Operations on Fractions – JSS1 Mathematics Lesson Note (First Term)

Operations on fractions are the four ways of combining fractions: addition, subtraction, multiplication and division.

Behavioural objectives

By the end of the lesson, students should be able to:

1. add and subtract fractions with the same denominator;
2. add and subtract mixed numbers by using the LCM of the denominators;
3. multiply fractions, including mixed numbers changed to improper fractions;
4. divide one fraction by another by multiplying by the inverse.

What is a fraction?

A fraction is a number that shows part of a whole, written as one number over another.

• Numerator: the top number. It shows how many parts are taken.
• Denominator: the bottom number. It shows how many equal parts make the whole.
• Example: in 3/5, the numerator is 3 and the denominator is 5.

What are the types of fractions?

There are three types of fractions: proper fractions, improper fractions and mixed numbers.

How do you change between mixed numbers and improper fractions?

You change a mixed number to an improper fraction by multiplying the whole number by the denominator and adding the numerator.

Mixed number to improper fraction

1. Multiply the whole number by the denominator.
2. Add the numerator to the answer.
3. Write the result over the same denominator.

• Example: 3 5/17 = (3 × 17 + 5)/17 = 56/17.

Improper fraction to mixed number

1. Divide the numerator by the denominator.
2. The quotient is the whole number.
3. The remainder is the new numerator. Keep the denominator.

• Example: 247/24: 247 ÷ 24 = 10 remainder 7, so 247/24 = 10 7/24.

What is the LCM of denominators?

The LCM (lowest common multiple) of numbers is the smallest number that each of them divides into exactly.

• Multiples of 4: 4, 8, 12, 16, 20, 24.
• Multiples of 8: 8, 16, 24.
• Multiples of 12: 12, 24.
• LCM of 4, 8 and 12: 24, the first number found in all three lists.

How do you add and subtract fractions?

To add or subtract fractions, make the denominators the same, then add or subtract the numerators.

Same denominators

1. Add or subtract the numerators.
2. Keep the denominator unchanged.
3. Simplify the answer if you can.

Different denominators

1. Find the LCM of the denominators.
2. Change each fraction to an equivalent fraction with the LCM as denominator.
3. Add or subtract the numerators.
4. Simplify, or change an improper fraction to a mixed number.

Example 1

Find the value of the following: (a) 4/9 + 5/9 + 7/9 (b) 5/8 – 3/8 (c) 1 + 4/7

Solution

• (a) 4/9 + 5/9 + 7/9 = 16/9. The denominators are the same, so add the numerators together.
• (b) 5/8 – 3/8 = 2/8 = 1/4. The denominator is the same, so subtract the numerators.
• (c) Try this one yourself.

Example 2

Simplify the following: (a) 33/4 + 5/8 + 17/12 (b) 2 + 4 2/5 – 5 3/5

Solution

(a) Find the LCM of the denominators 4, 8 and 12. The LCM is 24.

• 33/4 = 198/24 (24 ÷ 4 = 6, and 33 × 6 = 198).
• 5/8 = 15/24 (24 ÷ 8 = 3, and 5 × 3 = 15).
• 17/12 = 34/24 (24 ÷ 12 = 2, and 17 × 2 = 34).
• Add: (198 + 15 + 34)/24 = 247/24.
• Change to a mixed number: 247/24 = 10 7/24.

(b) Separate the whole numbers from the fractions.

• Whole numbers: 2 + 4 – 5 = 1.
• Fractions (same denominator): 2/5 – 3/5 = -1/5.
• Answer: 1 – 1/5 = 4/5.

How do you multiply fractions?

To multiply fractions, multiply the numerators together and multiply the denominators together.

1. Change any mixed number to an improper fraction.
2. Cancel any numerator and denominator that share a common factor.
3. Multiply the numerators, then multiply the denominators.
4. Change an improper answer to a mixed number.

How do you divide fractions?

To divide by a fraction, change the division sign to multiplication and turn the second fraction upside down.

• Inverse (reciprocal): a fraction turned upside down. The inverse of 4/9 is 9/4.
• Rule: a/b ÷ c/d = a/b × d/c.
• Only the second fraction is turned: the first fraction stays as it is.

Example 3

Simplify the following:

1. 4/5 × 2/3
2. Find the product of 3 5/17, 2 5/6 and 4/8.
3. 3/5 ÷ 4/9
4. 2 4/5 × 5 6/8 ÷ 5 1/9

Solution

(a) 4/5 × 2/3 = 8/15. Multiply the numerators together and the denominators together.

(b) Change the mixed numbers to improper fractions.

• 3 5/17 × 2 5/6 × 4/8 = 56/17 × 17/6 × 4/8.
• The 17s cancel. 4/8 becomes 1/2. 56 and 6 are both divisible by 2.
• Answer: 14/3 = 4 2/3.

(c) Try this one yourself.

(d) Change the mixed numbers to improper fractions.

• Change the division sign to multiplication and turn the last fraction upside down.
• 2 4/5 × 5 6/8 ÷ 5 1/9 = 14/5 × 46/8 × 9/46.
• Cancel the 46s.
• Answer: 63/20 = 3 3/20.

What are the rules for the four operations?

What mistakes should you avoid?

• Adding denominators: 1/5 + 2/5 is 3/5, not 3/10.
• Skipping the LCM: fractions with different denominators cannot be added directly.
• Turning the wrong fraction: in a division, only the second fraction is inverted.
• Cancelling across addition: cancel only when fractions are being multiplied.
• Forgetting to simplify: always reduce the answer or write it as a mixed number.

Evaluation

1. How do you add fractions that have the same denominator?
2. What do you do before adding fractions with different denominators?
3. Solve Example 1(c) under addition and subtraction: 1 + 4/7.
4. Solve 3/5 ÷ 4/9, and state how you changed the division sign.

Assignment

Simplify each of the following, showing your working:

1. 3/7 + 2/7
2. 5/6 – 1/4
3. 2/3 × 9/10
4. 3/4 ÷ 5/8