The average of 20 numbers is 80. The average of the first 10 numbers is 76.5 and that of the next 7 numbers is 82. The $$18^{th}$$ number is 3 more than the $$19^{th}$$ number but 3 less than the $$20^{th}$$ number. What is the average of $$18^{th}$$ and $$19^{th}$$ numbers?

The average of 20 numbers is 80.

Total of 20 numbers = $$80\times20$$ = 1600    Eq.(i)

The average of the first 10 numbers is 76.5 and that of the next 7 numbers is 82.

Sum of first 17 numbers = $$76.5\times10+82\times7$$
= 765+574

= 1339    Eq.(ii)

Sum of last 3 numbers = Eq.(i)-Eq.(ii) = 1600-1339

= 261

The $$18^{th}$$ number is 3 more than the $$19^{th}$$ number but 3 less than the $$20^{th}$$ number.

Let's assume the $$19^{th}$$ number is 'y'.

$$18^{th}$$ number = (y+3)

$$20^{th}$$ number = (y+3)+3 = (y+6)

So (y+3)+y+(y+6) = 261

3y+9 = 261

3y = 261-9 = 252

y = 84

Average of $$18^{th}$$ and $$19^{th}$$ numbers = $$\frac{y+\left(y+3\right)}{2}$$

= $$\frac{84+\left(84+3\right)}{2}$$

= $$\frac{84+87}{2}$$

= $$\frac{171}{2}$$

= 85.5