Number System Question 1 - CAT 2022
Consider six distinct natural numbers such that the average of the two smallest numbers is 14, and the average of the two largest numbers is 28. Then, the maximum possible value of the average of these six numbers is:
Step-by-Step Explanation
Let's arrange our six different counting numbers from smallest to biggest and call them A, B, C, D, E, and F.
- Find the sums we already know:
- The average of the two smallest (A and B) is 14, so their total sum is 14 × 2 = 28.
- The average of the two largest (E and F) is 28, so their total sum is 28 × 2 = 56.
- Make the middle numbers as big as possible:
To get the highest possible average for the whole group, we need the total sum of all six numbers to be as large as possible. The sums of the first two and the last two numbers are already locked in (28 and 56). So, we just need to make the middle numbers, C and D, as big as we can.
To give C and D room to grow, we should make E as small as possible. Since E and F must be different numbers and add up to 56, the closest they can get to each other is 27 and 29. So, let's set E = 27 and F = 29.
Since all six numbers must be different and in increasing order:
- The biggest value D can take is 26 (since it must be smaller than E = 27).
- The biggest value C can take is 25 (since it must be smaller than D = 26).
- Calculate the final answer:
Maximum Total Sum = (A + B) + C + D + (E + F) = 28 + 25 + 26 + 56 = 135
Maximum Average = (135)/(6) = 22.5
Correct Option: D
💡 FundaMakers Tip:
Think of this like a chain reaction. To make the middle numbers climb as high as possible, you have to push the larger numbers right next to them down as low as possible. Squishing the two largest numbers together at 27 and 29 leaves the perfect opening for the middle numbers to reach 25 and 26.