LCD Least Common Denominator

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Equivalent Fractions Calculator

Type any fraction and instantly see twelve fractions that equal it, its simplified lowest-terms form, and, if you give it a target denominator, the exact equivalent fraction that lands on that denominator.

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EQUIVALENT FRACTIONS Live

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Simplest Form

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×	Numerator	Denominator	Fraction

01 · DEFINITION

What Are Equivalent Fractions?

Equivalent fractions are different ways of writing the exact same amount. The numbers on top and bottom change, but the value the fraction represents never does.

Same Value, Different Numbers

¹₂, ²₄, ³₆, and ⁴₈ all describe half of something. Cut a pizza into 2 slices and take 1, or cut it into 8 slices and take 4. Either way, you have eaten exactly half the pizza. That shared value is what makes a whole family of fractions "equivalent" to one another.

Mathematically, a/b and c/d are equivalent whenever a × d equals b × c. Every fraction has an infinite number of equivalent forms, because you can always multiply the numerator and denominator by any whole number without changing what the fraction is worth. Among all of those equivalent forms, exactly one is the simplest: the version where the numerator and denominator share no common factor besides 1.

Seeing It as Equal Bars

Each bar below is the same length. The first is split into 2 equal parts with 1 shaded, the second into 4 parts with 2 shaded, and the third into 8 parts with 4 shaded. The shaded area, the actual amount, never changes, only the number of slices used to show it.

02 · METHODS

How to Find Equivalent Fractions

There are only two operations you ever need: scaling a fraction up by multiplying, or scaling it down by dividing out the greatest common factor.

Multiply Numerator and Denominator by the Same Number

Pick any whole number greater than 1 and multiply both the top and bottom of the fraction by it. Because that number divided by itself equals 1, you are really just multiplying the fraction by 1 in disguise. The value stays put even though the digits change.

Choose a multiplier, such as 4.
Multiply the numerator by that multiplier.
Multiply the denominator by the same multiplier.
The new fraction is equivalent to the original.

3/5 × 4/4 = 12/20

Divide Numerator and Denominator by the GCF

To move in the other direction, shrinking a fraction toward its simplest form, find the greatest common factor (GCF) of the numerator and denominator, then divide both by it. This is the exact reverse of multiplying and produces an equivalent fraction with smaller numbers.

Find the GCF of the numerator and denominator.
Divide the numerator by the GCF.
Divide the denominator by the GCF.
If the GCF is 1, the fraction is already in its simplest form.

GCF(8, 12) = 4 → 8/12 ÷ 4/4 = 2/3

03 · CONNECTION

Equivalent Fractions and the LCD

Finding a least common denominator is really just a search for the right equivalent fractions: ones that all happen to share the same denominator so they can be added, subtracted, or compared directly.

You cannot add or compare fractions that have different denominators without first rewriting them. The least common denominator (LCD) is the smallest number that every original denominator divides into evenly, and once you have it, each fraction is converted into its equivalent form over that shared denominator using the same multiply-the-top-and-bottom technique from Section 02.

Multiples of 4:

4 8 12 16

Multiples of 6:

6 12 18

LCD(1/4, 1/6) = LCM(4, 6) = 12

With the LCD found, rewrite each fraction as an equivalent fraction over 12:

Denominator 4 needs to become 12, so multiply by 3/3: 1/4 × 3/3 = 3/12.
Denominator 6 needs to become 12, so multiply by 2/2: 1/6 × 2/2 = 2/12.
Both fractions now share denominator 12: 3/12 and 2/12. They can be added directly to get 5/12.

Every step of finding an LCD depends on this same idea: an equivalent fraction is not a different number, just a different name for it, chosen so it lines up with the others.

04 · SIMPLIFYING

How to Simplify Fractions

Simplifying is the mirror image of building an equivalent fraction. Instead of scaling up, you divide out the largest number that fits evenly into both the numerator and the denominator.

Start with the fraction 8/12. To simplify it, find the greatest common factor of 8 and 12: their shared factors are 1, 2, and 4, so the GCF is 4. Dividing both the numerator and denominator by 4 produces the fraction's simplest form.

GCF(8, 12) = 4

8 ÷ 4 = 2 12 ÷ 4 = 3 → 2/3

A fraction is fully simplified, in "lowest terms," once its numerator and denominator have no common factor left besides 1. If you divide by a number that is not the full GCF, you will need a second round of dividing to finish the job, so it helps to find the largest shared factor up front rather than guessing.

