Adding and Subtracting Fractions with Unlike Denominators: Steps for Finding a Common Denominator, Examples and Practice Tips
The key to adding and subtracting fractions with unlike denominators is finding a common denominator: first find a common multiple of the two denominators (usually the least common multiple), convert both fractions to the same denominator, then add or subtract using the rules for fractions with the same denominator. Below are worked examples, along with common mistakes and practice methods.
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- Fractions with unlike denominators must first be converted to a common denominator before the numerators can be added or subtracted.
- The common denominator is usually the least common multiple, which keeps the numbers smaller.
- When converting, the numerator and denominator must both be multiplied by the same number.
- After calculating, check whether the answer can be simplified to its simplest form.
Why fractions with unlike denominators cannot be added or subtracted directly
The denominator of a fraction shows how many equal parts a whole is divided into, and the numerator shows how many of those parts are taken. When two fractions have different denominators, each part is a different size: for example, one part out of two is bigger than one part out of three, so you cannot simply add the numerators.
To make parts of different sizes the same size, you need a common denominator: rewrite both fractions with the same denominator. Once they share a denominator, each part is the same size, and only then can the numerators be added or subtracted.
This is like everyday measurement: to add a length in metres to a length in centimetres, you must first convert them to the same unit. Finding a common denominator for fractions is simply making the "unit" the same.
Step 1: Find a common denominator
A common denominator is a common multiple of the two denominators. The easiest approach is to use the least common multiple (LCM), because the numbers are smaller and you are less likely to make mistakes when simplifying.
How to find the least common multiple: list the multiples of the larger denominator, then find the first number that is also a multiple of the other denominator. For example, with denominators 4 and 6, the multiples of 4 are 4, 8, 12, 16… and the multiples of 6 are 6, 12, 18…; the first number that appears in both lists is 12, so the common denominator is 12.
If the two denominators are coprime (their only common factor is 1), such as 3 and 5, the least common multiple is simply their product, 15.
When finding a common denominator, don't rush to multiply the two denominators together. Using the least common multiple first keeps the numbers smaller and reduces the burden of simplifying and calculating.
Step 2: Convert the fractions by multiplying the numerator and denominator by the same number
Once you have found the common denominator, rewrite each fraction as an equivalent fraction with that denominator. The method: divide the common denominator by the original denominator to get a multiplier, then multiply both the numerator and the denominator by that multiplier.
For example, to convert two-thirds to a denominator of 12: 12 ÷ 3 = 4, so multiply both the numerator and the denominator by 4 to get eight-twelfths.
The key rule: the numerator and denominator must be multiplied by the same number so that the value of the fraction stays the same. Multiplying only the denominator or only the numerator changes the value of the fraction.
Step 3: Adding and subtracting
After converting, both fractions have the same denominator; you then simply add or subtract the numerators and keep the denominator unchanged.
Addition example: one-half plus one-third. The common denominator is 6; one-half equals three-sixths and one-third equals two-sixths; adding the numerators gives five-sixths.
Subtraction example: three-quarters minus one-sixth. The common denominator is 12; three-quarters equals nine-twelfths and one-sixth equals two-twelfths; subtracting the numerators gives seven-twelfths.
Mixed addition and subtraction example: first work out one-half plus one-third (five-sixths), then subtract one-sixth (five-sixths minus one-sixth); the result is four-sixths, which simplifies to two-thirds.
After calculating, check whether the result can be simplified. If the numerator and denominator have a common factor, simplify to the simplest form.
- PrimaryAdding unlike fractions12 + 13 = 36 + 26 = 56Make the denominators the same, then add the numerators
- PrimaryFactors and multiples12 = 2 × 2 × 3
18 = 2 × 3 × 3H.C.F. = 2 × 3 = 6, L.C.M. = 2 × 2 × 3 × 3 = 36 - PrimaryMixed numbers and improper fractions213 = 2 × 3 + 13 = 73Whole number times denominator, add the numerator, keep the denominator
Common mistakes and how to check
The most common mistake is adding the denominators directly, for example writing one-half plus one-third as two-fifths. This changes the size of each part, so the answer is always wrong.
Another common mistake is multiplying only the denominator and not the numerator when converting, or converting only one of the fractions. After each conversion, check in reverse by dividing: does the new numerator divided by the new denominator equal the original value of the fraction?
When subtracting, pay attention to the order when subtracting the numerators and whether borrowing is needed. If the numerator is too small to subtract from, first convert one fraction to a mixed number or adjust the common denominator, then calculate carefully.
Finally, get into the habit of simplifying. An answer that is not in its simplest form has the same value, but is usually considered incomplete.
Practice tips: from easy to hard
Practise in stages by difficulty: first do questions where the denominators are multiples of each other (such as 2 and 4, or 3 and 6), then questions where the denominators are coprime (such as 3 and 5), and only then questions that need simplifying first or involve mixed numbers.
Focus on one type of question per practice session, and require the working for the conversion to be written out, not just the answer. Writing the working lets parents and teachers see which step went wrong.
You can make your own flashcards with a question on one side and the conversion steps on the other, and do three to five questions a day; this is more effective than doing a large number of questions repeatedly.
If your child often makes a mistake at the same step, such as finding the least common multiple, practise that step on its own first before returning to the full calculation.
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How parents can help
Parents don't need to calculate for their child; instead, ask the child to explain the reason for each step: why this common denominator was chosen, what the numerator and denominator were multiplied by, and why the denominator stays the same. Being able to explain clearly usually shows real understanding.
If your child gets stuck, use objects or diagrams first: for example, fold a piece of paper into different numbers of parts to see the difference between one-half and one-third, then introduce the written method for finding a common denominator.
If you are considering extra support, first understand the different forms of tutoring: tutoring centres, private tutoring, home tutoring and online tutoring differ in scheduling and interaction, so choose according to your child's needs rather than looking only at the price.
Frequently asked questions
Must you use the least common multiple when finding a common denominator?
Not necessarily. Any common multiple can be used, but the least common multiple usually keeps the numbers smaller, making calculation and simplifying easier, so it is generally recommended to find the least common multiple first.
Can you multiply only the numerator or only the denominator?
No. The numerator and denominator of a fraction must both be multiplied by the same non-zero number for the value of the fraction to stay the same; multiplying only one of them changes its value.
Must the answer always be simplified to its simplest form?
It is generally recommended to simplify to the simplest form. Although the value is the same, an unsimplified answer is usually considered incomplete, and the simplest form makes it easier to compare sizes.