Equivalent Fractions Calculator
Result
Equivalent fractions (×2, ×3, …)
- Fraction in lowest terms
- 1/3
- As a decimal
- 0.333333
An equivalent fractions calculator lists other fractions that are equal to the one you type: 1/3 starts 2/6, 3/9, 4/12. They are equal because both parts of the fraction have been multiplied by the same number — multiply the numerator and the denominator of 1/3 by 2 and you get 2/6, by 3 and you get 3/9 — and changing the size of the pieces while keeping how many of them you have never changes the value, the same way cutting every slice of a cake in half leaves you with twice as many slices of half the size. Asking for five fractions gives the multiples 2, 3, 4, 5 and 6, so the count you set is the length of that run and not the number of denominators involved. The panel answers the other half of the question beside it: the lowest terms the fraction can be written in and its decimal value. Those two figures are worth reading together with the run, because the run deliberately stays unreduced. Writing 2/6 rather than 1/3 is the point of the exercise — they are the same number in different notation — so reducing every item in the run would print one value several times over and hide the very thing being shown. The lowest terms come from dividing the numerator and the denominator by their greatest common factor, which is the largest number that divides both of them exactly; a fraction already in lowest terms has no common factors left to remove, which is why 1/3 answers 1/3. The run starts from the fraction as you typed it rather than from its simplest form, so 2/6 and 1/3 share the same lowest terms but produce different runs — the first thing after 2/6 is 4/12, and after 1/3 it is 2/6. That is deliberate: the page answers the question about the notation you actually used, and a reader who types 7/21 should not be shown a list starting at 2/6. Use it to find another fraction equal to one you have, to check a comparison, or to make two fractions comparable when you would rather see the pieces than divide. Both parts have to be whole numbers here, and the count has an upper limit, because the multiplication is done in exact integer arithmetic and every figure on the panel is written as a numerator over a denominator. A decimal is a different notation for the same value, so the decimal reading is given as one more way to see what all of those fractions have in common.
Common fractions and their equivalent fractions
| Fraction | Equivalent fractions | In lowest terms |
|---|---|---|
| 1/2 | 2/4, 3/6, 4/8, 5/10 | 1/2 |
| 1/3 | 2/6, 3/9, 4/12, 5/15 | 1/3 |
| 3/4 | 6/8, 9/12, 12/16, 15/20 | 3/4 |
| 2/6 | 4/12, 6/18, 8/24, 10/30 | 1/3 |
| 6/4 | 12/8, 18/12, 24/16, 30/20 | 3/2 |
| 5/5 | 10/10, 15/15, 20/20, 25/25 | 1 |
| 0/3 | 0/6, 0/9, 0/12, 0/15 | 0 |
| -1/2 | -2/4, -3/6, -4/8, -5/10 | -1/2 |
Every row comes out of the same multiplication, and no row is a special case of it. The first row is the simplest case, 1/2 doubling and tripling into 2/4 and 3/6. The second and fourth rows are the pair worth studying together: 1/3 and 2/6 are the same number and both end in 1/3, yet their runs differ, because the run is built from the fraction as typed — the row for 1/3 begins with 2/6, and the row for 2/6 begins with 4/12. That is why the second figure cannot be read off the first. The third row is a fraction already in lowest terms, so its own answer repeats it. The fifth row is larger than one, and its lowest terms are improper too: no multiplication changes whether a fraction is more than one. The sixth row is a fraction whose parts divide exactly, and the panel writes that as 1 rather than as 5/5 — while the run beside it keeps its denominators, since the notation is the subject. The seventh row is a numerator of zero: every equivalent fraction of it is zero as well, and it is allowed here, even though a ratio with a zero first term is not. The last row is a negative fraction, and the minus sign stays on the numerator throughout: -2/4, -3/6, never 2/-4. All eight rows use a count of four, so the runs are the same length and can be compared directly.
