M.6 Find two numbers given their sum and product
Find two numbers given their sum and product
In mathematics, the sum of two numbers is the result when you add them together. The product of two numbers is the result when you multiply them. When a problem gives you the sum and the product, it provides two important clues to figure out exactly which two numbers are being described. This skill helps build a deeper understanding of how addition and multiplication are connected.
- The sum of two numbers is 9. The product of the same two numbers is 20. The numbers are 4 and 5 because 4 + 5 = 9 and 4 × 5 = 20.
- The sum of two numbers is 13. The product is 36. The numbers are 4 and 9 because 4 + 9 = 13 and 4 × 9 = 36.
- The sum of two numbers is 15. The product is 56. The numbers are 7 and 8 because 7 + 8 = 15 and 7 × 8 = 56.
Think of sum and product as a pair of clues. The sum tells you how the numbers combine when added, and the product tells you how they combine when multiplied. Together, they point to only one correct pair of numbers.
Understanding sum and product with factor pairs
A factor pair is a set of two numbers that multiply together to give a specific product. When finding two numbers from their sum and product, you can start by listing all the factor pairs of the product. Then, check which pair also adds up to the given sum. This method works for all whole-number problems.
- Product is 24. Factor pairs: 1 and 24, 2 and 12, 3 and 8, 4 and 6. If the sum is 10, the numbers are 4 and 6 because 4 × 6 = 24 and 4 + 6 = 10.
- Product is 30. Factor pairs: 1 and 30, 2 and 15, 3 and 10, 5 and 6. If the sum is 11, the numbers are 5 and 6 because 5 × 6 = 30 and 5 + 6 = 11.
- Product is 48. Factor pairs: 1 and 48, 2 and 24, 3 and 16, 4 and 12, 6 and 8. If the sum is 14, the numbers are 6 and 8 because 6 × 8 = 48 and 6 + 8 = 14.
When listing factor pairs, work in an organized way. Start with 1 times the product, then 2, then 3, and so on. Stop when the factors start to repeat. This ensures you do not miss any possible pair.
Using a systematic guess-and-check strategy
Sometimes the product is large, and listing every factor pair can take time. Another strategy is to think about the sum. If you know the sum, you can guess one number and subtract to find the other. Then multiply to see if the product matches. This is called a systematic guess-and-check. It helps you narrow down the possibilities quickly.
- Sum is 17, product is 72. Guess one number is 8. The other number would be 17 − 8 = 9. Multiply: 8 × 9 = 72. It matches! The numbers are 8 and 9.
- Sum is 20, product is 91. Guess one number is 7. The other number is 20 − 7 = 13. Multiply: 7 × 13 = 91. It matches! The numbers are 7 and 13.
- Sum is 25, product is 100. Guess one number is 10. The other number is 25 − 10 = 15. Multiply: 10 × 15 = 150, which is too high. Guess a different number, like 20. The other number is 5. Multiply: 20 × 5 = 100. It matches! The numbers are 5 and 20.
Start your guess with a number that seems reasonable. If the product of your guess and its partner is too high, choose a smaller first number. If the product is too low, choose a larger first number. Adjust until you find the correct pair.
Working with larger numbers and estimation
When numbers become larger, estimation becomes a helpful tool. You can use the sum to estimate the average of the two numbers. The average is the sum divided by 2. The two numbers will be one above and one below that average. Then, you can test pairs that are close to the average to see which pair gives the correct product.
- Sum is 46, product is 493. The average is 46 ÷ 2 = 23. Try numbers around 23, like 22 and 24. 22 × 24 = 528, which is too high. Try 21 and 25. 21 × 25 = 525, still too high. Try 17 and 29. 17 × 29 = 493. It matches! The numbers are 17 and 29.
- Sum is 62, product is 937. The average is 31. Try 30 and 32. 30 × 32 = 960, too high. Try 29 and 33. 29 × 33 = 957, still too high. Try 24 and 38. 24 × 38 = 912, too low. Try 25 and 37. 25 × 37 = 925, too low. Try 26 and 36. 26 × 36 = 936, very close. Try 27 and 35. 27 × 35 = 945. The correct pair is not found with whole numbers, which tells us the numbers might not be whole numbers or that we need to check factor pairs instead.
Not every sum and product combination will result in two whole numbers. If you have tried several reasonable pairs and none match exactly, the numbers may be fractions or decimals. In fourth grade, most problems use whole numbers.
Understanding the relationship between sum, product, and factors
The two numbers you are looking for are actually factors of the product. They are also addends of the sum. This means the numbers have a special relationship. When you multiply them, you get the product. When you add them, you get the sum. Recognizing this relationship helps you solve problems faster and prepares you for more advanced math like algebra.
- Sum = 12, Product = 35. The factor pairs of 35 are 1 and 35, 5 and 7. Which pair adds to 12? 5 + 7 = 12. The numbers are 5 and 7.
- Sum = 18, Product = 77. Factor pairs of 77 are 1 and 77, 7 and 11. Which pair adds to 18? 7 + 11 = 18. The numbers are 7 and 11.
- Sum = 23, Product = 120. Factor pairs of 120 include 8 and 15, 10 and 12. 8 + 15 = 23, so the numbers are 8 and 15.
If you know your multiplication facts well, finding factor pairs becomes much easier. Practice multiplication tables up to 12 × 12 to build fluency. This skill will make these problems feel like puzzles you can solve quickly.
Solving word problems with sum and product
In real-world situations, sum and product problems often appear in word problems. You may need to read carefully to identify the sum and the product. Key words like “total,” “in all,” “combined,” and “sum” point to addition. Key words like “multiply,” “product,” “times,” and “area” point to multiplication. Once you identify both, you can find the two unknown numbers.
