Adding fractions with different denominators - Tips for Teachers by Craig Barton

Add and Subtract Proper Fractions with Different Denominators

Atoms of Knowledge

(F) Common Denominator – When two fractions have the same denominator, they have a common denominator

(R) Finding Equivalent Fractions – Multiply both numerator and denominator by the same number

(R) Finding the Lowest Common Multiple (LCM) – The smallest number that both denominators divide into

(R) Adding Fractions with the Same Denominator – Add the numerators and keep the denominator the same

(R) Subtracting Fractions with the Same Denominator – Subtract the numerators and keep the denominator the same

(T) Simplifying Fractions – Dividing both numerator and denominator by their highest common factor

I Do – Worked Examples

Example 1

\(\frac{1}{3} + \frac{1}{4}\)

Step 1: Find the lowest common multiple (LCM) of the denominators
Multiples of 3: 3, 6, 9, 12, 15…
Multiples of 4: 4, 8, 12, 16…
LCM = 12

We need to find a common denominator so we can add the fractions. The LCM gives us the smallest common denominator possible.

Step 2: Convert \(\frac{1}{3}\) to an equivalent fraction with denominator 12
\(3 \times 4 = 12\), so multiply both parts by 4
\(\frac{1 \times 4}{3 \times 4} = \frac{4}{12}\)

To change the denominator from 3 to 12, we multiply by 4. We must multiply the numerator by 4 as well to keep the fraction’s value the same.

Step 3: Convert \(\frac{1}{4}\) to an equivalent fraction with denominator 12
\(4 \times 3 = 12\), so multiply both parts by 3
\(\frac{1 \times 3}{4 \times 3} = \frac{3}{12}\)

To change the denominator from 4 to 12, we multiply by 3. Again, we multiply the numerator by the same number.

Step 4: Add the fractions with the same denominator
\(\frac{4}{12} + \frac{3}{12} = \frac{7}{12}\)

Now that both fractions have the same denominator, we can add the numerators (4 + 3 = 7) and keep the denominator (12). The answer \(\frac{7}{12}\) is already in its simplest form.

Example 2

\(\frac{2}{3} – \frac{1}{4}\)

Step 1: Find the lowest common multiple (LCM) of the denominators
Multiples of 3: 3, 6, 9, 12, 15…
Multiples of 4: 4, 8, 12, 16…
LCM = 12

Notice that in this example, we’re subtracting instead of adding. However, we still need to find a common denominator first – the process for finding the LCM is exactly the same as in Example 1.

Step 2: Convert \(\frac{2}{3}\) to an equivalent fraction with denominator 12
\(3 \times 4 = 12\), so multiply both parts by 4
\(\frac{2 \times 4}{3 \times 4} = \frac{8}{12}\)

The conversion process is the same for subtraction as it is for addition. We still need equivalent fractions with the common denominator.

Step 3: Convert \(\frac{1}{4}\) to an equivalent fraction with denominator 12
\(4 \times 3 = 12\), so multiply both parts by 3
\(\frac{1 \times 3}{4 \times 3} = \frac{3}{12}\)

Again, converting to equivalent fractions works the same way whether we’re adding or subtracting.

Step 4: Subtract the fractions with the same denominator
\(\frac{8}{12} – \frac{3}{12} = \frac{5}{12}\)

The key difference between Example 1 and Example 2 is this final step – we subtract the numerators (8 – 3 = 5) instead of adding them. The rest of the process is identical. Whether adding or subtracting, we always need to find a common denominator first, then perform the operation on the numerators.

We Do – Guided Practice

\(\frac{1}{2} + \frac{1}{5}\)

Step 1: Find the LCM of 2 and 5 = 10

The multiples of 2 are: 2, 4, 6, 8, 10, 12… and the multiples of 5 are: 5, 10, 15, 20… The smallest number in both lists is 10.

Step 2: Convert first fraction: \(\frac{1 \times 5}{2 \times 5} = \frac{5}{10}\)

We multiply both numerator and denominator by 5 to change the denominator from 2 to 10.

Step 3: Convert second fraction: \(\frac{1 \times 2}{5 \times 2} = \frac{2}{10}\)

We multiply both numerator and denominator by 2 to change the denominator from 5 to 10.

Step 4: \(\frac{5}{10} + \frac{2}{10} = \frac{7}{10}\)

Now that both fractions have the same denominator, we add the numerators and keep the denominator.

You Do – Independent Practice

Challenge Questions

Question 1: Procedural Variation
Calculate \(\frac{3}{7} + \frac{2}{9}\)

Working:
LCM of 7 and 9 = 63
\(\frac{3 \times 9}{7 \times 9} = \frac{27}{63}\)
\(\frac{2 \times 7}{9 \times 7} = \frac{14}{63}\)
\(\frac{27}{63} + \frac{14}{63} = \frac{41}{63}\)

This question uses denominators (7 and 9) that have a larger LCM (63) than the examples we’ve seen. The process is exactly the same, but requires more careful arithmetic with the larger numbers.

Question 2: Representational Variation
A recipe needs \(\frac{1}{4}\) cup of sugar and \(\frac{1}{6}\) cup of honey. How much sweetener is needed in total?

