The average of 12 numbers is 47. The average of the first 5 numbers is 45 and the average of the next 4 numbers is 52. If the 10
th
number is 10 less than the 11
th
number and is 5 more than the 12
th
number, then what is the average value of the 11
th
and 12
th
numbers?

Step 1: Compute sums
Total sum of 12 numbers = 12×47=564

Sum of first 5 numbers = 5×45=225

Sum of next 4 numbers = 4×52=208

Sum of the remaining 3 numbers (10
th
+11
th
+12
th
)=564−(225+208)=564−433=131
Step 2: Use relative values of the final three numbers
Let the 10
th
number be x.

10
th
number =x

Since the 10
th
number is 10 less than the 11
th
number, the 11
th
number =x+10

Since the 10
th
number is 5 more than the 12
th
number, the 12
th
number =x−5

Forming the equation:

x+(x+10)+(x−5)=131
3x+5=131
3x=126⟹x=42
So the numbers are:

10
th
number =42

11
th
number =42+10=52

12
th
number =42−5=37

Step 3: Average of 11
th
and 12
th
numbers

Average=
2
52+37
​
=
2
89
​
=44.5