Number Navigators: How to Select Pairs with Targeted Sums and Differences

Example 1:

Pick two numbers that sum up to \(15\).

Solution Process:

Multiple pairs can sum up to \(15\), such as:

1. \(7 + 8\)

2. \(6 + 9\)

3. \(5 + 10\)

… and so on.

Answer:

One possible pair is \(7\) and \(8\).

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Example 2:

Pick two numbers that have a difference of \(5\).

Solution Process:

Again, multiple pairs can have a difference of \(5\), such as:

1. \(10 – 5\)

2. \(15 – 10\)

3. \(20 – 15\)

… and so on.

Answer:

One possible pair is \(10\) and \(5\).

Picking numbers with a specific sum or difference is a foundational skill in mathematics. It’s necessary for problem-solving, especially in algebra and arithmetic. By understanding the relationship between numbers and their operations, you can quickly identify pairs that meet the criteria. This skill is handy for various applications, from splitting bills to budgeting and planning. Keep practicing to build your ability to swiftly pick the right number pairs!

Practice Questions:

1. Find two numbers that sum up to \(20\).

2. Pick two numbers with a difference of \(7\).

3. Choose a pair of numbers that sum up to \(30\).

4. Identify two numbers with a difference of \(3\).

5. Select two numbers that, when added, give \(50\).

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Answers:

1. \(10\) and \(10\), \(11\) and \(9\), \(12\) and \(8\), etc.

2. \(10\) and \(3\), \(14\) and \(7\), \(20\) and \(13\), etc.

3. \(15\) and \(15\), \(20\) and \(10\), \(25\) and \(5\), etc.

4. \(7\) and \(4\), \(10\) and \(7\), \(13\) and \(10\), etc.

5. \(25\) and \(25\), \(30\) and \(20\), \(35\) and \(15\), etc.

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A Strategy for Pairing Numbers with Targeted Sums and Differences

“Find pairs of numbers that add to 14” sounds simple, but with a long list of choices, it pays to have a system. The same skill shows up in algebra later, when you need to factor a quadratic by finding numbers that sum to one value and multiply to another. Building the habit now pays off for years.

Step-by-Step Method

1. Sort the list. Arrange the numbers from smallest to largest. This lets you scan the list once instead of jumping around.
2. Pair from the ends. For a target sum, take the smallest and the largest. If their sum is too low, move the smaller pointer up. Too high, move the larger pointer down.
3. Cross off pairs you find. Don’t reuse the same number twice unless the problem allows it.
4. Check every pair. Recompute each sum or difference before writing the answer.

Worked Examples

Example 1: Sum target = 12

From {3, 5, 7, 8, 9, 10}: Sort already done. Smallest + largest = 3+10 = 13 (too high). Move down: 3+9 = 12 ✓. Continue: 5+8 = 13. 5+7 = 12 ✓. So pairs are (3,9) and (5,7).

Example 2: Difference target = 5

From {2, 4, 7, 9, 12}: Pair from ends. 12-2 = 10 (too high). 12-4 = 8. 12-7 = 5 ✓. 9-4 = 5 ✓. 9-2 = 7. Pairs: (12,7) and (9,4).

Common Mistakes to Avoid

• Forgetting to sort first, leads to jumping around and missing pairs.
• Reusing the same number when the problem says “distinct pairs.”
• Checking only the first match and stopping. List ALL pairs unless the problem asks for one.
• Adding when the problem says “difference.”

Related Topics

• Order of Operations
• One-Step Equations
• Multi-Step Equations
• The Ultimate 6th Grade Math Course

FAQ

How do you find two numbers if their sum and difference is given?

Add the sum and the difference, then divide by 2 to get the larger number. Subtract the larger number from the sum to get the smaller one. For a sum of 20 and a difference of 6: (20 + 6) ÷ 2 = 13, and 20 − 13 = 7. Check: 13 + 7 = 20 and 13 − 7 = 6. Both targets are met, so the pair is 13 and 7.

How do you find the sum and difference of two numbers?

The sum is what you get when you add the two numbers, and the difference is what you get when you subtract the smaller from the larger. For 18 and 11, the sum is 18 + 11 = 29 and the difference is 18 − 11 = 7. When picking pairs for a target, it helps to keep both in mind, since the same pair always has one sum and one difference.

How do you solve sum and difference word problems?

Find the total (the sum) and the gap (the difference) in the story, then use the add-and-halve method. Example: Ana and Ben have 30 stickers together, and Ana has 4 more than Ben. Ana has (30 + 4) ÷ 2 = 17 and Ben has 30 − 17 = 13. If the story gives only a sum, many pairs can work, so list a few and pick one that fits.

What two numbers have a sum of 1 and a difference of 35?

The numbers are 18 and −17. Use the same method: (1 + 35) ÷ 2 = 18 for the larger number, then 1 − 18 = −17 for the smaller one. Check: 18 + (−17) = 1 and 18 − (−17) = 35. When the difference is bigger than the sum, the smaller number has to be negative, which is a good clue to watch for.

Practice Problems

1. From {1, 4, 6, 8, 11}, find all pairs with sum = 12. (Answer: (1,11), (4,8))
2. From {3, 5, 9, 14, 18}, find pairs with difference = 9. (Answer: (14,5), (18,9))
3. From {2, 5, 7, 10, 13}, find pairs with sum = 15. (Answer: (2,13), (5,10))