Question

The average of all the non-negative integers upto 99 is : (a) \(50.49\) (b) \(49.50\) (d) 99 (c) \(50.50\)

Short answer

Answer: (b) 49.50

Step-by-step solution

List all the non-negative integers up to 99

The non-negative integers up to 99 are all the integers from 0 to 99, inclusive.

Find the sum of all the integers in the list

We can use the formula for the sum of the first n integers, which is given by: \(S_n = \frac{n(n + 1)}{2}\) In our case, n = 99, so we can calculate the sum as follows: \(S_{99} = \frac{99(99 + 1)}{2} = \frac{99 \cdot 100}{2} = 99 \cdot 50 = 4950\)

Determine the total number of integers in the list

Since the list includes integers from 0 to 99, there are a total of 100 integers in the list.

Calculate the average of the integers in the list

To find the average, we divide the sum of the integers by the total number of integers: \(Average = \frac{Sum}{Number\,of\,Integers} = \frac{4950}{100} = 49.5\)

Identify the correct option from the given choices

Comparing our calculated average (49.5) with the given options, we find that the correct answer is (b) \(49.50\).

Key concepts

Non-Negative Integers

Understanding non-negative integers is a basic concept in mathematics that helps in problem-solving. Non-negative integers include all whole numbers starting from zero, such as 0, 1, 2, 3, and so on. They are called non-negative because they are not below zero, essentially arranged in ascending order. Non-negative integers are crucial in counting and basic arithmetic operations.

• Start from zero and go up: 0, 1, 2, 3, ...
• No negative values included.
• Used in various calculations, like sums and averages.

Whether you're dealing with ranking, tallying scores, or solving problems, knowing these numbers is foundational for progressing in arithmetic and further mathematical studies.

Sum of First n Integers

Calculating the sum of the first n integers is a common mathematical task. It is often required in problems involving sequences of numbers. The formula to sum these integers is very straightforward:
Let’s use the formula:\[ S_n = \frac{n(n + 1)}{2} \]In this exercise, you have to calculate the sum of integers up to 99:- Set \( n = 99 \), so plug this into the formula: - \[ S_{99} = \frac{99 \times 100}{2} = 4950 \]Using this formula streamlines the process, making it easy to solve for sums without having to add numbers one by one. By recognizing this pattern, problems involving long lists of numbers become a lot more manageable.

Arithmetic Mean

The arithmetic mean is a simple but powerful concept often used in statistical calculations. When dealing with averages, the arithmetic mean provides a single value that represents an overall distribution of numbers. Here’s how to calculate it:

• Add up all numbers in the set to get the sum.
• Count the numbers in the set to get the total.
• Divide the sum by the total count.

This method gives an average by balancing out all values:For example, find the average of the integers from 0 to 99:- Sum = 4950- Total number = 100- Arithmetic Mean = \( \frac{4950}{100} = 49.5 \)Averages are crucial in understanding the overall trend or central tendency within any data set, especially in statistics and real-world data analysis.

Problem Solving in Mathematics

Problem-solving in mathematics is an essential skill that combines logical thinking and understanding of mathematical concepts. Tackling problems systematically helps in deriving solutions effectively. Here’s a methodical approach:

1. Understand the problem:
- Read the question carefully to identify key information and what is being asked.
2. Plan your approach:
- Identify formulas or methods necessary to find the solution.- Organize the information to apply these methods.
3. Execute the plan:
- Carry out the necessary calculations step-by-step.- Use formulas like \( S_n = \frac{n(n+1)}{2} \) for sums, or use averages to simplify calculations.
4. Review your solution:
- Check if the answer makes sense in the context of the problem.- Compare with possible solution options if available.
By practicing such steps, you enhance not only your ability to solve mathematical problems but also analytical skills applicable in everyday challenges.