Question

The average of 10 numbers is 25. The average of first three numbers is 22 1/3 and that of next five is 15 1/5. If the eighth be less than the ninth and tenth numbers by 5 and 11 respectively, then the tenth number is:

Answer 1

zesus chrises.. .. ... .

Answer 2

Answer:

$56\frac{1}{2}$

Step-by-step explanation:

Given :

To find the 10th number if,

The average of 10 numbers is 25,.

The average of 1st 3 numbers is 22$\frac{1}{3}$

The average of next 5 numbers is 15$\frac{1}{5}$

The  8th number is less than 9th by 5 and 10th by 11 respectively,.

Solution :

Let the numbers be $a_1,a_2,a_3,a_4 ,.. a_{10}$

Then,

Statement 1 :

The average of 10 numbers is 25,.

⇒ Average = $\frac{Sum \ of \ all \ the \ observations}{Total \ number \ of \ observations}$

⇒ $\frac{a_1 + a_2 + a_3 + a_4 + .. + a_{10}}{10} = 25$

⇒ $a_1 + a_2 + a_3 + a_4 + .. + a_{10} = 25 \times 10 = 250$ ..(i)

Statement 2 :

The average of 1st 3 numbers is 22$\frac{1}{3}$

⇒ $\frac{a_1 + a_2 + a_3}{3} = 22\frac{1}{3}=\frac{22 \times 3 + 1}{3}=\frac{66 + 1}{3}=\frac{67}{3}$

⇒ $a_1 + a_2 + a_3 =\frac{67}{3} \times 3 = 67$ ..(ii)

Statement 3 :

The average of next 5 numbers is 15$\frac{1}{5}$

⇒ $\frac{a_4 + a_5 + a_6 + a_7 + a_8}{5} = 15\frac{1}{5}=\frac{15 \times 5 + 1}{5}=\frac{75 + 1}{5}=\frac{76}{5}$

⇒ $a_4 + a_5 + a_6 + a_7 + a_8 =\frac{76}{5} \times 5 = 76$ ..(iii)

Statement 4 :

The  8th number is less than 9th by 5 and 10th by 11 respectively,.

Then,

⇒ $a_9 - a_8 = 5$ ⇒ $a_9 = a_8 + 5$

⇒ $a_{10} - a_8 = 11$ ⇒ $a_{10} = a_8 + 11$

By subtracting (ii) & (iii) from (i),

We get,

⇒ $(a_1 + a_2 + a_3 + a_4 + a_5 + a_6 + a_7 + a_8 + a_9 + a_{10}) - (a_1 + a_2 + a_3) - ( a_4 + a_5 + a_6 + a_7 + a_8) = 250 - 67 - 76$

⇒ $a_9 + a_{10} = 107$

⇒$(a_8 + 5) + (a_8 + 11) = 107$

⇒ $2a_8 + 16 = 107$

⇒ $2a_8 = 107 - 16$

⇒ $2a_8 = 91$

⇒ $a_8 = \frac{91}{2}$

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Then the 10th term,

$a_{10} = a_8 + 11 = \frac{91}{2} + 11 = \frac{91 + 22}{2} = \frac{113}{2}=56\frac{1}{2}$