Fractions with unlike denominators

Why you cannot add them as they are#

A fraction is a number of equal parts. In $\frac{1}{6}$ the denominator tells you how many parts the whole was cut into, and the numerator how many of them you took. You cannot add sixths to eighths directly for the same reason you cannot add three metres to four kilograms: they are different kinds of quantity.

So first you make the parts the same size. Bringing fractions to a common denominator does not change the numbers themselves: $\frac{1}{6}$ and $\frac{4}{24}$ are the same amount, just written in smaller pieces. The one rule of the conversion: whatever you multiply the denominator by, you multiply the numerator by the same number.

How to find a common denominator#

Any number that both denominators divide into will do. The most convenient is the least common multiple (LCM), because it keeps the numbers small.

Three ways to find the LCM, from the quickest to the one that always works:

First. If one denominator divides into the other, the larger one is the common denominator. For $\frac{1}{4}$ and $\frac{3}{12}$ it is 12.

Second. List multiples of the larger denominator until you hit one the smaller divides into. For 6 and 8: 8 is not divisible by 6, 16 is not, 24 is — that is the LCM.

Third, which always works. Break both denominators into prime factors and take each factor at its highest power. For 12 and 20: $12 = 2^2 \times 3$, $20 = 2^2 \times 5$, so the LCM is $2^2 \times 3 \times 5 = 60$.

Multiplying the denominators together also works, but then you will have to simplify the answer: for 12 and 20 you would get 240 instead of 60.

Example 1: adding#

Work out $\frac{1}{6} + \frac{3}{8}$.

Step 1. Find the common denominator. Multiples of eight: 8, 16, 24 — the first one divisible by 6 is 24.

Step 2. Find the scale factors: $24 \div 6 = 4$ for the first fraction and $24 \div 8 = 3$ for the second.

Step 3. Multiply the top and bottom of each fraction by its factor: $\frac{1}{6} = \frac{4}{24}$, $\frac{3}{8} = \frac{9}{24}$.

Step 4. Add the numerators and keep the denominator: $\frac{4 + 9}{24} = \frac{13}{24}$.

Step 5. Check whether the answer simplifies. 13 is prime and does not divide into 24, so the fraction is already in lowest terms.

Example 2: subtracting and simplifying the answer#

Work out $\frac{5}{12} - \frac{3}{20}$.

Step 1. We found the LCM of these denominators above: 60.

Step 2. Scale factors: $60 \div 12 = 5$ and $60 \div 20 = 3$.

Step 3. Convert: $\frac{5}{12} = \frac{25}{60}$, $\frac{3}{20} = \frac{9}{60}$.

Step 4. Subtract the numerators: $\frac{25 - 9}{60} = \frac{16}{60}$.

Step 5. Divide top and bottom by 4: $\frac{16}{60} = \frac{4}{15}$. Only the fraction in lowest terms counts as the final answer.

Example 3: comparing#

Which is larger, $\frac{5}{8}$ or $\frac{7}{12}$? You cannot tell at a glance: the first fraction has a smaller numerator, but also a smaller denominator.

Step 1. The common denominator of 8 and 12 is 24.

Step 2. Convert: $\frac{5}{8} = \frac{15}{24}$, $\frac{7}{12} = \frac{14}{24}$.

Step 3. The pieces are now the same size, so compare the numerators: $15 > 14$, which means $\frac{5}{8} > \frac{7}{12}$.

You can answer the same question by converting to decimals: $0.625$ against $0.5833…$. How to do that is covered in operations with decimals.

Mistakes that cost the answer#

Adding the denominators: $\frac{1}{6} + \frac{3}{8} = \frac{4}{14}$. Test it with numbers: by that "rule" $\frac{1}{2} + \frac{1}{2}$ would give $\frac{2}{4}$, a half instead of one whole.

Multiplying the denominator and forgetting the numerator. The fraction then changes its value, and the rest of the solution is wrong even if the addition is done carefully.

Finding a common denominator where none is needed. When you multiply or divide fractions, you do not need one: different rules apply there.

Not simplifying the answer, or simplifying across a plus sign: you can cancel only factors, not separate numbers inside a sum in the numerator.

How to check yourself#

Estimate the size. Are both fractions roughly a half? Then the sum is roughly one. In the first example $\frac{1}{6}$ is small and $\frac{3}{8}$ is a bit less than a half, so the sum should come out near a half: $\frac{13}{24}$ is just over $\frac{12}{24}$, which fits. The estimate takes seconds and catches gross errors such as a lost scale factor.

Step-by-step plan

1. Step 1 — LCMFind the least common multiple for ten pairs of numbers, both by listing multiples and by prime factors.
2. Step 2 — convertingConvert one fraction at a time to a given denominator, saying the scale factor out loud.
3. Step 3 — adding and subtractingTen examples, always simplifying the answer and writing improper fractions as mixed numbers.
4. Step 4 — comparingCompare pairs of fractions using a common denominator and check by estimating against one half.
5. Step 5 — mixed numbersAdd and subtract mixed numbers, including borrowing one from the whole part.

Check yourself

Sources

• Khan Academy — ArithmeticStep-by-step practice on common denominators with hints
  free
• OpenStax Prealgebra 2eFree textbook; chapter 4 covers fractions, with answers to odd-numbered problems
  free

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