Addition and Subtraction of Fractions
Now that you know all about fractions — what they are, their types, and how to compare them — it is time to learn how to add and subtract them. Imagine you eat 1/4 of a pizza and your friend eats 2/4 of the same pizza. How much pizza have you both eaten together? That is addition of fractions: 1/4 + 2/4 = 3/4. And if there was 3/4 left and someone eats 1/4 more, how much is left? That is subtraction: 3/4 - 1/4 = 2/4 = 1/2. Adding and subtracting fractions is a skill you will use in cooking (adding ingredients), in time calculations (adding durations), in measurements (combining lengths), and in countless other real-life situations. The key idea is this: you can only add or subtract fractions directly when they have the same denominator (like fractions). If the denominators are different (unlike fractions), you must first convert them to like fractions by finding a common denominator. In this chapter, you will learn step-by-step methods for adding and subtracting like fractions, unlike fractions, mixed numbers, and whole numbers with fractions. With practice, this will become as easy as adding whole numbers. Let us get started.
What is Addition and Subtraction of Fractions?
Addition and Subtraction of Like Fractions:
When fractions have the same denominator (like fractions), the process is straightforward:
Addition: a/n + b/n = (a + b)/n — add the numerators, keep the denominator.
Subtraction: a/n - b/n = (a - b)/n — subtract the numerators, keep the denominator.
Example: 3/7 + 2/7 = (3 + 2)/7 = 5/7.
Example: 5/7 - 2/7 = (5 - 2)/7 = 3/7.
This works because the denominator tells you the size of each part, and the numerator tells you how many parts. If parts are the same size, you can simply count them.
Addition and Subtraction of Unlike Fractions:
When fractions have different denominators, you must first make the denominators the same:
Step 1: Find the LCM (Least Common Multiple) of the denominators.
Step 2: Convert each fraction to an equivalent fraction with the LCM as the denominator.
Step 3: Now add or subtract the numerators (since the denominators are the same).
Step 4: Simplify the result if possible.
Example: 1/3 + 1/4.
Step 1: LCM of 3 and 4 = 12.
Step 2: 1/3 = 4/12 and 1/4 = 3/12.
Step 3: 4/12 + 3/12 = 7/12.
Answer: 1/3 + 1/4 = 7/12.
Addition and Subtraction of Mixed Numbers:
Method 1 — Convert to improper fractions first:
Convert each mixed number to an improper fraction, then add or subtract as usual.
Example: 2 1/3 + 1 1/4 = 7/3 + 5/4. LCM of 3 and 4 = 12. So 28/12 + 15/12 = 43/12 = 3 7/12.
Method 2 — Add whole parts and fraction parts separately:
Add the whole numbers together, add the fractions together, then combine.
Example: 2 1/3 + 1 1/4 = (2 + 1) + (1/3 + 1/4) = 3 + 7/12 = 3 7/12.
Adding a Whole Number and a Fraction:
Simply combine them as a mixed number: 3 + 2/5 = 3 2/5.
Or convert the whole number to a fraction: 3 = 15/5, then 15/5 + 2/5 = 17/5 = 3 2/5.
Addition and Subtraction of Fractions Formula
Key Formulas for Adding and Subtracting Fractions:
1. Like Fractions (same denominator):
a/n + b/n = (a + b)/n
a/n - b/n = (a - b)/n
2. Unlike Fractions (different denominators):
a/p + b/q = (a × q + b × p) / (p × q)
a/p - b/q = (a × q - b × p) / (p × q)
Note: Using LCM instead of p × q gives the simplest possible denominator, but the formula with p × q always works and is sometimes quicker.
3. Mixed Numbers:
Convert to improper fractions, then use the formulas above.
W a/b = (W × b + a) / b
4. Simplification:
Always reduce the answer to its simplest form by dividing numerator and denominator by their HCF.
5. Converting Result:
If the result is an improper fraction, convert to a mixed number for the final answer.
Types and Properties
Let us look at all the different cases you will encounter:
1. Adding Like Fractions
This is the simplest case — just add the numerators.
2/9 + 4/9 = 6/9 = 2/3 (simplified).
Think of it as: 2 ninths + 4 ninths = 6 ninths, just like 2 apples + 4 apples = 6 apples.
2. Subtracting Like Fractions
Just subtract the numerators.
7/10 - 3/10 = 4/10 = 2/5 (simplified).
Think of pizza: 7 slices minus 3 slices = 4 slices (out of 10).
3. Adding Unlike Fractions
First find a common denominator (LCM of the denominators), convert, then add.
1/2 + 1/3: LCM(2,3) = 6. Convert: 3/6 + 2/6 = 5/6.
