The average of thirteen numbers is 80. The average of the first five numbers is 74.5 and that of the next five numbers is 82.5. The $$11^{th}$$ number is 6 more than the $$12^{th}$$ number and the $$12^{th}$$ number is 6 less than the $$13^{th}$$ number. What is the average of the $$11^{th}$$ and the $$13^{th}$$ numbers ?

As per the question,

$$\dfrac{x_1+x_2+x_3....x_{13}}{13}=80$$

$$x_1+x_2+x_3....x_{13}=80\times 13$$------------(i)

As per the condition given in the question, average of first 5 number

$$\dfrac{x_1+x_2+x_3+x_3+x_4+x_5}{5}=74.5$$

$$x_1+x_2+x_3+x_3+x_4+x_5}=74.5\times 5$$-----------(ii)

Given that, average of next six number

$$\dfrac{x_6+x_7....x_{10}}{5}=82.5$$

$$x_6+x_7....x_{10}=82.5\times 5$$-----------(iii)

Let the 12th number is x,

As per the condition given in the question,

So 11th number will be x+6 and 13th number will be x+6,

Now substitute in the equation (i)

$$x_1+x_2+x_3....x_{13}=80\times 13$$

$$74.5\times 5+82.5\times 5+x+6+x+x+6=80\times 13$$

$$372.5+412.5+x+6+x+x+6=1040$$

$$3x=1040-372.5-412.5-12=1040-797=243$$

$$x=\dfrac{243}{3}=81$$

So, 11th and 13th numbers will be $$x+6=81+6=87$$

Hence the average of 11th and 13th number $$=\dfrac{87+87}{2}=87$$