Comparing Fractions Calculator
Enter two fractions or mixed numbers and this calculator tells you which is larger, smaller, or whether they are equal. It shows the comparison symbol (<, >, or =) and works through the problem three ways: converting to decimals, finding the least common denominator, and cross-multiplication. Toggle between simple fractions and mixed numbers using the mode selector. All steps update in real time as you type.
How to compare fractions
Two fractions can be compared by converting them to a common form. The three most widely taught methods are decimal conversion, the least common denominator (LCD) method, and cross-multiplication. This calculator shows all three so you can see how each technique arrives at the same answer. Decimal conversion is the fastest check: divide the top number by the bottom for each fraction and compare the results. If 3/4 = 0.75 and 2/3 = 0.667, then 3/4 is larger. The LCD method is preferred in classroom settings because it keeps everything in fraction form. First, find the smallest number that both denominators divide into evenly, then scale both fractions to that denominator and compare the numerators. Cross-multiplication skips the LCD step: multiply each numerator by the other fraction's denominator and compare the two products. If n1 x d2 is greater than n2 x d1, then the first fraction is larger.
Comparing mixed numbers
Mixed numbers such as 1 3/4 or 2 1/2 contain a whole-number part and a fractional part. The cleanest way to compare them is to convert each to an improper fraction first: multiply the whole number by the denominator, add the numerator, and keep the same denominator. So 1 3/4 becomes (1 x 4 + 3)/4 = 7/4, and 2 1/2 becomes (2 x 2 + 1)/2 = 5/2. With both values as improper fractions, any of the three comparison methods apply normally. Alternatively, compare the whole-number parts first. If they differ, the fraction with the larger whole number is larger and you need not look at the fractional parts at all. Only if the whole-number parts are equal do you need to compare the fractional portions.
Finding the least common denominator
The LCD of two fractions is the least common multiple (LCM) of their denominators. One systematic approach is prime factorization: factor each denominator into primes, then take the highest power of every prime that appears. For example, 12 = 2^2 x 3 and 8 = 2^3, so LCM(12, 8) = 2^3 x 3 = 24. A faster shortcut uses the relationship LCM(a, b) = (a x b) / GCD(a, b), where GCD is the greatest common divisor found with the Euclidean algorithm. Once the LCD is known, multiply each fraction's numerator and denominator by whatever factor turns its denominator into the LCD. After that, numerators can be compared directly because the denominators are the same.
Ordering more than two fractions
To sort a list of fractions from smallest to largest, find the LCD of all the denominators (the LCM of the whole set), convert every fraction to that common denominator, and then sort by numerator. Alternatively, convert all fractions to decimals and sort the decimal values. For fractions like 1/2, 2/5, 3/7 and 4/9, the LCD of 2, 5, 7, and 9 is 630. Converting each: 1/2 = 315/630, 2/5 = 252/630, 3/7 = 270/630, 4/9 = 280/630. Sorting by numerator gives 252 < 270 < 280 < 315, so the order is 2/5 < 3/7 < 4/9 < 1/2.
Common fraction comparison methods
| Method | How it works | Best used when |
|---|---|---|
| Decimal conversion | Divide numerator by denominator for each fraction, then compare the decimals | Quick estimate or when a calculator is handy |
| Least common denominator (LCD) | Find the LCM of the two denominators, rewrite both fractions with that denominator, then compare numerators | Exact comparison needed; denominators share common factors |
| Cross multiplication | Multiply each numerator by the other fraction's denominator; compare the two products | Speed comparison of two fractions without finding the LCD |
| Same denominator (shortcut) | If denominators are already equal, just compare numerators directly | Fractions already share a denominator |
| Same numerator (shortcut) | If numerators are equal, the fraction with the smaller denominator is larger | Fractions already share a numerator |
Three reliable methods for comparing any two fractions. All three should give the same answer.
Frequently asked questions
How do you compare fractions with different denominators?
The two most common approaches are the LCD method and cross-multiplication. For the LCD method, find the least common multiple of the two denominators, convert both fractions to equivalent fractions with that common denominator, then compare numerators. The larger numerator belongs to the larger fraction. For cross-multiplication, multiply the first numerator by the second denominator and the second numerator by the first denominator. Compare those two products: the larger product corresponds to the larger fraction. Both methods always give the same result.
Which is bigger: 3/4 or 2/3?
3/4 is bigger. Converting to decimals: 3/4 = 0.75 and 2/3 = 0.6667. Using the LCD method: LCD(4, 3) = 12, so 3/4 = 9/12 and 2/3 = 8/12. Since 9 > 8, we have 3/4 > 2/3. Cross-multiplication confirms this: 3 x 3 = 9 and 2 x 4 = 8, and 9 > 8.
How do you compare mixed numbers like 1 2/3 and 1 3/4?
First compare the whole-number parts. Both are 1, so you must compare the fractional parts: 2/3 vs 3/4. Using cross-multiplication: 2 x 4 = 8 and 3 x 3 = 9. Since 8 < 9, we have 2/3 < 3/4, so 1 2/3 < 1 3/4. Alternatively, convert both to improper fractions: 1 2/3 = 5/3 and 1 3/4 = 7/4. LCD(3, 4) = 12, giving 20/12 vs 21/12, confirming 5/3 < 7/4.
What does it mean when two fractions are equivalent?
Two fractions are equivalent if they represent the same value, even though their numerators and denominators are different. For example, 1/2, 2/4, 3/6, and 50/100 are all equivalent because they all equal 0.5. Equivalent fractions simplify to the same lowest-terms fraction: divide numerator and denominator by their greatest common divisor and you always get the same result.
How do you compare a fraction and a decimal?
Convert the fraction to a decimal by dividing its numerator by its denominator, then compare the two decimal values directly. For example, to compare 3/8 and 0.4: 3/8 = 0.375, and 0.375 < 0.4, so 3/8 < 0.4. You can also go the other direction and convert the decimal to a fraction before comparing.
What is the cross-multiplication method for comparing fractions?
To compare a/b and c/d using cross-multiplication, compute a x d (the first numerator times the second denominator) and c x b (the second numerator times the first denominator). If a x d is greater than c x b, then a/b is larger. If a x d equals c x b, the fractions are equal. If a x d is less than c x b, then a/b is smaller. This method works because it effectively scales both fractions to the same denominator (b x d) without actually computing it.