2/5 + 3/4: LCM(5,4) = 20. Convert: 8/20 + 15/20 = 23/20 = 1 3/20.
4. Subtracting Unlike Fractions
Same process as adding, but subtract the numerators at the end.
3/4 - 1/3: LCM(4,3) = 12. Convert: 9/12 - 4/12 = 5/12.
5/6 - 1/4: LCM(6,4) = 12. Convert: 10/12 - 3/12 = 7/12.
5. Adding Mixed Numbers
Method A (Improper fractions):
3 1/2 + 2 2/3 = 7/2 + 8/3. LCM(2,3) = 6. Convert: 21/6 + 16/6 = 37/6 = 6 1/6.
Method B (Separate parts):
Whole parts: 3 + 2 = 5.
Fraction parts: 1/2 + 2/3. LCM(2,3) = 6. 3/6 + 4/6 = 7/6 = 1 1/6.
Combine: 5 + 1 1/6 = 6 1/6. ✓ Same answer.
6. Subtracting Mixed Numbers
Method A (Improper fractions):
5 1/4 - 2 2/3 = 21/4 - 8/3. LCM(4,3) = 12. Convert: 63/12 - 32/12 = 31/12 = 2 7/12.
Method B (Separate parts) — be careful with borrowing:
Whole parts: 5 - 2 = 3.
Fraction parts: 1/4 - 2/3. LCM(4,3) = 12. 3/12 - 8/12 = -5/12 (negative!).
Since the fraction part is negative, borrow 1 from the whole part: 3 - 1 = 2, and add 12/12 to the fraction: -5/12 + 12/12 = 7/12.
Answer: 2 7/12. ✓ Same answer.
7. Adding a Whole Number and a Fraction
5 + 3/7 = 5 3/7. Or as improper: 35/7 + 3/7 = 38/7.
8. Subtracting a Fraction from a Whole Number
4 - 2/5. Convert 4 to fifths: 4 = 20/5. So 20/5 - 2/5 = 18/5 = 3 3/5.
Solved Examples
Example 1: Example 1: Adding like fractions
Problem: Find 3/8 + 5/8.
Solution:
Same denominator (8). Add the numerators:
3/8 + 5/8 = (3 + 5)/8 = 8/8 = 1.
Answer: 3/8 + 5/8 = 1 (one whole).
Example 2: Example 2: Subtracting like fractions
Problem: Find 11/15 - 4/15.
Solution:
Same denominator (15). Subtract the numerators:
11/15 - 4/15 = (11 - 4)/15 = 7/15.
Check: Is 7/15 in simplest form? HCF(7, 15) = 1. Yes, it is already simplified.
Answer: 7/15.
Example 3: Example 3: Adding unlike fractions
Problem: Find 2/3 + 1/5.
Solution:
Step 1: LCM of 3 and 5 = 15.
Step 2: Convert to equivalent fractions with denominator 15:
2/3 = (2 × 5)/(3 × 5) = 10/15.
1/5 = (1 × 3)/(5 × 3) = 3/15.
Step 3: Add: 10/15 + 3/15 = 13/15.
Check: HCF(13, 15) = 1. Already in simplest form.
Answer: 2/3 + 1/5 = 13/15.
Example 4: Example 4: Subtracting unlike fractions
Problem: Find 3/4 - 2/5.
Solution:
Step 1: LCM of 4 and 5 = 20.
Step 2: Convert:
3/4 = (3 × 5)/(4 × 5) = 15/20.
2/5 = (2 × 4)/(5 × 4) = 8/20.
Step 3: Subtract: 15/20 - 8/20 = 7/20.
Answer: 3/4 - 2/5 = 7/20.
Example 5: Example 5: Adding mixed numbers
Problem: Find 2 1/3 + 3 2/5.
Solution:
Method — Convert to improper fractions:
2 1/3 = (2 × 3 + 1)/3 = 7/3.
3 2/5 = (3 × 5 + 2)/5 = 17/5.
LCM of 3 and 5 = 15.
7/3 = 35/15.
17/5 = 51/15.
Add: 35/15 + 51/15 = 86/15.
Convert to mixed number: 86 ÷ 15 = 5 remainder 11. So 86/15 = 5 11/15.
Answer: 2 1/3 + 3 2/5 = 5 11/15.
Example 6: Example 6: Subtracting mixed numbers
Problem: Find 5 1/2 - 2 3/4.
Solution:
Convert to improper fractions:
5 1/2 = 11/2.
2 3/4 = 11/4.
LCM of 2 and 4 = 4.
11/2 = 22/4.
11/4 = 11/4.
Subtract: 22/4 - 11/4 = 11/4.