05 · EXAMPLES

Worked Equivalent Fraction Examples

Six starting fractions, three equivalent forms each, and the simplest form they all reduce back down to.

Example 1 ×2, ×3, ×4

¹ ₂

2 4 = 3 6 = 4 8

×2: 1 × 2 = 2, 2 × 2 = 4
×3: 1 × 3 = 3, 2 × 3 = 6
×4: 1 × 4 = 4, 2 × 4 = 8

Simplest Form 1/2

Already in lowest terms: the GCF of 1 and 2 is 1.

Example 2 ×2, ×3, ×4

³ ₄

6 8 = 9 12 = 12 16

×2: 3 × 2 = 6, 4 × 2 = 8
×3: 3 × 3 = 9, 4 × 3 = 12
×4: 3 × 4 = 12, 4 × 4 = 16

Simplest Form 3/4

Already in lowest terms: the GCF of 3 and 4 is 1.

Example 3 ×2, ×3, ×4

² ₅

4 10 = 6 15 = 8 20

×2: 2 × 2 = 4, 5 × 2 = 10
×3: 2 × 3 = 6, 5 × 3 = 15
×4: 2 × 4 = 8, 5 × 4 = 20

Simplest Form 2/5

Already in lowest terms: the GCF of 2 and 5 is 1.

Example 4 ×2, ×3, ×4

⁶ ₉

12 18 = 18 27 = 24 36

×2: 6 × 2 = 12, 9 × 2 = 18
×3: 6 × 3 = 18, 9 × 3 = 27
×4: 6 × 4 = 24, 9 × 4 = 36

Simplest Form 2/3

GCF(6, 9) = 3, so 6/ 9 reduces to 2/3.

Example 5 ×2, ×3, ×4

⁸ ₁₂

16 24 = 24 36 = 32 48

×2: 8 × 2 = 16, 12 × 2 = 24
×3: 8 × 3 = 24, 12 × 3 = 36
×4: 8 × 4 = 32, 12 × 4 = 48

Simplest Form 2/3

GCF(8, 12) = 4, so 8/ 12 reduces to 2/3.

Example 6 ×2, ×3, ×4

⁵ ₆

10 12 = 15 18 = 20 24

×2: 5 × 2 = 10, 6 × 2 = 12
×3: 5 × 3 = 15, 6 × 3 = 18
×4: 5 × 4 = 20, 6 × 4 = 24

Simplest Form 5/6

Already in lowest terms: the GCF of 5 and 6 is 1.

06 · FAQ

Equivalent Fractions: Frequently Asked Questions

What makes two fractions equivalent?

Two fractions are equivalent when they represent the exact same value even though the numerator and denominator are different numbers. Formally, a/b and c/d are equivalent when a × d = b × c. You can always create an equivalent fraction by multiplying (or dividing) both the numerator and denominator by the same non-zero number, because that operation is the same as multiplying by 1.

How do you find an equivalent fraction with a specific denominator?

Divide the target denominator by your current denominator. If it divides evenly, that quotient is your scaling factor. Multiply both the numerator and denominator by it. For example, to rewrite 3/5 with a denominator of 20, divide 20 by 5 to get 4, then multiply 3 × 4 and 5 × 4 to get 12/20. If the target denominator does not divide evenly by your current one, that exact denominator is not reachable through simple scaling.

Are 3/4 and 6/8 equivalent fractions?

Yes. Multiplying both the numerator and denominator of 3/4 by 2 gives 6/8, so the two fractions describe the same quantity. You can double check with cross-multiplication: 3 × 8 = 24 and 4 × 6 = 24. Since both products match, the fractions are equivalent.

How do you simplify a fraction?

Find the greatest common factor (GCF) of the numerator and denominator, then divide both by it. The result is the same fraction written with the smallest possible whole numbers, known as lowest terms or simplest form. A fraction is fully simplified once its numerator and denominator share no common factor other than 1.

What is the simplest form of 6/9?

The GCF of 6 and 9 is 3. Dividing both the numerator and denominator by 3 gives 2/3, which is the simplest form of 6/9. Every equivalent fraction of 6/9, including 12/18, 18/27, and 24/36, also simplifies down to that same 2/3.