S.5 Add and subtract fractions with unlike denominators
What are fractions with unlike denominators?
Fractions represent parts of a whole. The denominator (the bottom number) tells you how many equal parts the whole is divided into. When two fractions have denominators that are not the same number, they have unlike denominators.
- 12 and 14 have unlike denominators (2 and 4).
- 35 and 23 have unlike denominators (5 and 3).
- 78 and 12 have unlike denominators (8 and 2).
Fractions with unlike denominators represent parts of wholes that are divided into different-sized pieces. You cannot simply add or subtract the numerators until the pieces are the same size.
Why do we need a common denominator?
To add or subtract fractions, the parts being combined must be the same size. This means the fractions must have the same denominator, which is called a common denominator. Finding a common denominator allows you to rename the fractions as equivalent fractions with equal-sized pieces.
- Think about adding 12 (one half) and 14 (one quarter). Halves and quarters are different-sized pieces. To combine them, you cut the half into two quarters. Then you have 2 quarters + 1 quarter = 3 quarters.
- You cannot add 23 (thirds) and 16 (sixths) directly. You must first rename the thirds as sixths.
The common denominator is a multiple of both original denominators. Finding it is the key first step for all addition and subtraction problems with unlike denominators.
Finding a common denominator
One reliable method to find a common denominator is to find the least common multiple (LCM) of the two denominators. This LCM becomes the least common denominator (LCD). Using the LCD keeps the numbers smaller and simpler, but any common multiple will work.
- For 13 and 14, the denominators are 3 and 4. Their least common multiple is 12. So, the LCD is 12.
- For 56 and 18, list multiples: 6, 12, 18, 24... and 8, 16, 24... The LCM is 24. So, the LCD is 24.
If one denominator is a multiple of the other, the larger denominator is the LCD. For example, for 34 and 18, the LCD is 8.
Creating equivalent fractions
Once you have a common denominator, you must rewrite each original fraction as an equivalent fraction with that new denominator. To do this, multiply both the numerator and the denominator by the same number. This does not change the value of the fraction; it only changes its appearance.
- Rewrite 13 and 14 with denominator 12. Multiply 13 by 44 to get 412. Multiply 14 by 33 to get 312.
- Rewrite 25 and 110 with denominator 10. Multiply 25 by 22 to get 410. 110 already has the denominator 10, so it stays the same.
Always multiply the numerator and denominator by the same factor. The fraction you multiply by (22, 33, etc.) is equal to 1, so you are not changing the fraction's value.
Adding fractions with unlike denominators
To add fractions with unlike denominators, follow these steps: 1) Find a common denominator (preferably the LCD). 2) Rewrite each fraction as an equivalent fraction with that common denominator. 3) Add the numerators. The denominator stays the same. 4) Simplify the fraction if necessary.
-
Problem: 12 + 13
Step 1: Find the least common denominator (LCD). The denominators are 2 and 3. The LCD is the smallest number that both 2 and 3 divide into evenly, which is 6.
Step 2: Rewrite each fraction as an equivalent fraction with the denominator 6. Multiply the numerator and denominator of each fraction by the factor needed to reach 6. For 12, multiply by 3 to get 36. For 13, multiply by 2 to get 26.
Step 3: Add the numerators. The denominator stays the same. 36 + 26 = 56.
Step 4: Simplify if necessary. 56 has no common factors other than 1, so it is already in simplest form. -
Problem: 34 + 16
Step 1: Find the least common denominator (LCD). The denominators are 4 and 6. The LCD is the smallest number that both 4 and 6 divide into evenly. The multiples of 4 are 4, 8, 12, 16... and multiples of 6 are 6, 12, 18... The LCD is 12.
Step 2: Rewrite each fraction as an equivalent fraction with the denominator 12. For 34, multiply numerator and denominator by 3 to get 912. For 16, multiply numerator and denominator by 2 to get 212.
Step 3: Add the numerators. The denominator stays the same. 912 + 212 = 1112.
Step 4: Simplify if necessary. 1112 has no common factors other than 1, so it is already in simplest form.
Never add the denominators. The denominator represents the size of the pieces, which remains the same after you have a common denominator. Only the numerators (the number of pieces) are added.
