how many pairs of whole numbers have a sum of 110

2 min read 14-10-2024

how many pairs of whole numbers have a sum of 110

Finding the Pairs: A Journey to 110

Have you ever wondered how many different ways you can add two whole numbers to reach a specific sum? Let's explore this question, focusing on finding pairs of whole numbers that add up to 110. This problem is a great way to practice basic number theory and understand the concept of pairs.

The Power of Pairs

To start, we'll use a little bit of math magic. If we have two whole numbers, let's call them 'x' and 'y', and their sum is 110, we can write this mathematically as:

x + y = 110

Now, the key is to realize that for every value of 'x' we choose, there's only one corresponding value of 'y' that will make the equation true.

For example, if 'x' is 50, then 'y' must be 60 to make the sum 110.

This means finding all the pairs is just a matter of systematically listing out the possible values for 'x' and then calculating the corresponding value of 'y'.

Stepping Through the Solutions

Let's start with the smallest possible value for 'x', which is 0. In this case, 'y' would be 110.

x = 0, y = 110

Now, we can increase the value of 'x' by one and repeat the process:

x = 1, y = 109
x = 2, y = 108
x = 3, y = 107

We can continue this pattern until 'x' reaches 55. Notice that if 'x' is 55, then 'y' would also be 55.

x = 55, y = 55

After this point, we would just be repeating the pairs in reverse order. For example, if 'x' is 56, then 'y' is 54, which we already listed when 'x' was 54 and 'y' was 56.

Counting the Pairs

To find the total number of pairs, we can simply count the number of values 'x' took on, which is 56.

Therefore, there are 56 pairs of whole numbers that add up to 110.

A Note on Ordering

It's important to note that we are counting pairs, which means we are not distinguishing between (1, 109) and (109, 1). These are considered the same pair. If we were counting ordered pairs, where the order matters, there would be 111 different possibilities (from (0, 110) to (110, 0)).

Beyond 110

The same logic can be applied to any target sum. To find the number of pairs of whole numbers that sum to a specific number 'n', simply follow these steps:

Find the middle value: Divide 'n' by 2. If 'n' is even, the middle value is 'n/2'. If 'n' is odd, the middle value is the largest integer less than 'n/2'.
Count the pairs: The number of pairs is equal to the number of whole numbers from 0 up to the middle value (including the middle value if 'n' is even).

Additional Resources

For further exploration of number theory and related concepts, you can refer to resources like:

Stack Overflow: https://stackoverflow.com/ - A community-driven question and answer website for programmers.
Khan Academy: https://www.khanacademy.org/ - A free, online learning platform covering various subjects, including mathematics.
Brilliant.org: https://brilliant.org/ - A website offering interactive problem-solving challenges and educational content in mathematics and science.

Remember, exploring mathematics is a journey of discovery, and understanding the principles behind these concepts can open up a world of exciting possibilities!