Convert to mixed number: 11/4 = 2 3/4.
Answer: 5 1/2 - 2 3/4 = 2 3/4.
Example 7: Example 7: Subtracting a fraction from a whole number
Problem: Find 6 - 2/3.
Solution:
Convert 6 to a fraction with denominator 3:
6 = 18/3.
Subtract: 18/3 - 2/3 = 16/3.
Convert to mixed number: 16/3 = 5 1/3.
Answer: 6 - 2/3 = 5 1/3.
Example 8: Example 8: Pizza sharing word problem
Problem: Rahul ate 1/4 of a pizza, Priya ate 1/3 of the same pizza, and Sneha ate 1/6. How much pizza was eaten in total? How much is left?
Solution:
Total eaten: 1/4 + 1/3 + 1/6.
LCM of 4, 3, 6 = 12.
1/4 = 3/12, 1/3 = 4/12, 1/6 = 2/12.
Total eaten = 3/12 + 4/12 + 2/12 = 9/12 = 3/4.
Pizza left: 1 - 3/4 = 4/4 - 3/4 = 1/4.
Answer: 3/4 of the pizza was eaten. 1/4 is left.
Example 9: Example 9: Cooking measurement problem
Problem: A recipe needs 2 1/2 cups of flour and 1 3/4 cups of sugar. How much more flour than sugar is needed?
Solution:
Difference = 2 1/2 - 1 3/4.
Convert to improper fractions:
2 1/2 = 5/2.
1 3/4 = 7/4.
LCM of 2 and 4 = 4.
5/2 = 10/4.
7/4 = 7/4.
Subtract: 10/4 - 7/4 = 3/4.
Answer: 3/4 cup more flour than sugar is needed.
Example 10: Example 10: Distance word problem
Problem: Arun walked 3/5 km to school and then 1/4 km from school to the library. What is the total distance he walked?
Solution:
Total distance = 3/5 + 1/4.
LCM of 5 and 4 = 20.
3/5 = 12/20.
1/4 = 5/20.
Total = 12/20 + 5/20 = 17/20 km.
Answer: Arun walked 17/20 km in total.
Real-World Applications
Adding and subtracting fractions is used in countless everyday situations:
Cooking and Baking: When you combine ingredients from a recipe, you add fractions. Half a cup of oil plus one-third cup of vinegar for a dressing requires adding 1/2 + 1/3. When you want less salt than the recipe says, you subtract — 1 teaspoon minus 1/4 teaspoon = 3/4 teaspoon. Doubling or halving recipes involves these fraction operations constantly.
Time Calculations: Adding durations often requires fraction addition. If one task takes 1 1/4 hours and another takes 2 1/2 hours, the total time is 1 1/4 + 2 1/2 = 3 3/4 hours. Scheduling activities, planning travel, and managing deadlines all use fraction addition and subtraction.
Measurement and Construction: Carpenters frequently add and subtract fractional measurements. If a shelf needs to be 3 1/4 feet long and the board is 5 1/2 feet, how much must be cut off? 5 1/2 - 3 1/4 = 2 1/4 feet. Tailors, architects, and engineers do this constantly.
Money and Shopping: Calculating total costs and change often involves fractions. If an item costs Rs. 12 1/2 and another costs Rs. 8 3/4, the total is Rs. 21 1/4. Finding the difference between sale price and original price involves fraction subtraction.
Academic Performance: Combining test scores from different sections requires fraction addition. If a student scores 3/5 on Section A and 7/10 on Section B, the overall performance requires adding these fractions. Report cards sometimes express performance as fractions of total marks.
Sports and Fitness: Tracking distances — jogging 2 1/2 km in the morning and 1 3/4 km in the evening means running 4 1/4 km total. Nutrition tracking also uses fractions: eating 1/3 of daily protein at breakfast and 2/5 at lunch means 1/3 + 2/5 = 11/15 has been consumed, leaving 4/15 for dinner.
Key Points to Remember
- Like fractions (same denominator): add or subtract the numerators, keep the denominator. Example: 3/7 + 2/7 = 5/7.
- Unlike fractions (different denominators): find LCM of denominators, convert to equivalent like fractions, then add or subtract.
- Always simplify the result. Divide numerator and denominator by their HCF.
- If the result is an improper fraction, convert to a mixed number for the final answer.
- For mixed numbers: convert to improper fractions first, then add or subtract, then convert back to mixed numbers.
- Alternatively, add/subtract whole parts and fraction parts separately (but watch for borrowing in subtraction).
- To subtract a fraction from a whole number: convert the whole number to a fraction with the same denominator.
