Equivalent Fractions Calculator
Three tools in one: generate a chain of equivalent fractions for any fraction or mixed number, verify whether two fractions are equal, or solve for a missing numerator or denominator in a proportion. Enter your values and see the full working shown step by step.
Formula
Worked example
For 2/3: multiply by 2 gives 4/6, by 3 gives 6/9, by 4 gives 8/12, by 5 gives 10/15. Each equals 0.6667. The GCD of 2 and 3 is 1, so 2/3 is already in lowest terms. To check whether 4/6 and 8/12 are equivalent: cross-multiply to get 4 x 12 = 48 and 6 x 8 = 48. Equal, so confirmed.
What makes two fractions equivalent
Two fractions are equivalent when they represent exactly the same portion of a whole, even though they use different numbers. You create an equivalent fraction by multiplying, or dividing, the numerator and denominator by the same non-zero number. Because you scale the top and bottom by an identical factor, the ratio between them never changes. For example, 1/2, 2/4, 3/6, and 50/100 all describe one half. Visually, cutting a pizza into 4 slices and taking 2 is the same as cutting it into 8 slices and taking 4.
Three ways to check equivalence
The fastest method is cross-multiplication: for fractions a/b and c/d, compute a x d and b x c. If the products are equal, the fractions are equivalent. This avoids the need to find a common denominator first. A second method is to simplify both fractions to lowest terms by dividing by their greatest common divisors; if the simplified forms match, the fractions are equal. A third approach is to convert both fractions to decimals and compare the results. Cross-multiplication is generally fastest when the numbers are large.
Simplifying to lowest terms
The simplest form of a fraction is the one where the numerator and denominator share no common factor other than 1. To reach it, divide both numbers by their greatest common divisor (GCD). The GCD of two integers is found efficiently with the Euclidean algorithm: divide the larger by the smaller, take the remainder, and repeat until the remainder is zero; the last non-zero divisor is the GCD. Reducing a fraction makes it easier to read and compare, and it is usually the expected form for a final answer.
Solving for a missing numerator or denominator
When you know one fraction and part of an equivalent fraction, cross-multiplication gives you the missing value. Set up the proportion a/b = n/d and cross-multiply to get a x d = b x n. Rearrange to isolate whichever variable is unknown: n = (a x d) / b, or d = (b x n) / a. This is the same technique used in proportional reasoning, scaling recipes, reading maps, and converting between units. The solve mode in this calculator does exactly this arithmetic and shows every step.
Least common denominator and comparing fractions
When you need to add, subtract, or compare two fractions with different denominators, the least common denominator (LCD) is the smallest denominator that works for both. The LCD equals the least common multiple (LCM) of the two denominators. Once both fractions are rewritten over the LCD, their numerators can be compared or combined directly. For example, to compare 2/3 and 3/4, the LCD is 12: rewrite as 8/12 and 9/12. Now it is clear that 3/4 is larger. This calculator displays the LCD in comparison mode.
Common equivalent fraction chains
| Simplified | x 2 | x 3 | x 4 | x 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 |
| 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 |
Each row shows the same value expressed as a simplified fraction and four equivalents.
Frequently asked questions
How do I create an equivalent fraction?
Multiply the numerator and the denominator by the same non-zero whole number. For example, multiplying 3/4 by 2 gives 6/8, which has the same value. You can also divide both by a common factor to get a smaller equivalent fraction. The key rule is that whatever you do to the top, you must do identically to the bottom.
How do I check whether two fractions are equivalent?
Use cross-multiplication: for fractions a/b and c/d, multiply a x d and b x c. If both products are equal, the fractions are equivalent. For example, 2/3 and 4/6: 2 x 6 = 12 and 3 x 4 = 12, so they are equivalent. Alternatively, simplify both fractions to lowest terms and see if they match.
Are 2/3 and 4/6 equivalent?
Yes. Multiplying both the numerator and denominator of 2/3 by 2 gives 4/6. Cross-multiplying confirms it: 2 x 6 = 12 and 3 x 4 = 12. Both fractions equal 0.6667.
How do I know when a fraction is fully simplified?
A fraction is in lowest terms when the greatest common divisor of the numerator and denominator is 1, meaning no whole number other than 1 divides both evenly. For example, 6/9 is not fully simplified because both share the factor 3; dividing gives 2/3, which is fully simplified.
How do I find a missing numerator or denominator?
Set up the proportion and cross-multiply. For example, if 3/4 = ?/8, then ? = (3 x 8) / 4 = 6. This is the same as asking: what number gives 8 when multiplied by 4/3? Use the Solve mode in this calculator to handle these automatically.
What is the least common denominator and why does it matter?
The least common denominator (LCD) is the smallest number that is a multiple of both denominators. It matters when you need to add, subtract, or compare fractions with different bottoms: rewriting each fraction with the LCD lets you work with matching denominators. For 1/4 and 1/6, the LCD is 12, giving 3/12 and 2/12.
Can this calculator handle negative fractions?
Yes. Enter a negative numerator to represent a negative fraction. For example, -2/3 is a valid input in generate mode. The simplification always moves the negative sign to the numerator and keeps the denominator positive.