Add unlike fractions by making the parts the same size. Start with the visual model, then work through a mixed number and a missing-fraction problem.
- Choose a common denominator and rewrite fractions without changing their values.
- Add the numerators, then simplify or write an improper fraction as a mixed number.
- Use estimation and a missing-addend check to test whether an answer is reasonable.
The idea
A denominator tells you the size of each part. Before adding fractions, express both amounts using the same-sized parts. Equivalent fractions change the way an amount is written, not its value.
Before you begin: Recognise equivalent fractions and find common multiples.
Worked examples
01 Start with the method
Find the exact sum. Give your answer in simplest form.
The lowest common multiple of 3 and 4 is 12. Rewrite both fractions in twelfths, multiplying the numerator and denominator by the same number.
The parts now have the same size. Add the number of parts and keep the denominator.
The numerator and denominator share no factor other than 1. The answer is already in simplest form and is slightly less than one whole.
A mistake to avoid
Do not add the denominators. Adding thirds to quarters does not create sevenths; first choose a shared unit.
02When one denominator is already a multiple of the other
Add the fractions and give the answer in simplest form.
Ten is a multiple of both denominators. Rewrite the fifths as tenths; the second fraction can stay as it is.
Count the tenths together. The denominator stays ten because the parts remain the same size.
Seven and ten share no factor greater than one. The sum is less than one whole, which agrees with the two starting amounts.
03Adding to a mixed number
A recipe uses one and two thirds cups of flour, then another three quarters of a cup. Find the total.
Keep the whole cup separate and express the fractional parts in twelfths.
The fractional parts add to more than one whole.
Combine the whole cups. An exact mixed number or the equivalent improper fraction represents the same total.
Practice questions with answers
Work these out first, then open each solution to compare your method. These questions are untimed.
Question 1
Try a new pair before opening the solution.
Show solution 1
Use thirtieths as the common unit. The result is already in simplest form.
Question 2
Add and simplify.
Show solution 2
Use sixths: one half is three sixths and one third is two sixths. Add three and two, keeping the denominator six.
Question 3
Add without using a larger denominator than necessary.
Show solution 3
Eight is already a common denominator. One quarter is two eighths, so three eighths plus two eighths is five eighths.
Question 4
Find the sum and write it as a mixed number.
Show solution 4
Five sixths is ten twelfths. Adding seven twelfths gives seventeen twelfths: twelve twelfths make one whole, with five twelfths left.
Question 5
Find the missing fraction in simplest form.
Show solution 5
Undo the addition by subtracting two fifths from eleven twelfths. In sixtieths this is fifty-five minus twenty-four, giving thirty-one sixtieths. Adding twenty-four sixtieths back checks the result.
Common questions
Must I multiply the two denominators to find a common denominator?
That always gives a common multiple, but it may not be the smallest one. For fifths and tenths, tenths already work; fiftieths create larger numbers to simplify. The lowest common denominator often makes the calculation easier.
Why do I change the numerator as well as the denominator?
The fraction must keep the same value. Splitting each fifth into two equal pieces doubles both the number of pieces counted and the number in a whole. Changing only the denominator changes the amount instead of making an equivalent fraction.