Equivalent Fractions Calculator

An equivalent fractions calculator takes one fraction and lists the other fractions that name the same number. Enter 3/4 and it returns 6/8, 9/12, 12/16, 15/20 and more, all the way to 36/48. Every one of those equals 0.75, or 75 percent. The second mode checks whether any two fractions you type are equivalent.

Three neighboring tools cover the rest of the fraction toolkit: the Simplify Fractions Calculator moves in the opposite direction and reduces a fraction to lowest terms, the Fraction Calculator adds, subtracts, multiplies and divides two fractions, and the GCF Calculator finds the greatest common factor that reduction depends on. Use this page when you need the family of equal fractions rather than a single answer.

How equivalent fractions work

Two fractions are equivalent when they occupy the same point on the number line. The rule that builds them is one line long:

The reason it is safe is that k/k equals 1, and multiplying a number by 1 leaves it alone. Scaling 3/4 by 5 is really 3/4 × 5/5, which is 3/4 × 1. The same identity read backwards is division: if the numerator and the denominator share a factor, dividing both by it also preserves the value, which is how a fraction gets reduced.

To test a pair of fractions instead of building new ones, use the cross-product form of the same idea:

That version uses only whole-number multiplication, so it never rounds and never needs a common denominator. It is the test this calculator runs in check mode.

Worked example: every equivalent of 3/4

Start with 3/4. Multiply the top and the bottom by 2: the numerator becomes 3 × 2 = 6 and the denominator becomes 4 × 2 = 8, giving 6/8. Multiply by 3 instead and you get 9/12; by 4, 12/16. Carrying that through k = 12 produces the eleven fractions in the table below. The decimal column is the proof that nothing moved: every row divides out to 0.75.

The list does not stop at 12. Continuing gives 39/52, 42/56, 45/60 and, at k = 25, 75/100 - which is exactly why 3/4 is 75 percent. There are infinitely many members, and 3/4 itself is the only one in lowest terms.

Second worked example: going down instead of up

Equivalence runs in both directions. Take 12/18. The greatest common divisor of 12 and 18 is 6, so dividing both parts by 6 gives 12 ÷ 6 = 2 over 18 ÷ 6 = 3, that is 2/3. Both fractions equal 0.666667 to six places. Once you know the lowest-terms form, the entire family is just its multiples: 2/3, 4/6, 6/9, 8/12, 10/15, 12/18, 14/21, 16/24, 18/27, 20/30, 22/33, 24/36. Notice that the fraction you started with is simply the sixth member. That is what the calculator means when it highlights one entry in the family list: your input is one rung on a ladder that runs forever in one direction and stops at lowest terms in the other.

Equivalent fractions chart

The chart below scales twelve common fractions by 2 through 6. Read across a row to find a form with the denominator you need: to rewrite 5/8 with a denominator of 24, the row shows 15/24. The last two columns give the same value as a decimal and a percent.

Checking a pair with cross products

Cross-multiplying answers the yes-or-no question in one step. Write the two fractions side by side, multiply the first numerator by the second denominator, then the first denominator by the second numerator, and compare. The table shows both products for six pairs.

The 2/5 and 6/20 row is the classic trap. The numerator was tripled but the denominator was quadrupled, so the value slipped from 0.4 down to 0.3. Whenever the two scale factors differ, the fractions are not equivalent, and the cross products catch it immediately.

Filling in a missing number

School worksheets usually ask the question in the form 3/5 = ?/45. Solve it in two steps. First find the scale factor by dividing the new denominator by the old one: 45 ÷ 5 = 9. Then apply that factor to the numerator: 3 × 9 = 27, so the answer is 27/45. If the blank is on the bottom instead, work from the numerators: in 4/5 = 28/?, the factor is 28 ÷ 4 = 7, so the denominator is 5 × 7 = 35. Either way, the scale factor is the same number on top and bottom, which is the whole point of equivalence. You can confirm the answer here by entering the original fraction and reading down the list until the denominator you want appears.

How to use this calculator

1. Pick a mode. Find equivalents takes one fraction and returns a list; Check two fractions takes a pair and returns a verdict.
2. Type the numerator and the denominator. Whole numbers only, and the denominator cannot be 0. Negative values are allowed, and the sign moves to the numerator automatically.
3. Choose how many to list. Five is enough for a quick homework check; nineteen gives you a long chart to scan for a specific denominator.
4. Read the headline. The first four equivalents appear at the top with the shared decimal and percent value directly underneath.
5. Scan the table. Each row shows the arithmetic in full, so you can copy the working into a worksheet rather than just the answer.
6. Check the family strip. It lists the first twelve multiples of the lowest-terms form and highlights the fraction you entered, which makes it obvious where your fraction sits.

