Number System Question 8 - CAT 2020

Step-by-Step Explanation

Any 4-digit number of the form 'aabb' can be written using place-value expansion:

Value = 1000a + 100a + 10b + b = 1100a + 11b

Let's find the valid options for the digits a and b:

• The number must be an even natural number, so the units digit b must be even: b in 0,2,4,6,8 (5 options).
• The thousands digit a must be greater than 0: a in 1,2,3,4,5,6,7,8,9 (9 options).

This gives a total of 9 × 5 = 45 numbers. Because the choices for a and b are completely independent and distributed uniformly:

• The average value of digit a = (1 + 9)/(2) = 5
• The average value of digit b = (0 + 8)/(2) = 4

We can find the overall mean of these numbers by plugging the average values of the digits directly into the place-value formula:

Mean = 1100( Average of a ) + 11( Average of b ) = 1100(5) + 11(4) = 5500 + 44 = 5544

Correct Option: D

💡 FundaMakers Tip:

Do not spend time calculating the total sum of all 45 numbers. When digits vary uniformly and independently, the mean of the numbers is always equal to the value formed by using the individual averages of those digits.