Tri-Fractional Operations: How to Add and Subtract Three Fractions with Different Denominators

In this guide, we’ll walk you through the steps to add or subtract three fractions, even when they have different denominators.

Step-by-step Guide to Add and Subtract Three Fractions with Different Denominators:

1. Basics of Fractions: 

Recall that a fraction consists of a numerator (top number) and a denominator (bottom number). The denominator indicates the total number of equal parts, while the numerator tells us how many of those parts we’re considering.

2. Identifying Different Denominators: 

If the fractions you’re working with don’t have the same denominator, they have different denominators. For instance, in the fractions \(\frac{1}{2}\), \(\frac{3}{4}\), and \(\frac{5}{6}\), the denominators 2, 4, and 6 are all different.

3. Finding the Least Common Denominator (LCD): 

The LCD is the smallest number into which all the denominators can divide. This ensures that the fractions are of comparable sizes.

4. Adjusting Each Fraction to the LCD: 

Multiply the numerator and denominator of each fraction by the necessary factor to achieve the LCD.

5. Performing the Operation: 

With the same denominator in place, either add or subtract the numerators of the fractions to get the final result.

Example 1 (Addition): 

Add \(\frac{1}{3}\), \(\frac{1}{4}\), and \(\frac{1}{5}\). 

Solution: 

The LCD for 3, 4, and 5 is 60. Adjusting the fractions:

– \(\frac{1}{3}\) becomes \(\frac{20}{60}\).

– \(\frac{1}{4}\) becomes \(\frac{15}{60}\).

– \(\frac{1}{5}\) becomes \(\frac{12}{60}\). 

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Adding them up, the result is \(\frac{47}{60}\).

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Example 2 (Subtraction): 

Subtract \(\frac{1}{6}\) and \(\frac{1}{8}\) from \(\frac{1}{4}\). 

Solution: 

The LCD for 6, 8, and 4 is 24. Adjusting the fractions:

– \(\frac{1}{6}\) becomes \(\frac{4}{24}\).

– \(\frac{1}{8}\) becomes \(\frac{3}{24}\).

– \(\frac{1}{4}\) becomes \(\frac{6}{24}\). 

Subtracting, the result is \(\frac{6 – 4 – 3}{24} = \frac{-1}{24}\).

Practice Questions: 

1. Add \(\frac{1}{7}\), \(\frac{2}{9}\), and \(\frac{3}{11}\).

2. Subtract \(\frac{2}{8}\) and \(\frac{3}{12}\) from \(\frac{1}{6}\).

3. Add \(\frac{1}{10}\), \(\frac{2}{15}\), and \(\frac{3}{20}\).

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Answers: 

1. \(\frac{442}{693}\)

2. \(-\frac{1}{3}\)

3. \(\frac{23}{60}\)

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Subtracting Three Fractions: A Full Walkthrough

Let us solve 7/8 − 1/3 − 1/6 one step at a time.

1. Find the LCD of 8, 3 and 6. Multiples of 8 are 8, 16, 24. Both 3 and 6 divide 24, so the LCD is 24.
2. Rewrite each fraction. 7/8 = 21/24 (multiply top and bottom by 3). 1/3 = 8/24 (multiply by 8). 1/6 = 4/24 (multiply by 4).
3. Subtract the numerators from left to right. 21 − 8 = 13, then 13 − 4 = 9. The result is 9/24.
4. Simplify. 9 and 24 share a factor of 3, so 9/24 = 3/8.

A shortcut that also works: subtracting two fractions in a row is the same as subtracting their sum. So 7/8 − 1/3 − 1/6 = 7/8 − (1/3 + 1/6). Since 1/3 + 1/6 = 1/2, the problem becomes 7/8 − 4/8 = 3/8. Same answer.

Watch out for this mistake: 7/8 − 1/3 − 1/6 is not the same as 7/8 − (1/3 − 1/6). Each minus sign belongs only to the fraction right after it. Do the steps left to right unless parentheses tell you otherwise.

Mixed signs: for 2/3 + 3/4 − 1/2, use the LCD 12. That gives 8/12 + 9/12 − 6/12 = 11/12. If the result ever comes out negative, as in 1/4 − 1/6 − 1/8 = −1/24, keep the minus sign in front of the fraction.

Recommended EffortlessMath Books

For a complete fractions workbook that drills three-fraction operations into a full skill set, the Grade 5 Common Core Math for Beginners (resource currently unavailable) walks through every grade-5 fraction topic with worked examples. For state-test prep, the Grade 6 Common Core Math for Beginners (resource currently unavailable) picks up the topic with the more advanced applications students see in grade 6.

Three Ways to Find the LCD of Three Denominators

Finding one denominator that fits all three fractions is the step that slows most students down. Pick whichever of these methods feels easiest for the numbers in front of you.