Formula
Equivalent fraction = (numerator × k) ÷ (denominator × k), for k = 2, 3, 4, … up to one more than the count Lowest terms = numerator and denominator divided by their greatest common factor Decimal value = numerator ÷ denominator
- Fraction
- The numerator and the denominator you type. The denominator may not be zero, and it is the numerator that carries the minus sign if the fraction is negative. A numerator of zero is allowed, and every equivalent fraction of it is zero too.
- Multiplier k
- The whole number that both parts are multiplied by. It runs from 2 up to one more than the count you asked for, so asking for five gives the multipliers 2, 3, 4, 5 and 6. It never takes the value 1, because that is the fraction as typed.
- Equivalent fraction
- One of the fractions in the run: the numerator and the denominator each multiplied by the same k. The value is unchanged, only the size of the pieces is, and the items in the run are printed as they come rather than reduced, so 2/6 appears as 2/6.
- Lowest terms
- The same value written with the smallest possible numerator and denominator, obtained by dividing both parts by their greatest common factor. A fraction that is already in lowest terms is its own answer, and a fraction whose parts divide exactly is printed as a whole number: 5/5 is 1.
- Decimal value
- The numerator divided by the denominator, rounded to six decimal places. It is the third way of writing the same number, and unlike the run it cannot show that two fractions are equal by inspection: 1/3 and 2/6 both come out as 0.333333.
The first use is school work on fractions themselves, where the point is that a value has many names: typing 2/6 and seeing 4/12, 6/18 and 8/24 makes it plain that reducing to lowest terms changes the writing and not the number. The second is comparison, since two fractions are easy to compare once they share a denominator: 2/3 against 3/5 is hard to judge at a glance, and the equivalent fractions 10/15 and 9/15 are not, which is the same conversion that adding fractions needs. The third is checking a reduction someone else did: if a fraction is said to simplify to 3/4, then the run from 3/4 should contain the original, and if it does not, the two are not the same number after all. The fourth is when a fraction has to be reported in a particular denominator — sixteenths of an inch, twelfths of an hour — and you want to see what the value is in the units the work is measured in. When a fraction is already in lowest terms the panel repeats it, which is itself the answer to whether it can be simplified any further.
Worked examples
1/3, five fractions
- Both parts of 1/3 multiplied by 2: 2/6
- By 3: 3/9, and by 4: 4/12
- By 5: 5/15, and by 6: 6/18 — five fractions, since the count was five
- Lowest terms: 1 and 3 share no factor, so the answer is 1/3 again
- Decimal value: 1 ÷ 3 = 0.333333
The default case, and the one that shows why the multiplier starts at 2: multiplying by 1 would print 1/3 itself as the first item, which is not another fraction at all. The lowest terms matching the input says the fraction was already as simple as it can be written, and the decimal repeats because thirds have no exact decimal form.
2/6, three fractions
- Both parts of 2/6 multiplied by 2: 4/12
- By 3: 6/18, and by 4: 8/24
- Lowest terms: 2 divides both parts, so 2/6 is 1/3
- Decimal value: 2 ÷ 6 = 0.333333, the same as 1 ÷ 3
This is the case that explains the second figure on the panel. 2/6 and 1/3 are the same number, so they have the same lowest terms and the same decimal value, and yet their runs differ: here the first item is 4/12, and the run for 1/3 starts at 2/6. The page builds the run from the fraction as typed, so the list you get depends on which of the two ways of writing that value you put in.
5/5, which is a whole number
- Both parts of 5/5 multiplied by 2: 10/10
- By 3: 15/15 — two fractions, since the count was two
- Lowest terms: 5 divides both parts, giving 1
- Decimal value: 5 ÷ 5 = 1
A fraction whose denominator divides its numerator is a whole number, and the panel writes it as 1 rather than 5/5 or 5/5 reduced to 1/1. The run beside it keeps its denominators even so, because the point of the run is the notation: 10/10 and 15/15 are how that same value looks with smaller pieces.
