Ratios & Proportions
How To Calculate Fractions
Fraction arithmetic is ordinary arithmetic with one extra step: the denominators have to agree before the numerators can be combined. Once that is handled, the four operations reduce to a handful of rules.
Quick Answer
a/b + c/d = (a x d + c x b) / (b x d)
- a/b
- The first fraction, numerator over denominator
- c/d
- The second fraction
- b x d
- A common denominator that always works
One half plus one third is five sixths. Cross-multiply to put both over a common denominator of six, add the numerators, and the answer is 5/6, or about 0.8333.
What Is Fractions?
A fraction is a way of writing a division. The numerator, on top, is the count, and the denominator, underneath, says how many parts make a whole. One half means one of two equal parts.
Adding and subtracting need a common denominator, because you can only count parts that are the same size. One third plus one half cannot be added directly, since the parts are different sizes.
The simplest common denominator is the product of the two denominators. It is not always the smallest possible one, but it always works, and simplifying at the end removes the extra factor.
Once both fractions share a denominator, the numerators are added or subtracted and the denominator is left alone. One third is two sixths and one half is three sixths, so the sum is five sixths.
Multiplication is easier than addition and needs no common denominator. Multiply the numerators together and the denominators together: one half times one third is one sixth.
Division is multiplication by the reciprocal. Flipping the second fraction and multiplying turns a division into a multiplication, which is why dividing by one half doubles a number.
The greatest common divisor is the largest number that divides both the numerator and the denominator exactly. Dividing both by it reduces the fraction to lowest terms without changing its value.
Euclid's algorithm finds the greatest common divisor quickly by repeated remainder-taking. The remainder of the larger divided by the smaller replaces the larger, and the process repeats until the remainder is zero.
A fraction is in lowest terms when the greatest common divisor is one. The fraction three sixths is not in lowest terms because both parts divide by three, giving one half.
An improper fraction has a numerator at least as large as its denominator. Three halves is improper, and it is often rewritten as the mixed number one and a half for readability.
A mixed number combines a whole number and a proper fraction. Converting back is a multiplication and an addition: one and a half is one times two plus one, over two, which is three halves.
A decimal is the numerator divided by the denominator. Fractions whose denominators are built from twos and fives terminate, while others repeat, which is why one third is 0.333 recurring.
Fractions are easier to compare once they share a denominator, which is why the common-denominator step is worth doing even when only an ordering is needed. Three quarters against four fifths becomes fifteen twentieths against sixteen twentieths, and the larger one is obvious at a glance.
Estimating before calculating catches most slips. One half plus one third should land a little above three quarters, so an answer of two fifths is plainly wrong and an answer of five sixths is plainly right. The habit costs nothing and catches the common denominator error immediately.
The calculator takes two fractions and an operation, and returns the simplified fraction, the decimal value, and the mixed-number form where one applies. It also reports the divisor used in the simplification.
Formula
a/b ± c/d = (a x d ± c x b) / (b x d)
Cross-multiply to reach a common denominator, combine the numerators, and simplify.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| a, c | Numerators | count | The counts above the line. |
| b, d | Denominators | count | How many parts make a whole. |
a/b = (a / g) / (b / g) where g = gcd(a, b)
Dividing the numerator and denominator by their greatest common divisor gives the equivalent fraction in lowest terms.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| a | Numerator | count | The count above the line. |
| b | Denominator | count | How many parts make a whole. |
| g | Greatest common divisor | count | The largest number dividing both a and b. |
How To Calculate Fractions
- 1
Work out the operation
Addition and subtraction need a common denominator. Multiplication and division do not.
- 2
Cross-multiply for addition or subtraction
The new denominator is b times d. The new numerator is a times d plus or minus c times b.
- 3
Multiply straight across for multiplication
Numerator times numerator and denominator times denominator, with no common denominator needed.
