Fractions
Equivalent Fractions
Equivalent fractions are different-looking fractions that name the same amount: 1/2, 2/4, 3/6 and 50/100 all sit at the same point on the number line. You make one by multiplying or dividing the top and bottom by the same number. This guide covers how to find them, how to fill in a missing number, how to test two fractions, and a chart of the common ones.
What are equivalent fractions?
Two fractions are equivalent when they have the same value. Cut a pizza into 4 slices and eat 2, or cut it into 8 and eat 4: you have eaten 2/4 and 4/8 of the pizza, and both are exactly half. The numbers in the fraction changed because the slices changed size, but the amount did not.
There are three ways to see that two fractions are equal, and each is useful in a different situation:
- Same decimal.
3 ÷ 4 = 0.75and6 ÷ 8 = 0.75, so 3/4 and 6/8 are equivalent. - Same simplest form. 6/8 and 9/12 both reduce to 3/4.
- Same point on the number line. Mark quarters and eighths on the same line from 0 to 1, and the 3/4 mark lands on the 6/8 mark. The number line guide shows how to plot fractions this way.
Seeing it with a folded strip
Take a strip of paper, fold it in half and shade one half. Fold it in half again without unfolding: the shaded part now covers 2 of 4 equal sections. Fold once more and it covers 4 of 8. Every fold doubles the number of sections and doubles the number shaded, so the fraction becomes 1/2, then 2/4, then 4/8, while the shaded area never moves. That picture is the whole rule in one object: the top and bottom were both multiplied by 2, and the amount stayed the same.
Every fraction has infinitely many equivalent fractions, because you can always multiply the top and bottom by a bigger number. Exactly one of them is in simplest form, with no common factor left between top and bottom.
How to find equivalent fractions
Multiply the top and bottom by the same number
To make a larger equivalent fraction, multiply the numerator and the denominator by the same whole number. Any number works, as long as you use it on both.
Example: fractions equivalent to 2/3
All five fractions, 2/3 included, equal 0.666…, which rounds to 0.6667.
Divide the top and bottom by a common factor
To make a smaller equivalent fraction, divide the numerator and the denominator by a number that goes into both. Repeat until nothing else divides both, and you have the simplest form.
Example: fractions equivalent to 12/18
Dividing by the greatest common factor reaches simplest form in one move. The GCF calculator finds that factor, and the guide to simplifying fractions covers this half of the topic in full.
Why this never changes the value
Multiplying the top and bottom by 3 is the same as multiplying the whole fraction by 3/3, and 3/3 = 1. Multiplying any number by 1 leaves it unchanged. That is all an equivalent fraction is: the original fraction multiplied by 1 in disguise. The same logic is why multiplying fractions by n/n produces a new name and not a new amount.
It also explains why adding does not work. Adding 1 to the top and bottom of 1/2 gives 2/3, and 0.5 is not 0.667. Addition is not multiplication by 1, so it changes the value.
Finding the missing number in equivalent fractions
Most worksheet and test questions give one complete fraction and one with a blank: 3/4 = ?/20. Find what the known part was multiplied (or divided) by, then do the same to the other part.
Example: 3/4 = ?/20
Example: 2/3 = 10/?
Example: 18/24 = 3/?
When the scaling number is not obvious, cross multiplication always works. For 5/8 = ?/24, the missing top is 5 × 24 ÷ 8 = 15. That is the method behind any equation of the form a/b = c/d, and the proportion calculator solves it with the blank in any position.
How to tell if two fractions are equivalent
Three tests, each one enough on its own.
Cross multiply
Multiply the top of each fraction by the bottom of the other. If the two products match, the fractions are equivalent.
This is the fastest test for pairs of fractions with awkward numbers, because it needs no division.
Simplify both
Reduce each fraction to lowest terms. If they end up identical, they were equivalent. 4/6 and 6/9 both reduce to 2/3. The simplify fractions calculator does the reducing and shows the GCF it used.
Convert both to decimals
Divide top by bottom for each. 2 ÷ 5 = 0.4 and 4 ÷ 10 = 0.4, so 2/5 and 4/10 are equivalent. This is quick with terminating decimals; with repeating ones such as 0.333… it is easy to be fooled by rounding, so prefer one of the other tests. The fraction to decimal calculator shows the repeating part exactly.
Equivalent fractions chart
Each row is one fraction in simplest form, followed by the equivalents you get by multiplying the top and bottom by 2, 3, 4 and 5, and the decimal they all share.
