Basic 7 · Mathematics · Fractions
Adding and Subtracting Unlike and Mixed Fractions
In this topic you will explain and carry out the addition and subtraction of two or three unlike and mixed fractions, and solve real-life problems involving these operations.
By the end of these notes you should be able to:
- Add two or three unlike fractions using the LCM of the denominators
- Subtract unlike and mixed fractions, including with regrouping
- Solve word problems involving addition and subtraction of fractions
What you already know
- Equivalent fractions and simplest form
- Multiples and LCM (Upper Primary)
Copy everything that has a blue line on its left into your notebook. The other parts are for reading.
Lesson 1
- Strand
- Number
- Sub-strand
- Fractions
- Indicator
- B7.1.3.2.1 – Explain the process of addition and subtraction of two or three unlike and mixed fractions
Introduction
You cannot add ½ and ⅓ by adding the tops and bottoms. First, the pieces must be the same size. We do this by finding a common denominator.
1. Like and Unlike Fractions
Like fractions have the same denominator, such as ⅖ and ⅗. Unlike fractions have different denominators, such as ½ and ⅓. Like fractions are added by adding the numerators: ⅖ + ⅗ = 5/5 = 1.
2. Using the LCM
- Find the LCM (lowest common multiple) of the denominators.
- Change each fraction to an equivalent fraction with the LCM as denominator.
- Add the numerators and keep the denominator.
- Simplify the answer and change improper fractions to mixed numbers.
Add two unlike fractions
Calculate ½ + ⅓.
- LCM of 2 and 3 is 6.
- ½ = 3/6 and ⅓ = 2/6.
- 3/6 + 2/6 = 5/6.
Answer: 5/6
Add three unlike fractions
Calculate ¾ + ⅚ + ⅔.
- LCM of 4, 6 and 3 is 12.
- ¾ = 9/12; ⅚ = 10/12; ⅔ = 8/12.
- 9 + 10 + 8 = 27, so 27/12 = 9/4.
Answer: 2¼
3. Adding Mixed Numbers
Add mixed numbers
Calculate 2⅓ + 1¾.
- Add whole numbers: 2 + 1 = 3.
- Add fractions: ⅓ + ¾ = 4/12 + 9/12 = 13/12 = 1 1/12.
- Total: 3 + 1 1/12.
Answer: 4 1/12
4. Why the Same Denominator?
Denominators show the size of the pieces. A half is bigger than a third, so we cannot simply count them together. Changing both into sixths makes the pieces the same size, so they can be counted.
New words
- unlike fractions
- Fractions with different denominators.
- LCM
- The lowest common multiple of numbers.
- common denominator
- A denominator shared by fractions.
- mixed number
- A whole number and a fraction together, such as 2⅓.
Summary
- Unlike fractions are added by first changing them to a common denominator using the LCM.
- Add the numerators, then simplify and convert improper fractions to mixed numbers.
- For mixed numbers, add the whole numbers and the fractions separately, then combine.
Class activity: Fraction Strip Addition
- Use fraction strips to show ½ + ¼ and ⅓ + ⅙.
- Check your results by calculation using the LCM.
- Make up and solve two more additions.
You need: Fraction strips, Bottle tops, Graph paper
Exercises
Answer in your exercise book first, then check your answers.
1. ¼ + ½ =
- A. ¾
- B. ⅔
- C. ⅙
- D. 2/6
Show answer
Answer: ¾¼ + 2/4.2. ⅖ + ⅓ =
- A. 11/15
- B. 3/8
- C. 2/15
- D. 5/15
Show answer
Answer: 11/156/15 + 5/15.3. To add unlike fractions, we first find a common denominator.
- A. True
- B. False
Show answer
Answer: TrueSame-size pieces.4. The lowest common multiple is written as ______.
Show answer
Answer: LCMLowest common multiple.5. Calculate 1½ + 2⅖.
Show answer
Answer: 3 9/10½ + ⅖ = 5/10 + 4/10.6. Calculate ⅝ + ¾ + ½ and give the answer as a mixed number.
Show answer
Answer: 1⅞5/8 + 6/8 + 4/8 = 15/8.
Lesson 2
- Strand
- Number
- Sub-strand
- Fractions
- Indicator
- B7.1.3.2.1 – Explain the process of addition and subtraction of two or three unlike and mixed fractions
Introduction
If you have ¾ of a bag of rice and use ⅓ of the bag, how much is left? Subtraction of fractions follows the same idea as addition: make the pieces the same size first.
1. Subtracting Unlike Fractions
Subtract
Calculate ¾ − ⅓.
- LCM of 4 and 3 is 12.
- ¾ = 9/12 and ⅓ = 4/12.
- 9/12 − 4/12 = 5/12.
Answer: 5/12
2. Subtracting Mixed Numbers
Subtract mixed numbers
Calculate 5⅚ − 2¼.
- Whole numbers: 5 − 2 = 3.
- Fractions: ⅚ − ¼ = 10/12 − 3/12 = 7/12.
Answer: 3 7/12
3. Subtracting with Regrouping
Regroup
Calculate 4⅕ − 1⅔.
- Fractions: ⅕ = 3/15 and ⅔ = 10/15. 3/15 is smaller than 10/15.
- Regroup: 4 3/15 = 3 + 1 + 3/15 = 3 18/15.
- Now 3 18/15 − 1 10/15 = 2 8/15.
Answer: 2 8/15
4. Another Method: Improper Fractions
You can also change mixed numbers to improper fractions first: 4⅕ = 21/5 and 1⅔ = 5/3. Then 21/5 − 5/3 = 63/15 − 25/15 = 38/15 = 2 8/15. Both methods give the same answer; choose the one you find easier.