- The cross-multiplication shortcut: a/p + b/q = (aq + bp) / pq. This always works but may not give the simplest denominator.
- Check your answer: the result of adding two proper fractions could be proper or improper. The result of subtracting should be less than the first fraction.
- In word problems, identify what needs to be added or subtracted, convert all fractions to the same denominator, and simplify.
Practice Problems
- Find: (a) 2/7 + 3/7, (b) 9/11 - 4/11, (c) 5/13 + 6/13.
- Find: (a) 1/2 + 1/3, (b) 3/4 - 1/6, (c) 2/5 + 3/10.
- Add: 2 1/4 + 3 2/3.
- Subtract: 4 1/3 - 1 5/6.
- Find: 8 - 3/5.
- A tank is 2/5 full. After adding more water, it becomes 4/5 full. What fraction of the tank was filled?
- Meera walked 3/4 km, then 1/2 km, then 2/3 km. What is the total distance?
- A recipe needs 1 1/2 cups of wheat flour and 2/3 cup of gram flour. Find the total flour needed.
Frequently Asked Questions
Q1. How do you add fractions with different denominators?
First, find the LCM (Least Common Multiple) of the denominators. Convert each fraction to an equivalent fraction with this LCM as the new denominator. Then add the numerators and keep the common denominator. For example, 1/3 + 1/4: LCM of 3 and 4 is 12. So 1/3 = 4/12 and 1/4 = 3/12. Adding: 4/12 + 3/12 = 7/12.
Q2. Why can you only add fractions with the same denominator?
The denominator tells you the size of each part. You can only add parts that are the same size — just like you can add 3 apples + 2 apples = 5 apples, but you cannot directly add 3 apples + 2 oranges without a common unit. When fractions have the same denominator, the parts are the same size and can be counted together. When denominators differ, the parts are different sizes and must be made equal first.
Q3. How do you subtract mixed numbers?
The safest method is to convert both mixed numbers to improper fractions, find a common denominator, subtract, and convert back. For example, 4 1/3 - 2 3/4: convert to 13/3 - 11/4, LCM = 12, so 52/12 - 33/12 = 19/12 = 1 7/12.
Q4. What is the LCM and why do we need it?
LCM stands for Least Common Multiple — the smallest number that is a multiple of both denominators. We need it because it gives us the smallest common denominator, making calculations easier. For denominators 4 and 6, the LCM is 12 (not 24, which also works but gives larger numbers). Finding the LCM keeps fractions as small and manageable as possible.
Q5. Should I always simplify my answer?
Yes, always simplify (reduce) your answer to its simplest form. Divide both the numerator and denominator by their HCF (Highest Common Factor). For example, if your answer is 6/8, simplify to 3/4 (divide both by 2). If your answer is an improper fraction, also convert it to a mixed number. A simplified answer is always preferred in mathematics.
Q6. Can I use cross multiplication for adding fractions?
Yes! For a/b + c/d, you can use the shortcut: (a x d + c x b) / (b x d). For example, 2/3 + 1/4 = (2 x 4 + 1 x 3) / (3 x 4) = (8 + 3) / 12 = 11/12. This always works but may give a denominator larger than the LCM. You might need to simplify afterwards.
Q7. How do I subtract a fraction from a whole number?
Convert the whole number to a fraction with the same denominator as the fraction you are subtracting. For example, 5 - 2/3: convert 5 to thirds: 5 = 15/3. Then 15/3 - 2/3 = 13/3 = 4 1/3. In general, n - a/b = (nb - a)/b.
Q8. What if the answer comes out as an improper fraction?
If your answer is an improper fraction (numerator >= denominator), convert it to a mixed number. Divide the numerator by the denominator: the quotient is the whole part and the remainder is the new numerator. For example, 17/5 = 3 2/5 (because 17 divided by 5 = 3 remainder 2). In exams, mixed numbers are usually the preferred final answer.
Q9. Is 1/2 + 1/3 equal to 2/5?
No! This is one of the most common mistakes. You CANNOT add numerators and denominators separately. 1/2 + 1/3 is NOT 2/5. The correct answer is: LCM of 2 and 3 = 6, so 1/2 = 3/6 and 1/3 = 2/6, giving 3/6 + 2/6 = 5/6. Always find a common denominator first.
Q10. How do you check if your fraction addition or subtraction is correct?
For addition: subtract one of the original fractions from your answer — you should get the other original fraction. For example, if 2/3 + 1/4 = 11/12, then 11/12 - 2/3 should equal 1/4. Check: 11/12 - 8/12 = 3/12 = 1/4. Correct! For subtraction: add the result to the fraction you subtracted — you should get the fraction you started with.
