Equivalent Fractions Calculator

What makes two fractions equivalent

A fraction is a division problem with a different hat on. The top is the amount, the bottom is the size of the pieces, and 3/4 means 3 divided by 4, which is 0.75. Two fractions are equivalent when they name the same number even though the digits differ. The list 1/2, 2/4, 4/8 and 50/100 all equal 0.5, so all four are equivalent to each other.

Picture a pizza. Cut it in half and take one piece, then cut a second pizza into eight slices and take four. The amount on your plate is identical. Nothing was added or removed, the pieces were cut smaller and more of them were needed to fill the same space.

The idea earns its keep when fractions have to be added, subtracted or ranked, because those jobs need the same size pieces. Renaming a fraction changes how it looks, never what it is worth.

Multiply the top and bottom by the same number

Pick a fraction and a whole number above zero, then multiply the numerator and the denominator by it. The result is equivalent because you multiplied by k/k, and any number over itself equals 1. Multiplying by 1 never changes a value.

Start with 3/4. Multiply both parts by 2: 3 x 2 = 6 and 4 x 2 = 8, so 3/4 = 6/8. By 3 it becomes 9/12, by 4 it becomes 12/16, and by 5 it becomes 15/20. Each of those equals 0.75.

Renaming is not the same as doubling a recipe. 3/4 cup renamed is 6/8 cup, still three quarters of a cup. Doubling the batch means 3/4 x 2 = 6/4, which is 1 1/2 cups, a different amount of food. Keep the two moves apart and most kitchen mistakes disappear.

Simplify by dividing out the greatest common factor

Going the other way shrinks the numbers. Find the largest number that divides both parts evenly, called the greatest common factor, and divide top and bottom by it.

Both parts of 15/45 divide by 15, so 15/45 = 1/3. Both parts of 6/8 divide by 2, so 6/8 = 3/4. Both parts of 9/12 divide by 3, so 9/12 = 3/4. When the only shared factor is 1, the fraction is already in lowest terms, as with 7/12 and 5/8.

Two shortcuts carry most of the work. Even numbers share a factor of 2, so halve first and look again. If both parts end in 0 or 5, divide by 5. Repeat until nothing is left to divide out, and the answer matches one pass with the greatest common factor.

Cross multiplication: the yes or no test

To test two fractions without a calculator, multiply along the diagonals: the top of the first times the bottom of the second, then the bottom of the first times the top of the second. Matching products mean matching values.

For 6/8 and 9/12: 6 x 12 = 72 and 8 x 9 = 72, so the two fractions are equivalent. For 5/8 and 3/5: 5 x 5 = 25 and 8 x 3 = 24, so they are not, and since 25 is the bigger product, 5/8 is the larger fraction.

The test works because cross multiplying is the same as handing both fractions a common denominator and then comparing the tops. Multiply 6/8 by 12/12 and 9/12 by 8/8 and both turn into 72/96.

Common denominators for adding and subtracting

Only pieces of the same size can be added. The least common denominator is the smallest number both bottoms divide into, which is the least common multiple of the two denominators.

For 3/4 + 1/6: the multiples of 4 run 4, 8, 12 and the multiples of 6 run 6, 12, so 12 comes first in both lists. Rename each fraction: 3/4 = 9/12 and 1/6 = 2/12. Now the tops add: 9/12 + 2/12 = 11/12, about 0.9167 of a cup.

When the two denominators share no factor, the least common multiple is just their product. 3 and 5 give 15, so 1/3 + 1/5 = 5/15 + 3/15 = 8/15.

Subtraction runs the same steps. 5/6 - 1/4 uses 12 as the least common denominator, giving 10/12 - 3/12 = 7/12.

Rulers, tape measures and recipe cups

Fractions live on every tape measure in the country, and the tick marks are equivalent fractions printed in a row. On a tape marked in sixteenths, 3/8 in sits on the sixth small mark because 3/8 = 6/16. Half an inch is the eighth mark, since 1/2 = 8/16, and 3/4 in is the twelfth, since 3/4 = 12/16.

Adding two lengths means renaming to the smallest shared unit. 5/16 in plus 3/8 in becomes 5/16 + 6/16 = 11/16 in, which a caliper reads as 0.6875 in. Doing it the other way, 3/8 plus 5/16 in decimals, gives 0.375 + 0.3125 = 0.6875, the same answer.

Cups behave the same way. A recipe calling for 2/3 cup of one liquid and 1/4 cup of another needs 8/12 + 3/12 = 11/12 cup in total, just shy of a full cup. Most cup sets skip twelfths, so measure 2/3 cup and 1/4 cup separately rather than hunting for an 11/12 measure.

Missing number problems

Worksheets ask for the blank in 3/5 = ?/20. Cross multiply the parts you have and divide by the one left over. The bottom grew from 5 to 20, which is 5 x 4 = 20, so the top grows the same way: 3 x 4 = 12, and 12/20 checks out at 0.6.

When the bottom shrinks, divide instead. 18/24 = ?/4 needs the bottom cut from 24 to 4, which is 24 / 6 = 4, so the top is 18 / 6 = 3 and the answer is 3/4.

The check is simple and catches most errors: whatever happens on the bottom has to happen on the top. If 5 turned into 20 by a factor of 4, then 3 has to meet the same factor of 4, and 3 x 4 = 12 confirms it.

Where people go wrong

Adding tops to tops and bottoms to bottoms is the classic slip. 1/2 + 1/3 is not 2/5. The real sum is 5/6, since 3/6 + 2/6 = 5/6, and 5/6 is 0.8333 while 2/5 is only 0.4.

Second slip: changing one part and not the other. Renaming 3/4 by touching only the top gives 6/4, which is twice the value. The bottom has to move as well, which is how you land on 6/8.

Third: dividing by different numbers. Cutting 8/12 down to 4/6 divides both parts by 2 and keeps the value. Cutting it to 4/8 would halve the top and cut the bottom by a third, which changes the number.

Fourth: ranking fractions by the top number alone. 2/3 beats 3/5 even though 2 is smaller than 3, because 2/3 = 0.6667 and 3/5 = 0.6. Cross multiplication settles it: 2 x 5 = 10 against 3 x 3 = 9.

For the classroom view, Teaching Fractions According to the Common Core Standards (Hung-Hsi Wu, University of California, Berkeley, 2011, revised 2014) treats cross multiplication as a result of common denominators rather than a trick, and Math Resources: Fractions (National University, checked October 2026) works the reduce-and-rename steps, including 15/45 reducing to 1/3. A fifth grade math curriculum (Williamson County Schools, Tennessee, 2022) states the same standard with 2/3 plus 5/4 becoming 8/12 plus 15/12, which adds to 23/12.