| Simplest form | × 2 | × 3 | × 4 | × 5 | Decimal |
|---|---|---|---|---|---|
| 1/2 | 2/4 | 3/6 | 4/8 | 5/10 | 0.5 |
| 1/3 | 2/6 | 3/9 | 4/12 | 5/15 | ≈ 0.3333 |
| 2/3 | 4/6 | 6/9 | 8/12 | 10/15 | ≈ 0.6667 |
| 1/4 | 2/8 | 3/12 | 4/16 | 5/20 | 0.25 |
| 3/4 | 6/8 | 9/12 | 12/16 | 15/20 | 0.75 |
| 1/5 | 2/10 | 3/15 | 4/20 | 5/25 | 0.2 |
| 2/5 | 4/10 | 6/15 | 8/20 | 10/25 | 0.4 |
| 3/5 | 6/10 | 9/15 | 12/20 | 15/25 | 0.6 |
| 4/5 | 8/10 | 12/15 | 16/20 | 20/25 | 0.8 |
| 1/6 | 2/12 | 3/18 | 4/24 | 5/30 | ≈ 0.1667 |
| 5/6 | 10/12 | 15/18 | 20/24 | 25/30 | ≈ 0.8333 |
| 1/8 | 2/16 | 3/24 | 4/32 | 5/40 | 0.125 |
| 3/8 | 6/16 | 9/24 | 12/32 | 15/40 | 0.375 |
| 5/8 | 10/16 | 15/24 | 20/32 | 25/40 | 0.625 |
| 7/8 | 14/16 | 21/24 | 28/32 | 35/40 | 0.875 |
| 1/10 | 2/20 | 3/30 | 4/40 | 5/50 | 0.1 |
| 3/10 | 6/20 | 9/30 | 12/40 | 15/50 | 0.3 |
Read down any column and the denominators line up: in the × 4 column, halves, quarters and eighths become 4/8, 4/16 and 4/32. That alignment is what makes a common denominator possible. For sixteenths and more decimal places, see the fraction to decimal chart.
Where equivalent fractions are used
Adding and subtracting fractions
You can only add fractions that are cut into the same size pieces. Equivalent fractions are how you get there: rewrite each fraction over a common denominator, then add the tops.
The least common denominator calculator finds the smallest shared bottom number and rewrites every fraction over it, and the guide to adding fractions covers the rest of the method.
Comparing fractions
Which is bigger, 3/5 or 5/8? Rewrite both over 40: 3/5 = 24/40 and 5/8 = 25/40. Now the comparison is plain: 5/8 is larger by 1/40.
Percentages
A percentage is an equivalent fraction with 100 on the bottom. 3/4 = 75/100 = 75%, and 2/5 = 40/100 = 40%. When the denominator divides 100 evenly this is the quickest route from a score to a percentage, as in x out of y as a percentage.
Ratios
Equivalent ratios work exactly like equivalent fractions. 2:3, 4:6 and 10:15 describe the same mix, and the ratio calculator simplifies or scales them the same way.
Common mistakes
- Adding instead of multiplying. Adding 1 to the top and bottom of 1/2 gives 2/3, which is not equal to 1/2. Only multiplication or division keeps the value.
- Changing only one part. Doubling the bottom of 3/4 without doubling the top gives 3/8, which is half as big.
- Dividing by a number that does not go into both. 9/12 can be divided by 3 (giving 3/4) but not by 2, because 9 is odd.
- Stopping short of simplest form. 6/9 is equivalent to 12/18 but is not its simplest form; 2/3 is.
- Trusting rounded decimals. 0.33 and 0.333 look close, but 33/100 is not equivalent to 1/3. Use cross multiplication for an exact answer.
- Treating mixed numbers carelessly. 1 1/2 is equivalent to 3/2 and 6/4, not to 1/2 or 2/4. Convert to an improper fraction first if you need to scale it.
Frequently asked questions
What fractions are equivalent to 2/3?
4/6, 6/9, 8/12, 10/15 and 12/18 are all equivalent to 2/3. Each comes from multiplying the top and bottom of 2/3 by the same number: 2, 3, 4, 5 and 6. They all equal about 0.6667, and each one simplifies back to 2/3 when you divide by the common factor.
What fractions are equivalent to 3/4?
6/8, 9/12, 12/16, 15/20 and 75/100 are equivalent to 3/4. Multiply the top and bottom of 3/4 by any whole number to make another one. They all equal 0.75, which is why 3/4 is the same as 75%. Cross multiplying any pair, such as 3 × 16 = 4 × 12 = 48, confirms it.
Are 4/6 and 6/9 equivalent fractions?
Yes. Cross multiplying gives 4 × 9 = 36 and 6 × 6 = 36, and equal products mean equal fractions. Both also simplify to 2/3: divide 4/6 by 2, and 6/9 by 3. Neither one is obtained from the other by multiplying by a whole number, but they are still equivalent through 2/3.
How do you find an equivalent fraction with a given denominator?
Divide the new denominator by the old one to get the scaling number, then multiply the numerator by it. To write 3/5 over 20, work out 20 ÷ 5 = 4, then 3 × 4 = 12, so 3/5 = 12/20. If the new denominator is not a multiple of the old one, no whole-number equivalent exists.
How many equivalent fractions does a fraction have?
Infinitely many. You can multiply the top and bottom by 2, 3, 4 or any other whole number, and there is no largest one. 1/2 is equivalent to 2/4, 50/100, 500/1,000 and so on without end. Only one equivalent fraction is in simplest form, though: the one whose top and bottom share no factor except 1.
What is 5/8 as an equivalent fraction with a denominator of 24?
It is 15/24. The denominator goes from 8 to 24, which is multiplying by 3, so the numerator goes from 5 to 5 × 3 = 15. Both fractions equal 0.625. Checking by cross multiplication: 5 × 24 = 120 and 8 × 15 = 120.
Why do equivalent fractions have the same value?
Because multiplying the top and bottom by the same number is the same as multiplying the whole fraction by 1. For example, 3/4 × 2/2 = 6/8, and 2/2 equals 1. Multiplying by 1 never changes a number, so 6/8 has to equal 3/4. The pieces are smaller, but you have proportionally more of them.
Calculators for this
Every one of these shows the working, so you can check the method as well as the answer.
More guides
https://fourfunctioncalculator.com/equivalent-fractions/