Which is larger: three-fifths or five-eighths? It is easy to look at the five and assume five-eighths must win, but fractions do not work like separate whole numbers. The denominator tells you how many equal parts make one whole, and the numerator tells you how many of those parts you have.
There are several dependable ways to compare fractions. If the denominators match, the answer is immediate. If they differ, you can find a common denominator, convert to decimals or use cross multiplication. The best method is the one that makes the meaning clear and gives you an answer you can check.
Start with what a fraction represents
In a fraction such as 3/5, the bottom number, 5, is the denominator. It divides the whole into five equal parts. The top number, 3, is the numerator, so you have three of those pieces.
A single fifth is larger than a single eighth because five equal parts of the same whole are bigger than eight equal parts. That is why comparing denominators in isolation is misleading.
Always assume the fractions refer to the same-sized whole. Half a very large pizza can contain more food than three-quarters of a tiny pizza; the fractions alone compare proportions, not differently sized objects.
When denominators are the same
Compare 2/7 with 5/7. Both fractions divide the whole into seven equal parts. Five of those pieces are more than two, so 5/7 is larger.
This is the easiest case. With matching positive denominators, the larger numerator makes the larger fraction. You do not need to find decimals or multiply anything.
If the numerators and denominators are identical, the fractions are equal. Write 2/7 < 5/7 for less than, 5/7 > 2/7 for greater than, and 3/7 = 3/7 for equality.
When the numerators match
Compare 3/4 and 3/8. Both describe three pieces, but quarters are bigger than eighths. Therefore 3/4 is greater.
For positive fractions smaller than or equal to one, with equal positive numerators, the fraction with the smaller denominator is larger. Three pieces each measuring a quarter outrank three pieces each measuring an eighth.
This shortcut is particularly helpful with simple everyday fractions such as 1/2, 1/3 and 1/4. Do not extend it blindly when numerator signs or other conditions change.
Use a common denominator
Suppose you need to compare 3/5 and 5/8. A common denominator is 40, because 5 and 8 both divide into 40.
Multiply 3/5 by 8/8 to obtain 24/40. Multiply 5/8 by 5/5 to obtain 25/40. Because 25/40 is larger than 24/40, five-eighths is larger than three-fifths.
Multiplying the top and bottom by the same nonzero number does not change the value of the fraction. That equivalent-fractions property is explained in OpenStax's prealgebra materials.
Find the smallest convenient common denominator
You can always multiply denominators, but it may create unnecessarily large numbers. For 3/4 and 5/6, multiplying 4 by 6 gives 24, which works. Yet 12 is a smaller common denominator.
Convert 3/4 to 9/12 and 5/6 to 10/12. It becomes clear that 5/6 is greater without handling larger figures.
The smallest positive common multiple of the denominators is the least common denominator. Finding it saves arithmetic, but any positive common denominator gives the correct comparison.
Cross multiplication as a shortcut
For fractions a/b and c/d with positive denominators, compare a × d with c × b. This uses the same idea as common denominators but avoids writing the intermediate fractions.
To compare 3/5 and 5/8, calculate 3 × 8 = 24 and 5 × 5 = 25. Because 24 is smaller, 3/5 < 5/8.
Write the products in a way that keeps their origins clear. Many wrong answers come from multiplying the two top numbers together instead of diagonally across.
Convert fractions into decimals
You can divide each numerator by its denominator. For example, 3/5 = 0.6 and 5/8 = 0.625. Since 0.625 is larger than 0.600, five-eighths wins.
This method is useful with a calculator or familiar fractions. It is also good when the final answer needs to be expressed in decimal form.
Be careful with recurring decimals. One-third is 0.3333... rather than exactly 0.33. Rounding too early can make nearly equal fractions appear identical.
Use a benchmark like one-half
Sometimes you can compare without full calculations. Consider 2/5 and 5/9. Two-fifths is 0.4, clearly below one-half. Five-ninths is a little more than one-half, since half of nine is 4.5.
You already know 5/9 > 2/5. This mental technique is convenient when one fraction lies clearly on each side of a familiar benchmark.
One, one-half and one-quarter are useful reference points. Benchmarks are less helpful when both fractions are extremely close together, such as 49/100 and 50/101.
What if one fraction is negative?
A negative fraction is less than a positive one. To compare -1/2 and -3/4, remember that -0.5 is greater than -0.75 because it sits closer to zero on the number line.
You can convert both to decimals or work with common denominators. With equal positive denominators, the usual ordering of negative numerators applies: -2 is greater than -5.
If a denominator is negative, rewrite the negative sign in the numerator or in front of the entire fraction first. This keeps your comparison method consistent.
Compare mixed numbers
A mixed number combines a whole part and a fraction, such as 2 1/3. Start by comparing whole parts. Any positive mixed number beginning with 3 is larger than one beginning with 2.
If the whole parts match, compare fractional parts. For 2 3/5 versus 2 5/8, the comparison reduces to 3/5 versus 5/8, so the second mixed number is larger.
Alternatively convert mixed numbers into improper fractions, where the numerator is at least as large as the denominator. This works well with cross multiplication.
Compare fractions of real quantities
Suppose one recipe uses 3/4 cup of oats and another uses 2/3 cup. Convert to twelfths: 9/12 versus 8/12. The first recipe uses more.
For discounts, a shop offering 1/4 off has a smaller percentage reduction than one offering 1/3 off, assuming the starting price is the same.
When comparing rates such as kilometres per hour, make sure units match before interpreting a fraction. A mathematically correct fraction comparison cannot fix mismatched units.
Practise with a few checks
Try 7/10 versus 2/3. Convert to thirtieths: 21/30 versus 20/30, so 7/10 is larger. Next, compare 4/9 with 1/2: 4/9 is less than half.
Finally compare 6/8 and 3/4. Simplifying 6/8 by dividing top and bottom by 2 gives 3/4, so they are equal. Different-looking fractions can represent the same amount.
Explain your answer in a sentence rather than just writing a symbol. 'Seven-tenths is larger because twenty-one thirtieths exceeds twenty thirtieths' demonstrates that you understand the result.
The best method to choose
If denominators are the same, compare numerators. If numerators are the same, consider piece size. Otherwise find a common denominator or cross multiply, with decimals useful as an alternative.
A quick reasonableness check is worth doing. Fractions describe parts of a whole, so an answer that contradicts an obvious benchmark often points to a sign or multiplication error.