Working:
LCM of 4 and 6 = 12
\(\frac{1 \times 3}{4 \times 3} = \frac{3}{12}\)
\(\frac{1 \times 2}{6 \times 2} = \frac{2}{12}\)
\(\frac{3}{12} + \frac{2}{12} = \frac{5}{12}\)
Answer: \(\frac{5}{12}\) cup

This is a word problem that requires you to recognize that “in total” means addition. You need to translate the real-world context into a mathematical operation, then apply the same fraction addition process.

Question 3: Backward
I subtracted \(\frac{1}{6}\) from a fraction and got \(\frac{1}{3}\). What was the original fraction?

Working:
We need to find: ? – \(\frac{1}{6}\) = \(\frac{1}{3}\)
This means: ? = \(\frac{1}{3}\) + \(\frac{1}{6}\)
LCM of 3 and 6 = 6
\(\frac{1 \times 2}{3 \times 2} = \frac{2}{6}\)
\(\frac{2}{6} + \frac{1}{6} = \frac{3}{6}\)
\(\frac{3}{6} = \frac{1}{2}\) (simplified)
Answer: \(\frac{1}{2}\)

This question works backwards from the answer. To find the starting fraction, we need to reverse the subtraction by adding. This tests whether you understand that addition and subtraction are inverse operations.

Question 4: Validity
Sam says: “To add \(\frac{1}{4}\) and \(\frac{1}{5}\), I can just add the numerators (1+1=2) and the denominators (4+5=9) to get \(\frac{2}{9}\).” Is Sam correct? Explain why or why not, and show the correct method.

Working:
Sam is INCORRECT.
You cannot add fractions by adding numerators and denominators separately.
Correct method:
LCM of 4 and 5 = 20
\(\frac{1 \times 5}{4 \times 5} = \frac{5}{20}\)
\(\frac{1 \times 4}{5 \times 4} = \frac{4}{20}\)
\(\frac{5}{20} + \frac{4}{20} = \frac{9}{20}\)

This is a common misconception. Fractions represent parts of a whole, and you can only add or subtract them when the parts are the same size (same denominator). Sam’s method would mean adding quarters and fifths as if they were the same size, which they’re not. The correct answer \(\frac{9}{20}\) is very different from Sam’s incorrect answer of \(\frac{2}{9}\).

Question 5: Boundary
What is \(\frac{5}{6} – \frac{2}{3}\)? Notice that the second denominator divides exactly into the first.

Working:
LCM of 6 and 3 = 6 (since 3 divides into 6)
\(\frac{5}{6}\) already has denominator 6
\(\frac{2 \times 2}{3 \times 2} = \frac{4}{6}\)
\(\frac{5}{6} – \frac{4}{6} = \frac{1}{6}\)

This is a special case where one denominator divides exactly into the other. When this happens, the larger denominator is already the LCM, so you only need to convert one fraction. This makes the calculation slightly easier than the general case.

Question 6: Cognitive
Explain why we need to find a common denominator before adding or subtracting fractions with different denominators.

Explanation:
We need a common denominator because fractions represent parts of a whole. Different denominators mean the parts are different sizes. For example, \(\frac{1}{3}\) represents thirds (large pieces) while \(\frac{1}{4}\) represents quarters (smaller pieces). You can’t add or subtract thirds and quarters directly because they’re not the same size – it’s like trying to add apples and oranges. By converting to a common denominator (like twelfths), we make all the pieces the same size, so we can combine them. Once the pieces are the same size, we can add or subtract by counting how many pieces we have in total.

Understanding why we need a common denominator is crucial. The denominator tells us what size pieces we’re working with. If the pieces are different sizes, we need to recut them into equal-sized pieces before we can combine them. This is what finding equivalent fractions with a common denominator does – it recuts both fractions into pieces of the same size.

Question 7: Sequencing
Calculate \(\frac{1}{2} + \frac{1}{3} – \frac{1}{4}\)

Working:
LCM of 2, 3, and 4 = 12
\(\frac{1 \times 6}{2 \times 6} = \frac{6}{12}\)
\(\frac{1 \times 4}{3 \times 4} = \frac{4}{12}\)
\(\frac{1 \times 3}{4 \times 3} = \frac{3}{12}\)
\(\frac{6}{12} + \frac{4}{12} – \frac{3}{12} = \frac{7}{12}\)

This question involves three fractions instead of two, and includes both addition and subtraction. The key is to convert all three fractions to the same common denominator first. Then you work from left to right, performing the operations in order: first add, then subtract. The process is the same as with two fractions, just extended to three.

Question 8: Simplification
Calculate \(\frac{3}{4} + \frac{1}{6}\) and simplify your answer.

Working:
LCM of 4 and 6 = 12
\(\frac{3 \times 3}{4 \times 3} = \frac{9}{12}\)
\(\frac{1 \times 2}{6 \times 2} = \frac{2}{12}\)
\(\frac{9}{12} + \frac{2}{12} = \frac{11}{12}\)
Answer: \(\frac{11}{12}\) (already in simplest form)

After adding or subtracting fractions, always check if the answer can be simplified. In this case, \(\frac{11}{12}\) cannot be simplified further because 11 and 12 have no common factors other than 1. If the answer had been \(\frac{10}{12}\), we would simplify it to \(\frac{5}{6}\) by dividing both numerator and denominator by 2.