- 4
Flip and multiply for division
Dividing by c over d is the same as multiplying by d over c. Turn the second fraction upside down and multiply.
- 5
Simplify with the greatest common divisor
Find the largest number dividing both parts and divide them by it. Then convert an improper result to a mixed number if a whole-number form reads better.
Examples
Example 1: 1/2 plus 1/3
- First numerator
- 1
- First denominator
- 2
- Second numerator
- 1
- Second denominator
- 3
- Operation
- Add (+)
| Step | Calculation | Result |
|---|---|---|
| Common denominator | 2 x 3 | 6 |
| Numerator | 1 x 3 + 1 x 2 | 5 |
| Simplified fraction | 5 over 6 | 5/6 |
| Decimal | 5 / 6 | 0.833333 |
Result: 1/2 plus 1/3 is 5/6, which is 0.833333 as a decimal.
Example 2: 3/4 divided by 1/2
- First numerator
- 3
- First denominator
- 4
- Second numerator
- 1
- Second denominator
- 2
- Operation
- Divide (÷)
| Step | Calculation | Result |
|---|---|---|
| Flip and multiply | 3/4 x 2/1 | 6/4 |
| Numerator | 3 x 2 | 6 |
| Denominator | 4 x 1 | 4 |
| Simplified fraction | 6 / 2 over 4 / 2 | 3/2 |
| Mixed number | three halves | 1 1/2 |
| Decimal | 3 / 2 | 1.5 |
Result: 3/4 divided by 1/2 is 6/4, which simplifies to 3/2, or 1 1/2, which is 1.5 as a decimal.
Calculator
Prefer a full-width tool? Open the Fractions calculator page.
Common Mistakes
Adding the denominators
One half plus one third is not two fifths. Denominators describe the size of the parts and are never added; only the numerators combine.
Forgetting to simplify
Six eighths is a correct answer but not a finished one. Divide both parts by their greatest common divisor to reach three quarters.
Cross-multiplying for multiplication
The cross-multiplication rule is only for addition and subtraction. Multiplying fractions is straight across, and using the addition rule gives a wrong answer.
Dividing without flipping
Dividing by a fraction means multiplying by its reciprocal. Dividing by one half is multiplying by two, not by one half.
Dropping a negative sign
A negative sign belongs to the whole fraction. Losing it during cross-multiplication flips the answer's sign with no other symptom.
Dividing by zero
A denominator of zero makes the fraction undefined. Any operation that produces a zero denominator has no numeric answer.
Mixing up mixed numbers and multiplication
One and a half means one plus a half, not one times a half. Convert to an improper fraction before multiplying or dividing.
FAQ
How do I add fractions with different denominators?
Cross-multiply to get a common denominator, then add the numerators. One half plus one third becomes three sixths plus two sixths, which is five sixths.
What is the greatest common divisor?
The largest number that divides both the numerator and the denominator exactly. Dividing both by it puts the fraction in lowest terms.
How do I convert an improper fraction to a mixed number?
Divide the numerator by the denominator. The whole-number part of the quotient is the whole number, and the remainder over the denominator is the fraction.
Why does dividing by a fraction make the answer bigger?
Because dividing asks how many of the second quantity fit into the first. There are two halves in one whole, so dividing by one half doubles the value.
Which fractions give terminating decimals?
Those whose denominators, in lowest terms, have only two and five as prime factors. One half, one quarter and one fifth terminate, while one third and one seventh repeat.
Can a fraction have a negative denominator?
It can, but the convention is to move the sign to the numerator so the denominator stays positive. Minus one over two and one over minus two are the same value.
References
- [1]Maths Is Fun, Fractions — https://www.mathsisfun.com/fractions.html
- [2]Wikipedia, Greatest common divisor — https://en.wikipedia.org/wiki/Greatest_common_divisor
- [3]Encyclopaedia Britannica, Euclidean algorithm — https://www.britannica.com/science/Euclidean-algorithm