In check mode the result card shows both cross products side by side, the decimal value of each fraction, and the lowest-terms form of both. When the verdict is no, the card also reports the size of the gap between the two values.

Who this calculator is for

• Students in grades 3 through 7 working through equivalent-fraction worksheets and needing the steps, not just the answer.
• Parents and tutors checking homework quickly and explaining why 6/8 and 3/4 are the same amount.
• Teachers generating practice sets or a printable chart of equivalents for a specific denominator.
• Cooks scaling recipes, who need to know that 3/4 cup is 12/16 cup when the only measure on hand is a sixteenth.
• Carpenters and machinists converting between eighths, sixteenths and thirty-seconds on a tape measure or a drill index.
• Anyone comparing odds or ratios, where 3 out of 4 and 15 out of 20 have to be recognized as the same rate.

Key terms

• Numerator: the top number, counting how many parts you have.
• Denominator: the bottom number, saying how many equal parts make one whole. It can never be 0.
• Equivalent fractions: fractions written differently that represent the same value, such as 3/4 and 27/36.
• Simplest form (lowest terms): the one member of the family whose numerator and denominator share no factor other than 1.
• Greatest common divisor (GCD or GCF): the largest whole number dividing both parts exactly. Dividing by it reduces the fraction in a single step.
• Common denominator: a shared bottom number that lets two fractions be added or compared directly.
• Cross product: the diagonal product a × d or b × c used to test whether a/b and c/d are equal.
• Scale factor: the number k that both parts are multiplied by. The same k must apply to the top and the bottom.

Why bigger equivalents matter: common denominators

Reducing gets most of the classroom attention, but scaling up is what real fraction arithmetic runs on. To add 3/4 and 5/6 you first need a common denominator. The least common multiple of 4 and 6 is 12, so rewrite each fraction over 12: 3/4 becomes 9/12 (scale factor 3) and 5/6 becomes 10/12 (scale factor 2). Now the addition is trivial, 9 + 10 = 19, giving 19/12, which is 1 and 7/12. Comparison works the same way: 5/8 versus 7/12 is hard to eyeball, but rewritten over 24 they become 15/24 and 14/24, so 5/8 is larger. The LCM Calculator finds the denominator to scale to, and this page produces the scaled numerators.

What changes the result

• The starting fraction. If your input is not in lowest terms, the family also contains smaller fractions than the one you typed. Entering 12/18 puts you six rungs up a ladder that begins at 2/3.
• The multiplier range. Listing 5 equivalents stops at k = 6; listing 19 runs to k = 20, which is often what you need to hit a denominator such as 100.
• Sign. A negative fraction keeps its sign through every multiplication, so -3/4 generates -6/8, -9/12 and so on.
• Improper fractions. When the numerator is larger than the denominator the calculator also prints the mixed-number form, because 15/4 and 3 3/4 are the same value.
• Zero on top. 0/5 is a legal fraction equal to 0, and every equivalent is 0 over something. Zero on the bottom is not legal and is rejected.

Tips for working with equivalent fractions

• Multiply, do not add. Adding 1 to the top and bottom of 1/2 gives 2/3, a completely different value.
• Reduce first when you can. Working from 2/3 is easier than working from 12/18, and the family is the same.
• To reach a target denominator, divide the target by the current denominator. If the division is not exact, no whole-number multiplier gets you there.
• To convert a simple fraction to a percent, scale the denominator to 100 when possible: 3/4 becomes 75/100, so 75 percent.
• Use the cross-product test rather than decimals when the numbers are large, because it never rounds.
• Sanity-check with a picture. Four quarters of a pizza and eight eighths of the same pizza cover the same area.

Limitations

• Inputs must be whole numbers. For 2.5/10 or 0.375, convert with the Decimal to Fraction Calculator first.
• The calculator lists up to nineteen equivalents. The true family is infinite, so a very large target denominator may not appear in the visible list.
• Numerators and denominators are capped at 100,000 to keep the arithmetic exact in standard JavaScript integers.
• Decimals are shown rounded to six places for display. The equivalence decision itself never uses them.
• Mixed numbers are shown as output but are not accepted as input; convert with the Mixed Number Calculator.

Which fraction tool to use

Each page answers a different question, so pick by what you actually need:

• To reduce one fraction to lowest terms with the steps shown, use the Simplify Fractions Calculator.
• To add, subtract, multiply or divide two fractions, use the Fraction Calculator.
• To find the common factor behind any reduction, use the GCF Calculator; for the denominator to scale up to, use the LCM Calculator.
• To turn a decimal such as 0.375 into a fraction, use the Decimal to Fraction Calculator.
• To convert between improper fractions and mixed numbers, use the Mixed Number Calculator.
• For the same scaling idea written as a:b, use the Ratio Calculator, and to solve a/b = c/d for one unknown, the Proportion Calculator.