Working with fractions, step by step
Why adding fractions needs a common denominator and multiplying does not, how dividing by a fraction became "flip and multiply", and where fractions beat decimals.
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In short
- Addition and subtraction need a common denominator. Multiplication and division do not, which is why they are easier.
- Dividing by a fraction is multiplying by its reciprocal, because dividing by a half means asking how many halves fit.
- Fractions are exact where decimals are not. One third is 0.333… forever, and rounding it introduces error that compounds.
- Always reduce to lowest terms by dividing both parts by their greatest common divisor.
- Recipes, tools, timber, music and shares are all still measured in fractions, and converting them to decimals loses information.
A fraction represents a division that has not been carried out. The numerator on top is what is being divided; the denominator below is what it is being divided into.
3/4 means three divided by four, and it is exactly 0.75. 1/3 also means a division — and it has no exact decimal form at all, which is the first reason fractions still exist.
Adding and subtracting
You cannot add quarters to thirds any more than you can add metres to seconds. Both parts have to be expressed in the same unit first, and that unit is the common denominator.
1/4 + 1/3
The least common multiple of 4 and 3 is 12, so rewrite both:
3/12 + 4/12 = 7/12
Multiply each numerator by whatever you multiplied its denominator by, then add the numerators and leave the denominator alone. Adding the denominators — the single most common mistake — would give 2/7, which is smaller than either starting fraction, and adding two positive quantities cannot make something smaller than both.
Subtraction works identically: 3/4 − 1/6 = 9/12 − 2/12 = 7/12.
The least common multiple keeps the numbers small, but any common multiple works. Multiplying the two denominators together always gives one, and the result simply needs reducing afterwards.
Multiplying
Multiplication needs no common denominator at all. Multiply across the top, multiply across the bottom:
2/3 × 3/5 = 6/15 = 2/5
"Of" means multiply. Two thirds of three fifths is the same operation as two thirds times three fifths, which is why "half of a half is a quarter" is the same statement as 1/2 × 1/2 = 1/4.
Cancelling before multiplying keeps the numbers manageable: in 2/3 × 3/5 the 3 on top and the 3 below cancel, leaving 2/1 × 1/5 = 2/5 directly.
Note that multiplying by a fraction below one makes things smaller, which contradicts the intuition that multiplication increases. That intuition comes from whole numbers and does not survive contact with fractions.
Dividing, and why you flip
1/2 ÷ 1/4 = 1/2 × 4/1 = 4/2 = 2
The rule is to multiply by the reciprocal of the second fraction, and it is worth understanding rather than memorising.
Division asks how many of the second thing fit into the first. How many quarters fit in a half? Two. The answer is larger than what you started with, because the thing you are dividing by is smaller than one.
The formal justification: dividing by 1/4 is the same as multiplying by whatever undoes 1/4, and that is 4/1, since 1/4 × 4/1 = 1.
This is also why dividing by a fraction below one produces a bigger number — the mirror image of multiplication making things smaller.
Reducing to lowest terms
18/24 is correct and untidy. Divide both parts by their greatest common divisor, 6, to get 3/4.
Finding the GCD by inspection works for small numbers. The Euclidean algorithm handles any size: divide the larger by the smaller, replace the pair with the smaller and the remainder, repeat until the remainder is zero, and the last non-zero remainder is the GCD.
An improper fraction has a numerator at least as large as its denominator — 7/4. A mixed number writes the same value as 1 3/4. Neither is more correct; mixed numbers read better in a recipe and improper fractions calculate better in arithmetic, so convert to improper before working and back afterwards if you like.
Comparing two fractions
Which is larger, 5/8 or 7/11?
Cross-multiply. 5 × 11 = 55 and 7 × 8 = 56. The second product is larger, so 7/11 is larger. This works because it is the common-denominator comparison without writing the denominator down.
Or convert to decimals. 0.625 against 0.6364. Same answer, and easier when the numbers are awkward.
The cross-multiplication trick is faster mentally and avoids rounding, which matters when the two are very close together.
Where fractions beat decimals
Exactness. One third is 0.333… and any decimal you write down is wrong. In a calculation that repeats, that error accumulates. 1/3 × 3 = 1 exactly; 0.333 × 3 = 0.999.
Recipes. Halving 3/4 cup gives 3/8 cup, which is a measurement a set of cups can produce. 0.375 cups is not.
Construction and tools. Imperial spanners, drill bits and timber are sized in fractions of an inch — 3/8, 7/16, 1/2. American plans and any imported hardware still use them, and rounding a 7/16 bolt to 0.44 inches loses the identity of the part.
Music. Note values are fractions of a bar. Time signatures are fractions.
Probability and shares. 1/6 for a die is exact where 0.1667 is not, and ownership is naturally expressed as parts of a whole.
Ratios and rates. A gear ratio of 3:1 and a map scale of 1/50,000 are fractions doing work that a decimal would obscure.
Converting between the forms
Fraction to decimal: divide. 5/8 = 0.625.
Terminating decimal to fraction: write the digits over the matching power of ten and reduce. 0.36 = 36/100 = 9/25.
Repeating decimal to fraction: the algebraic method. For 0.777…, let x = 0.777…, so 10x = 7.777…, and subtracting gives 9x = 7, therefore x = 7/9. Every repeating decimal is a fraction, and every fraction is either terminating or repeating.
A fraction terminates as a decimal exactly when its reduced denominator has no prime factors other than 2 and 5 — which is why halves, quarters, fifths and eighths are tidy and thirds, sevenths and ninths are not.
Fractions, ratios, decimals and percentages
Four notations for one idea, each suited to a different job.
3/4 = 3:1 as a ratio of part to remainder = 0.75 = 75%.
A fraction compares a part to the whole. A ratio usually compares parts to
each other — a 3:1 mix is three parts to one, which is 3/4 of the total, and
confusing the two is the most frequent error with ratios.
A decimal is a fraction with a denominator that is a power of ten, which is why it is convenient for calculation and unable to represent thirds.
A percentage is a fraction with a denominator of one hundred, which makes different fractions directly comparable at a glance.
Converting between them is mechanical, and choosing the wrong one for the job is what causes trouble: mixing paint by decimals, expressing a probability as a ratio, or reporting a gear ratio as a percentage all obscure what is being described.
Where fraction arithmetic goes wrong
Adding denominators. 1/2 + 1/3 is not 2/5. The denominator names the unit
and units do not add.
Cancelling across a sum. In (3 + 6)/3 the threes cannot be cancelled — only
factors of the whole numerator can, not terms inside a sum. Cancelling here gives
1 + 6 = 7 instead of the correct 3.
Forgetting to reduce. 6/8 is not wrong, and in most contexts 3/4 is the
expected answer.
Mishandling mixed numbers. 2 1/3 means two plus a third, not two multiplied
by a third. Convert to 7/3 before doing anything with it.
Assuming a bigger denominator means a bigger fraction. 1/8 is smaller than
1/3, because dividing into more pieces makes each piece smaller. This is
counterintuitive precisely because it is the reverse of how whole numbers behave.
What this calculator assumes
- Fractions are entered as numerator and denominator; mixed numbers should be converted to improper form first.
- Results are reduced to lowest terms using the greatest common divisor.
- A zero denominator is undefined and reported as an error rather than as infinity.
- Negative fractions carry the sign on the numerator internally, so
−1/2and1/−2are treated as the same value. - Decimal equivalents are shown to a fixed precision and may be rounded; the fraction is the exact value.