5. Checking Your Answer
Check subtraction by addition: 2 8/15 + 1⅔ = 2 8/15 + 1 10/15 = 3 18/15 = 4 3/15 = 4⅕. The answer is correct.
6. Common Errors
A common error is to subtract the smaller fraction from the larger one regardless of order, for example writing ⅔ − ⅕ for 4⅕ − 1⅔. Another is forgetting to change the whole number after regrouping. Write each step clearly and check by addition.
New words
- improper fraction
- A fraction with a numerator larger than its denominator.
- regroup
- To change 1 whole into a fraction, such as 1 = 15/15.
- difference
- The result of a subtraction.
- convert
- To change from one form to another.
Summary
- Subtract unlike fractions after changing them to a common denominator.
- For mixed numbers, regroup a whole when the first fraction is smaller.
- Improper fractions offer another method; check by addition.
Class activity: Subtraction Stories
- Write three subtraction problems about sharing food or cloth using fractions.
- Solve them using both methods.
- Check each answer by addition.
You need: Fraction strips, Bottle tops, Graph paper
Exercises
Answer in your exercise book first, then check your answers.
1. ⅚ − ½ =
- A. ⅓
- B. ⅔
- C. ⅙
- D. 4/6
Show answer
Answer: ⅓5/6 − 3/6 = 2/6.2. 3½ − 1¾ =
- A. 1¾
- B. 2¼
- C. 1¼
- D. 2¾
Show answer
Answer: 1¾Regroup.3. Subtraction of fractions can be checked by addition.
- A. True
- B. False
Show answer
Answer: TrueInverse operations.4. A fraction with a numerator larger than the denominator is ______.
Show answer
Answer: impropere.g. 21/5.5. Calculate 6⅜ − 2⅚.
Show answer
Answer: 3 13/246 9/24 − 2 20/24 → 5 33/24 − 2 20/24.6. Explain when you need to regroup in subtracting mixed numbers.
Show answer
Answer: When the fraction in the first number is smaller than the fraction being subtracted, change 1 whole into a fraction.Regrouping rule.
Lesson 3
- Strand
- Number
- Sub-strand
- Fractions
- Indicator
- B7.1.3.2.2 – Solve problems involving addition or subtraction of fractions
Introduction
Fractions appear in cooking, farming, sewing and sharing. Solving real problems shows why these skills matter.
1. Steps for Fraction Word Problems
- Read the problem and identify the fractions.
- Decide whether to add or subtract (or both).
- Use the LCM to find a common denominator.
- Calculate and simplify.
- Write the answer with units and check that it makes sense.
2. Worked Problems
Farm problem
A farmer planted ⅖ of his land with maize and ⅓ with cassava. What fraction of the land is planted? What fraction is left?
- Planted: ⅖ + ⅓ = 6/15 + 5/15 = 11/15.
- Left: 1 − 11/15 = 4/15.
Answer: 11/15 planted and 4/15 left
Cloth problem
Esi bought 5½ m of cloth. She used 2¾ m for a dress and 1⅔ m for a blouse. How much cloth remained?
- Used: 2¾ + 1⅔ = 2 9/12 + 1 8/12 = 3 17/12 = 4 5/12 m.
- Remained: 5½ − 4 5/12 = 5 6/12 − 4 5/12 = 1 1/12 m.
Answer: 1 1/12 m
3. Time Problems
Time spent
Kwame spends ⅓ of his day sleeping and ¼ of his day at school. What fraction of the day is left for other activities?
- ⅓ + ¼ = 4/12 + 3/12 = 7/12.
- 1 − 7/12 = 5/12.
Answer: 5/12 of the day (10 hours)
4. Making Sense of Answers
A fraction of something cannot be more than the whole, unless there is more than one whole. If your answer for land left is more than 1, check your working. Always write units such as metres, hours or bags.
New words
- word problem
- A mathematics question in a real-life setting.
- whole
- The complete amount, written as 1.
- remainder
- What is left after some has been used.
- units
- Labels such as metres or hours.
Summary
- Identify the fractions and whether to add, subtract or both.
- Use common denominators, simplify and include units.
- Check that answers make sense compared with the whole.
Class activity: Fraction Problem Fair
- Write two real-life fraction problems from your home or community.
- Swap with another group and solve.
- Present one solution to the class.
You need: Fraction strips, Bottle tops, Graph paper
Exercises
Answer in your exercise book first, then check your answers.
1. A tank is ¾ full. ⅓ of a tank is used. What fraction remains?
- A. 5/12
- B. ½
- C. ¼
- D. ⅔
Show answer
Answer: 5/129/12 − 4/12.2. Ama read ⅕ of a book on Monday and ⅖ on Tuesday. What fraction has she read?
- A. ⅗
- B. ⅖
- C. 2/10
- D. 3/10
Show answer
Answer: ⅗Like fractions.3. The whole is written as 1.
- A. True
- B. False
Show answer
Answer: TrueAll parts together.4. Labels such as metres or hours are called ______.
Show answer
Answer: unitsMeasurement labels.5. A farmer used ⅜ of his money on seeds and ¼ on fertiliser. What fraction did he spend?
Show answer
Answer: ⅝3/8 + 2/8.6. Kofi had 3½ bags of cement. He used 1⅔ bags on Monday and ¾ bag on Tuesday. How much is left?
Show answer
Answer: 1 1/12 bags3 6/12 − (1 8/12 + 9/12) = 3 6/12 − 2 5/12.
Homework
- Solve three addition and three subtraction problems with mixed numbers.
- Write a fraction word problem about your home and solve it.