1. Check multiples of the largest denominator

Start with the biggest denominator and count up by it until the other two divide in evenly. For 4, 6 and 8, the multiples of 8 are 8, 16, 24. Both 4 and 6 divide 24, so the LCD is 24.

2. List multiples of each denominator

Write a few multiples of each number and look for the first one on all three lists.

• 3: 3, 6, 9, 12, 15
• 4: 4, 8, 12, 16
• 6: 6, 12, 18

The first number on all three lists is 12.

3. Use prime factors

Break each denominator into primes, then take each prime the greatest number of times it appears in any one denominator.

The most 2s in any row is two, and the most 3s is two. So the LCD is 2 × 2 × 3 × 3 = 36.

Any common denominator gives the right answer, even 4 × 6 × 9 = 216. The LCD just keeps the numbers small and saves simplifying at the end.

Practice Quiz: 10 Questions

Frequently Asked Questions

How do you add and subtract three fractions with different denominators?

Find the least common denominator of all three fractions, rewrite each fraction with that denominator, then combine the numerators from left to right. For 1/2 + 1/3 − 1/4, the LCD is 12, so you get 6/12 + 4/12 − 3/12 = 7/12. Finish by simplifying if the numerator and denominator share a factor.

Can you add fractions with different denominators?

Yes, but not directly. First rewrite the fractions so they share a common denominator, because you can only add pieces of the same size. For example, 1/3 + 1/6 becomes 2/6 + 1/6 = 3/6, which simplifies to 1/2. The same idea works when a problem has three fractions, as long as the common denominator fits all three of them.

How do you add and subtract 3 or more fractions with unlike denominators?

Find one denominator that every denominator divides into evenly. A quick way is to start with the largest denominator and check its multiples. For 2/3 + 1/4 − 5/6, the multiples of 6 are 6, 12, 18, and 12 is the first one that 3 and 4 also divide, so the problem becomes 8/12 + 3/12 − 10/12 = 1/12.

How do you add and subtract three mixed fractions with unlike denominators?

Change each mixed number to an improper fraction, then follow the usual steps. For 1 1/2 + 2 1/3 − 1 1/4, rewrite the problem as 3/2 + 7/3 − 5/4. The LCD of 2, 3 and 4 is 12, giving 18/12 + 28/12 − 15/12 = 31/12, which is 2 7/12 as a mixed number.

What is 3/12 + 2/4?

3/12 + 2/4 equals 3/4. The LCD of 12 and 4 is 12, so rewrite 2/4 as 6/12. Then 3/12 + 6/12 = 9/12, and dividing the top and bottom by 3 simplifies it to 3/4. If you add a third fraction such as 1/6, rewrite it as 2/12 as well: 9/12 + 2/12 = 11/12.

Where can I find worksheets for adding and subtracting 3 fractions with unlike denominators?

Effortless Math has fraction worksheets and grade 5 and grade 6 workbooks with this kind of practice. Begin with sets where one denominator is a multiple of the others, like 1/2 + 1/4 + 1/8, so the LCD is easy to spot. Then move on to problems such as 1/3 + 1/4 − 1/6, where you need to build the LCD of 12 yourself.

How do I find the LCD of three denominators?

Start with the largest denominator and check its multiples until the other two divide in evenly. For 4, 6 and 9, the multiples of 9 are 9, 18, 27, 36, and 36 is the first one that 4 and 6 also divide. You can also multiply the highest power of each prime factor: 2 × 2 × 3 × 3 = 36.

What if two of the denominators are the same?

Then you only need a common denominator for two different numbers. For 1/4 + 3/4 − 1/6, the denominators are 4 and 6, so the LCD is 12. Rewrite as 3/12 + 9/12 − 2/12 = 10/12, which simplifies to 5/6. You can also combine the matching fractions first: 1/4 + 3/4 = 1, then 1 − 1/6 = 5/6.

Can I do the operations in any order?

Only when every sign is a plus. Addition can be done in any order, so 1/2 + 1/3 + 1/6 always equals 1. With subtraction, work from left to right. In 3/4 − 1/2 − 1/8, going left to right gives 1/4 − 1/8 = 1/8. Doing the last two first would wrongly give 3/4 − 3/8 = 3/8.

How do I handle a negative result?

A negative answer is fine when you take away more than you started with. Subtract the numerators as usual and keep the sign. For 1/4 − 1/6 − 1/8, the LCD is 24, so 6/24 − 4/24 − 3/24 = −1/24. Write the answer as −1/24, with the minus sign in front of the whole fraction.

Related EffortlessMath Lessons

If a topic on this page feels rusty, these short lessons go deeper:

• Add fractions with unlike denominators
• Fraction word problems
• Break fractions and mixed numbers apart
• Multiplying fractions and whole numbers
• Free Grade 5 FSA Math practice test