Understanding the Problem
To solve the problem, we first need to understand the properties of odd numbers and how their mean is calculated.
Identifying the Consecutive Odd Numbers
- The mean of 10 consecutive odd numbers is given as 120.
- The formula for the mean is the sum of all numbers divided by the count of numbers.
- Therefore, the sum of these 10 odd numbers can be calculated as:
Sum = Mean × Count = 120 × 10 = 1200
- Let the first odd number in this series be x. The 10 consecutive odd numbers can be represented as:
x, x+2, x+4, x+6, x+8, x+10, x+12, x+14, x+16, x+18
Calculating the Sum
- The sum of these numbers can also be expressed as:
Sum = 10x + (2 + 4 + 6 + 8 + 10 + 12 + 14 + 16 + 18)
- The sum of the first 9 even numbers (2, 4, ... , 18) is 90.
- Thus, we have:
10x + 90 = 1200
- Solving for x gives:
10x = 1110 → x = 111
First Five Odd Numbers
- The first five odd numbers are:
111, 113, 115, 117, 119
Calculating Their Mean
- The sum of these five numbers is:
111 + 113 + 115 + 117 + 119 = 575
- The mean of these five numbers can be calculated as:
Mean = Sum / Count = 575 / 5 = 115
Conclusion
Therefore, the mean of the first five odd numbers among the 10 consecutive odd numbers is 115.