/2

• Using sum of first n natural numbers is n(n+1)/2.

⇒ 1200 = 10x+ 2[9×5]

⇒1200 = 10x + 90

⇒ 10x = 1200-90

⇒10x = 1110

⇒x = 1110/10

⇒x = 111

Hence value of x is 111 .

Hence mean of first 5 is ,

= x+(x+2)+(x+4)+(x+6)+(x+8)/5

= 5x + 20/5

= 5x/5 + 20/5

= x + 4

= 111+4

= 115 .

Hence the required answer is 115.

Answer 2

Let ten consecutive odd number be

2x+1,2x+3,......,2x+19

Hence, (2x+1)+(2x+3)+.....+(2x+19)

=10×120(sum of observations=Mean × No. of observation)

⇒20x+100=12

⇒x=1100÷20=55

Mean of first five odd numbers

=(2x+1)+(2x+3)+....+(2x+9)÷5

=10×55×25÷5=575÷5=115