P.2 Find the missing numerator or denominator in equivalent fractions
What are equivalent fractions?
Equivalent fractions are fractions that name the same amount or part of a whole, even though they have different numerators (the top number) and denominators (the bottom number).
- One half of a pizza is the same amount as two fourths of a pizza.
- We can show this as: 12 = 24
- Another example: 34 = 68
Think of equivalent fractions as different ways to write the same number. If you shade a shape, equivalent fractions will cover the exact same area, just broken into smaller or larger pieces.
Finding a missing numerator
To find a missing numerator, you must determine what number was used to multiply or divide the known denominator to get the new denominator. Then, apply that same operation to the numerator.
If the denominators are getting larger, you are multiplying. If the denominators are getting smaller, you are dividing.
- Find the missing numerator: 34 = ?8
The denominator 4 was multiplied by 2 to become 8. Multiply the numerator 3 by the same number (2) to find the missing numerator. 3 × 2 = 6. So, 34 = 68. - Find the missing numerator: 812 = ?3
The denominator 12 was divided by 4 to become 3. Divide the numerator 8 by the same number (4). 8 ÷ 4 = 2. So, 812 = 23.
Always check your work. Multiply the numerator and denominator of the original fraction by the same number to ensure you get the new fraction.
Finding a missing denominator
To find a missing denominator, look at the numerators first. Determine what number was used to multiply or divide the known numerator to get the new numerator. Then, apply that same operation to the denominator.
- Find the missing denominator: 25 = 6?
The numerator 2 was multiplied by 3 to become 6. Multiply the denominator 5 by the same number (3). 5 × 3 = 15. So, 25 = 615. - Find the missing denominator: 912 = 3?
The numerator 9 was divided by 3 to become 3. Divide the denominator 12 by the same number (3). 12 ÷ 3 = 4. So, 912 = 34.
If the known numerator is smaller than the new numerator, you are multiplying. If it is larger, you are dividing. This helps you decide which operation to use.
Using multiplication to find missing numbers
When both the numerator and denominator are larger in the second fraction, you use multiplication to find the missing number. Find the factor that relates the known part of the fraction.
- Find the missing number: 47 = ?28
Look at the denominators: 7 × 4 = 28. The factor is 4. Multiply the numerator 4 by 4 to find the missing numerator: 4 × 4 = 16. So, 47 = 1628. - Find the missing number: 38 = 15?
Look at the numerators: 3 × 5 = 15. The factor is 5. Multiply the denominator 8 by 5 to find the missing denominator: 8 × 5 = 40. So, 38 = 1540.
This method is like finding equivalent fractions by scaling up. The factor you multiply by must be a whole number.
Using division to find missing numbers (simplifying)
When both the numerator and denominator are smaller in the second fraction, you use division to find the missing number. This is sometimes called simplifying or finding a fraction in lowest terms.
- Find the missing number: 610 = 3?
Look at the numerators: 6 ÷ 2 = 3. The divisor is 2. Divide the denominator 10 by 2 to find the missing denominator: 10 ÷ 2 = 5. So, 610 = 35. - Find the missing number: 1218 = ?3
Look at the denominators: 18 ÷ 6 = 3. The divisor is 6. Divide the numerator 12 by 6 to find the missing numerator: 12 ÷ 6 = 2. So, 1218 = 23.
Division is the opposite of multiplication. When you simplify a fraction, you are finding an equivalent fraction with smaller numbers.
Using cross multiplication to check your work
After you find a missing numerator or denominator, you can check your answer using cross multiplication. Multiply the numerator of the first fraction by the denominator of the second. Then, multiply the denominator of the first fraction by the numerator of the second. The products should be equal.
- Check if 23 = 69 is correct.
Cross multiply: 2 × 9 = 18 and 3 × 6 = 18. Since 18 = 18, the fractions are equivalent. - Check your work from a missing number problem: 58 = 2032
5 × 32 = 160 and 8 × 20 = 160. The products match, so the missing number was found correctly.
Cross multiplication is a reliable tool to verify your answers. It works for any pair of fractions you believe are equivalent.
Solving word problems with missing numbers
Sometimes, you will see equivalent fractions in real-world situations or word problems. The strategy remains the same: identify what operation (multiplication or division) was used to change one part of the fraction, then apply it to the missing part.
- Problem: A recipe calls for 34 cup of flour. If Mia wants to make a larger batch using ?12 cups of flour, what is the missing numerator to keep the recipe the same?
Solution: The denominators: 4 × 3 = 12. Multiply the numerator 3 by 3: 3 × 3 = 9. Mia needs 912 cup of flour. - Problem: A student solved 824 = 1?. What is the missing denominator?
Solution: The numerators: 8 ÷ 8 = 1. Divide the denominator 24 by 8: 24 ÷ 8 = 3. The missing denominator is 3. So, 824 = 13.
When solving word problems, first identify the two fractions. Then decide if the missing number is in the numerator or denominator before applying the correct operation.
Common Core alignment: CCSS.MATH.CONTENT.4.NF.A.1 – Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) and use this principle to recognize and generate equivalent fractions.
Notes for teachers
This lesson is aligned with CCSS.MATH.CONTENT.4.NF.A.1. All content is 100% free, use it for whole-class instruction, independent study and practice, or homework.