The average of 24 numbers is 65. The average of first 11 numbers is 67 and the average of last 10 numbersis 70.If the 12th number is 13 less than the 13th number, and the 14th number is one more than the 13th number, then the average of 12th and 14th number is:
Latek

Since we have given that

Numbers = 24

Average = 65

So, Sum of 24 observations = $$24\times\ 65=1560$$

Average of first 11 numbers = 67

So, Sum of first 11 observations = $$67\times\ 11=737$$

Average of last 10 number = 70

So, Sum of last 10 observations = $$10\times\ 70=700$$10*70=700

As, conditions given in the question,

Let the 13th number be 'x'.

Let the 12th number be 'x-13'

Let the 14 th number be 'x+1'

So, it becomes,

Sum of first 11 numbers + 12^th number + 13^th number + 14^th number + sum of last 10 numbers = 1560

$$737+x+\left(x-13\right)+\left(x-1\right)+700=1560$$

$$3x=1560-1437+12$$

$$x=\frac{135}{3}$$

x=45

So, the 12th number = 45-13 = 32

The 14th number = 45+1=46

So, the average of 12th and 14th number is

$$\frac{\left(46+32\right)}{2}$$

=39

Hence answer is option a