6/4, a fraction larger than one
- Both parts of 6/4 multiplied by 2: 12/8
- By 3: 18/12, by 4: 24/16 and by 5: 30/20 — four fractions
- Lowest terms: 2 divides both parts, so 6/4 is 3/2
- Decimal value: 6 ÷ 4 = 1.5
The value is more than one, and it is answered as 3/2 rather than as a mixed number, so that every figure on the panel is written the same way. The equivalent fractions are all improper as well, which follows from multiplying both parts of 3/2 by the same number: no multiplication can turn a fraction greater than one into a proper one.
Limitations
Both inputs must be whole numbers, so 0.75 is not something this page will work from: convert it to 3/4 first, and the reason is that the whole calculation is exact integer arithmetic, which stops being exact once a decimal is involved. The denominator has an upper limit and may not be zero, and the numerator has one too, because every item in the run is a product. The count is capped, which matters less than it sounds: the fractions get large quickly, and 20 items of a fraction like 7/21 already run into the hundreds. The run is built from the fraction as typed rather than from its lowest terms, so a value written two ways gives two different runs — that is a choice, not a slip, but it does mean the output depends on the notation in the input. Nothing in the run is reduced, so an item like 6/18 sits beside 1/3 in the same panel; if you want the smallest denominator, the second figure on the panel is the one to read. The result is written as a fraction and never as a mixed number, and a negative fraction keeps its minus sign on the numerator. This page lists fractions equal to a fraction and does not simplify a fraction for its own sake: a page for that shows the divisor and the working, where this one only prints the answer. Two things it does not do at all are arithmetic between fractions and conversion from a decimal, which are separate pages.
Frequently asked questions
- How do I find equivalent fractions for a fraction I have?
- Multiply the numerator and the denominator by the same whole number. Starting from 1/3 and multiplying both parts by 2, 3 and 4 gives 2/6, 3/9 and 4/12, and every one of them is equal to 1/3. That is the whole formula, and it works in the other direction too: dividing both parts by a common factor gives another fraction equal to the same value.
- Why does the list start at 2/6 and not at 1/3?
- Because multiplying by 1 would give the fraction you typed, which is not another fraction to compare against. The run is the multiples 2, 3, 4 and so on, and the count you set is how many of them are listed.
- Why are the listed fractions not reduced?
- Reducing them would print the same number again and again, which is the opposite of the point. 2/6, 3/9 and 4/12 are all 1/3, and seeing them written differently is what makes it clear that the value has not changed. The lowest terms of the value you typed are shown separately on the panel.
- What are the lowest terms?
- The same value written with the smallest possible numerator and denominator, found by dividing both parts by their greatest common factor. If the two parts have no common factors, the fraction is already in lowest terms and the panel simply repeats it, as it does for 1/3.
- Is the decimal value the same for all of them?
- It is, and that is exactly what equality means here: 1/3, 2/6, 3/9 and 4/12 all come out as 0.333333. The decimal is the quickest way to check that a run really is equivalent, but it is also the reading that hides the difference, since it shows one number where the fractions show four.
- Can I use a fraction larger than one, or a negative one?
- Both work, and both keep their character in the run. A fraction like 6/4 has 3/2 as its lowest terms, and 12/8, 18/12 and the rest are all larger than one as well. A negative fraction is written with the minus sign on the numerator, so -1/2 lists -2/4 and -3/6.
- Can I put in a decimal instead of a fraction?
- No. Both parts have to be whole numbers, so 0.75 is rejected; convert it to 3/4 first. The multiplication that builds the run is done in exact integer arithmetic, and a decimal would break that exactness as soon as the value cannot be written as a fraction over a power of ten.
References
- Fraction — two fractions are equal when one is obtained from the other by multiplying numerator and denominator by the same number, which is the operation this page performs and repeats — Wolfram MathWorld (United States)
- Reduced Fraction — a fraction whose numerator and denominator have no common factor, which is the second figure on the panel and the form the run deliberately does not take — Wolfram MathWorld (United States)
- Greatest Common Divisor — the largest integer dividing both parts, which is what the lowest terms are obtained by dividing out — Wolfram MathWorld (United States)
- Denominator — the number below the line in a fraction, which is the part two fractions have to agree on before they can be rewritten over one another — Wolfram MathWorld (United States)