How many pairs of whole numbers numbers have a sum of 40
The correct answer and explanation is:
To determine how many pairs of whole numbers have a sum of 40, you need to find all pairs of non-negative integers (a, b) that satisfy the equation: a+b=40a + b = 40
where both aa and bb are whole numbers (non-negative integers).
Step-by-step explanation:
- Range of values for aa: Since a+b=40a + b = 40, aa must be a whole number between 0 and 40 (inclusive). If a=0a = 0, then b=40b = 40. If a=1a = 1, then b=39b = 39, and so on. The largest possible value for aa is 40, where b=0b = 0.
- Counting the pairs: For each possible value of aa, there is exactly one corresponding value of bb. Specifically, b=40−ab = 40 – a. Therefore, for a=0,1,2,…,40a = 0, 1, 2, \dots, 40, the corresponding values of bb are 40,39,38,…,040, 39, 38, \dots, 0.
- Conclusion: The possible values for aa are 0 through 40, inclusive, giving a total of 41 different values for aa. Each value of aa corresponds to a unique pair (a,b)(a, b). Therefore, the number of pairs of whole numbers that sum to 40 is:
41\boxed{41}
Explanation:
This problem is a simple counting problem where you are asked to determine how many different ways you can pair two non-negative integers that sum to 40. Since the relationship between aa and bb is linear (i.e., b=40−ab = 40 – a), and both aa and bb are bounded between 0 and 40, the number of solutions corresponds directly to the number of possible values for aa, which are 41 in total.