- Verify each equation in the table to determine which ones sum to 10.
- Identify the incorrect equations: \(8 + 3 = 10\) and \(1 + 3 = 10\).
- Correct these equations to satisfy the property \(a + b = 10\), such as \(8 + 2 = 10\) and \(3 + 7 = 10\).
- The final corrected list of pairs includes: \((1, 9), (10, 0), (8, 2), (7, 3), (5, 5), (4, 6)\) which all satisfy \(\boxed{a + b = 10}\).
Explanation
- Analyzing the Problem and Data
The goal is to identify and list all number pairs \((a, b)\) such that \(a + b = 10\), where \(a\) and \(b\) are non-negative integers. We are provided with a table of six pairs and need to check if they each sum to 10. If an equation is incorrect, it must be replaced with a correct number partner for 10. The set of all number partners for 10 includes pairs like \((0, 10), (1, 9), (2, 8), (3, 7), (4, 6), (5, 5), (6, 4), (7, 3), (8, 2), (9, 1), (10, 0)\).
- Verifying the Provided Equations
We verify each equation in the provided table:
- \(1 + 9 = 10\) is correct because \(1 + 9 = 10\).
- \(10 + 0 = 10\) is correct because \(10 + 0 = 10\).
- \(8 + 3 = 10\) is incorrect because \(8 + 3 = 11\). The correct partner for 8 is 2, since \(8 + 2 = 10\).
- \(1 + 3 = 10\) is incorrect because \(1 + 3 = 4\). The correct partner for 1 is 9, since \(1 + 9 = 10\).
- \(5 + 5 = 10\) is correct because \(5 + 5 = 10\).
- \(8 + 2 = 10\) is correct because \(8 + 2 = 10\).
- Correcting and Finalizing the List
To complete the list, we identify the missing or corrected pairs:
- The pair \(8 + 3 = 10\) should be corrected to \(8 + 2 = 10\) (or \(7+3=10\)).
- The pair \(1 + 3 = 10\) should be corrected to \(1 + 9 = 10\) (or \(3+7=10\)).
Based on the existing valid entries, we can finalize the list of correct number partners for 10 that satisfy the requested table format. The complete set of correct pairs (number partners) are:
\(0 + 10 = 10\), \(1 + 9 = 10\), \(2 + 8 = 10\), \(3 + 7 = 10\), \(4 + 6 = 10\), \(5 + 5 = 10\).
Examples
Number partners are essential for quick mental math and building a strong foundation for understanding base-10 arithmetic. For example, if you are shopping and want to know how many more items you need to reach a total of 10, recognizing that 7 items need 3 more to reach 10 allows for faster transactions and better planning. This skill is used daily in situations ranging from counting change to organizing collections.