Subtracting fractions with unlike denominators
Subtracting fractions with unlike denominators follows the same essential process as addition. 1) Find a common denominator. 2) Rewrite both fractions with that denominator. 3) Subtract the numerators. The denominator stays the same. 4) Simplify the fraction if necessary.
-
Problem: 56 - 14
Step 1: Find the least common denominator (LCD). The denominators are 6 and 4. The LCD is the smallest number that both 6 and 4 divide into evenly. The multiples of 6 are 6, 12, 18... and multiples of 4 are 4, 8, 12, 16... The LCD is 12.
Step 2: Rewrite each fraction as an equivalent fraction with the denominator 12. For 56, multiply numerator and denominator by 2 to get 1012. For 14, multiply numerator and denominator by 3 to get 312.
Step 3: Subtract the numerators. The denominator stays the same. 1012 - 312 = 712.
Step 4: Simplify if necessary. 712 has no common factors other than 1, so it is already in simplest form. -
Problem: 23 - 25
Step 1: Find the least common denominator (LCD). The denominators are 3 and 5. Since 3 and 5 have no common factors other than 1, the LCD is their product: 3 × 5 = 15.
Step 2: Rewrite each fraction as an equivalent fraction with the denominator 15. For 23, multiply numerator and denominator by 5 to get 1015. For 25, multiply numerator and denominator by 3 to get 615.
Step 3: Subtract the numerators. The denominator stays the same. 1015 - 615 = 415.
Step 4: Simplify if necessary. 415 has no common factors other than 1, so it is already in simplest form.
When subtracting, be careful to subtract the smaller numerator from the larger one after finding the common denominator. If the problem involves a mixed number, you may need to borrow from the whole number.
Working with mixed numbers
A mixed number contains a whole number and a fraction, such as 213. To add or subtract mixed numbers with unlike denominators, you have two options: 1) Add or subtract the whole numbers and the fractions separately, finding a common denominator for the fractions. 2) Convert each mixed number to an improper fraction, find a common denominator, and then add or subtract.
- Addition (Separate method): 112 + 213
Add whole numbers: 1 + 2 = 3.
Add fractions: LCD of 2 and 3 is 6 → 36 + 26 = 56.
Combine: 3 + 56 = 356. - Subtraction (Improper fraction method): 312 - 123
Convert: 72 and 53.
LCD of 2 and 3 is 6 → 216 - 106 = 116 = 156.
When subtracting mixed numbers, if the fraction in the first number is smaller than the fraction in the second number, you must borrow 1 from the whole number. Convert that 1 into a fraction with the common denominator and add it to the existing fraction.
Simplifying your answer
After adding or subtracting, your answer should always be in simplest form. This means the fraction has no common factors other than 1 between the numerator and denominator. If the answer is an improper fraction (numerator larger than denominator), convert it to a mixed number.
- 48 simplifies to 12 (divide numerator and denominator by 4).
- 106 is an improper fraction. Simplify by dividing 10 ÷ 6 = 1 with remainder 4 → 146. Then simplify the fraction: 123.
- 1512 simplifies to 54 (divide by 3), which is improper. Convert to 114.
Always check if your final fraction can be reduced. Look for the greatest common factor (GCF) of the numerator and denominator to simplify efficiently.
Real-world application
Adding and subtracting fractions with unlike denominators is a crucial skill for many real-life situations. You use it when cooking (measuring ingredients), building projects (measuring lengths), managing time, and handling money.
- A recipe calls for 34 cup of flour and 23 cup of sugar. The total dry ingredients are 34 + 23 = 912 + 812 = 1712 = 1512 cups.
- You have a board that is 58 of a yard long and you cut off a piece that is 14 yard. The remaining length is 58 - 28 = 38 yard.
When solving word problems, identify the operation (addition or subtraction) by looking for keywords like "total," "sum," "in all," "difference," "how much more," or "how much less."
Common Core alignment: CCSS.MATH.CONTENT.5.NF.A.1 – Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators.
Notes for teachers
This lesson is aligned with CCSS.MATH.CONTENT.5.NF.A.1, a key fourth and fifth grade standard for fraction operations. All content is 100% free, use it for whole-class instruction, independent study and practice, or homework. Encourage students to master finding the least common denominator and simplifying fractions for a strong foundation in algebra-readiness.