The average of ten numbers is 72. The average of the first four numbers is 69 and that of the next three numbers is 74. The $$8^{th}$$ number is 6 more than the $$9^{th}$$ number and 12 more than the $$10^{th}$$ number. What is the average of the $$8^{th}$$ and $$9^{th}$$ numbers?

Let the ten numbers be $$A_1,A_2,A_3,.......,A_{10}$$

Sum of ten numbers = $$A_1+A_2+A_3+.......+A_{10}=72\times10=720$$ ---------------(i)

Similarly, $$A_1+A_2+A_3+A_4=69\times4=276$$ -----------(ii)

and $$A_5+A_6+A_7=222$$ ---------------(iii)

Now, subtracting sum of equations (ii) and (iii) from (i), we get :

=> $$A_8+A_9+A_{10}=720-276-222=222$$ -------------(iv)

Also, $$A_8=6+A_9$$ and $$A_8=12+A_{10}$$ ----------(v)

Substituting value of $$A_9$$ and $$A_{10}$$ in equation (iv)

=> $$A_8+(A_8-6)+(A_8-12)=222$$

=> $$3A_8=222+18=240$$

=> $$A_8=\frac{240}{3}=80$$

Thus, substituting in equation (iv), => $$A_9=80-6=74$$

$$\therefore$$ Average of the $$8^{th}$$ and $$9^{th}$$ numbers = $$\frac{80+74}{2}=77$$

=> Ans - (D)