- Maria bought two kinds of pencils. The total number of pencils she bought was 14. The product of the two numbers of pencils was 48. How many of each kind did she buy? Factor pairs of 48: 6 and 8. 6 + 8 = 14. She bought 6 of one kind and 8 of the other.
- In a rectangule, the sum of the length and width is 22 meters. The product of the length and width is 120 square meters. What are the length and width? Factor pairs of 120 that add to 22 are 10 and 12. The length could be 12 meters and the width 10 meters.
- A farmer has two types of apple trees. The total number of trees is 19. The product of the two numbers is 84. How many of each type does the farmer have? Factor pairs of 84: 7 and 12, 6 and 14, 4 and 21. 7 + 12 = 19, so the farmer has 7 of one type and 12 of the other.
When solving word problems, underline the numbers and circle the words that tell you whether to add or multiply. This helps you separate the important information from the extra details.
Checking your answer for accuracy
After finding two numbers that you believe match the sum and product, always check your work. First, add the two numbers to make sure they equal the given sum. Second, multiply the two numbers to make sure they equal the given product. If both check out, your answer is correct. If not, review your factor pairs or guess again.
- Problem: Sum = 24, Product = 143. Solution: Numbers are 11 and 13. Check: 11 + 13 = 24. 11 × 13 = 143. Correct.
- Problem: Sum = 34, Product = 225. Solution: Numbers are 9 and 25. Check: 9 + 25 = 34. 9 × 25 = 225. Correct.
- Problem: Sum = 30, Product = 216. Solution: Numbers are 12 and 18. Check: 12 + 18 = 30. 12 × 18 = 216. Correct.
Checking your answer is a habit of strong mathematicians. It takes only a few seconds and helps you catch mistakes before you submit your work. Always show your check in your written work.
Common mistakes and how to avoid them
When finding two numbers from their sum and product, students sometimes make predictable errors. One common mistake is confusing sum with product, adding when they should multiply or multiplying when they should add. Another mistake is listing factor pairs incorrectly, missing a pair or stopping too early. A third mistake is forgetting to check both conditions.
- Mistake: Sum = 12, Product = 27. A student says the numbers are 4 and 8 because 4 + 8 = 12, and they forget to check the product.
Correction: 4 × 8 = 32, not 27. So 4 and 8 cannot be correct. The correct factor pairs of 27 are 1 and 27, and 3 and 9. Only 3 + 9 = 12, and 3 × 9 = 27. The correct numbers are 3 and 9. - Mistake: Sum = 18, Product = 56. A student lists factor pairs of 56 as 1 and 56, 2 and 28, and 4 and 14. They stop there and choose 4 and 14 because 4 + 14 = 18. They think they have found the answer.
Correction: The student stopped too early. The complete list of factor pairs for 56 is: 1 and 56, 2 and 28, 4 and 14, and 7 and 8. While 4 and 14 do add to 18, the student must also check the product: 4 × 14 = 56, which matches. So 4 and 14 is actually correct. The mistake was not a wrong answer but incomplete work. To be thorough, always list all factor pairs before choosing. - Mistake: Sum = 21, Product = 104. A student finds factor pairs of 104: 1 and 104, 2 and 52, 4 and 26, 8 and 13. They choose 4 and 26 because they multiply to 104, but they forget to check the sum.
Correction: 4 + 26 = 30, which does not equal 21. The correct pair is 8 and 13 because 8 × 13 = 104 and 8 + 13 = 21. Another common mistake is choosing 2 and 52. While 2 × 52 = 104, 2 + 52 = 54, which is also incorrect. Always verify both the sum and the product before finalizing your answer. Both conditions must be satisfied.
To avoid these mistakes, work in an organized way. Use a table to list factor pairs. Write the sum next to each pair. Circle the pair that matches. Always perform the final check with both addition and multiplication.
Extending the concept to three or more numbers
While fourth grade focuses on two numbers, the concept of sum and product can extend to three or more numbers. For example, you might be given the sum of three numbers and the product of the same three numbers. The strategy changes because there are more combinations to consider. However, understanding the two-number case builds a strong foundation for these more advanced problems in later grades.
- Three numbers have a sum of 10 and a product of 30. One possible set is 2, 3, and 5 because 2 + 3 + 5 = 10 and 2 × 3 × 5 = 30.
- Three numbers have a sum of 13 and a product of 36. One possible set is 2, 2, and 9 because 2 + 2 + 9 = 13 and 2 × 2 × 9 = 36.
- Three numbers have a sum of 16 and a product of 108. One possible set is 3, 4, and 9 because 3 + 4 + 9 = 16 and 3 × 4 × 9 = 108.
For now, focus on mastering two-number sum and product problems. This skill will help you understand how numbers work together and prepare you for algebra, where you will learn to solve such problems using variables and equations.
Common Core alignment: CCSS.MATH.CONTENT.4.OA.B.4 – Find all factor pairs for a whole number in the range 1–100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1–100 is a multiple of a given one-digit number. This lesson also builds foundational skills for CCSS.MATH.CONTENT.4.OA.A.3 – Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations.
Notes for teachers
This lesson is aligned with CCSS.MATH.CONTENT.4.OA.B.4 and supports CCSS.MATH.CONTENT.4.OA.A.3. All content is 100% free. Use it for whole-class instruction, small group work, independent study, or homework. Encourage students to use factor pair lists and systematic guessing to build number sense and problem-solving skills.