Comparing Fractions Calculator
Free Comparing fractions Calculator for fractions. Enter values to get step-by-step solutions with formulas and graphs.
Reviewed by Manoj Kumar, Mathematics Educator
Formula
Compare a/b and c/d: if ad > bc then a/b > c/d
Cross multiplication provides the fastest comparison: multiply each numerator by the other denominator. The fraction whose cross product is larger is the greater fraction. Alternatively, convert both fractions to the same denominator (LCD) and compare numerators directly.
Worked Examples
Example 1: Comparing Unlike Fractions
Problem:Which is larger: 3/7 or 5/9?
Solution:Method 1 - Cross Multiplication:\n3 x 9 = 27 and 5 x 7 = 35\nSince 27 < 35, we know 3/7 < 5/9\n\nMethod 2 - Common Denominator:\nLCD(7, 9) = 63\n3/7 = 27/63 and 5/9 = 35/63\n27/63 < 35/63, so 3/7 < 5/9\n\nMethod 3 - Decimals:\n3/7 = 0.4286 and 5/9 = 0.5556\n0.4286 < 0.5556
Result:3/7 < 5/9 | Difference: 8/63 = 0.1270
Example 2: Checking Fraction Equivalence
Problem:Are 4/6 and 10/15 equivalent fractions?
Solution:Method 1 - Simplify both:\n4/6: GCD(4,6) = 2, so 4/6 = 2/3\n10/15: GCD(10,15) = 5, so 10/15 = 2/3\nBoth simplify to 2/3, so they ARE equivalent\n\nMethod 2 - Cross multiply:\n4 x 15 = 60 and 10 x 6 = 60\nSince 60 = 60, the fractions are equal\n\nMethod 3 - Decimals:\n4/6 = 0.6667 and 10/15 = 0.6667
Result:4/6 = 10/15 (both equal 2/3 = 0.6667)
Frequently Asked Questions
How do you compare fractions with different denominators?
There are three main methods to compare fractions with different denominators. The most common is finding a common denominator by calculating the Least Common Denominator (LCD), converting both fractions, and comparing numerators. For example, to compare 3/4 and 5/6: LCD = 12, giving 9/12 and 10/12, so 5/6 is larger. The second method is cross multiplication: multiply each numerator by the other denominator. For 3/4 vs 5/6: 3 x 6 = 18 and 5 x 4 = 20, and since 18 < 20, we know 3/4 < 5/6. The third method converts to decimals: 3/4 = 0.75 and 5/6 = 0.833, making the comparison immediate.
What is the cross multiplication method for comparing fractions?
Cross multiplication is a quick and reliable shortcut for comparing two fractions. Given fractions a/b and c/d, multiply diagonally: compute a x d and c x b. If a x d is greater than c x b, then a/b is the larger fraction. 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 is algebraically equivalent to converting both fractions to a common denominator (b x d) and comparing numerators. The beauty of cross multiplication is that you never need to find the LCD or convert fractions, making it extremely fast for simple comparisons. It is particularly useful in standardized tests and mental math.
How do you compare fractions with the same denominator?
When fractions have the same denominator, comparison is trivial: simply compare the numerators. The fraction with the larger numerator is the larger fraction. For example, 7/12 is greater than 5/12 because 7 is greater than 5. This works because the denominator represents the size of each piece, and when pieces are the same size, more pieces mean a larger value. This principle is why finding a common denominator is the foundation of fraction comparison. It reduces the problem to comparing whole numbers, which is intuitive. This same logic extends to ordering multiple fractions: once all share a common denominator, rank them by their numerators.
Can you compare fractions by converting to decimals?
Yes, converting fractions to decimals is often the most practical comparison method, especially when dealing with fractions that have large or complex denominators. Simply divide each numerator by its denominator to get a decimal value, then compare the decimals. For example, 7/11 = 0.6364 and 5/8 = 0.6250, so 7/11 is larger. This method is especially useful with a calculator available, as it avoids the mental effort of finding LCD values. However, be aware that some fractions produce repeating decimals (like 1/3 = 0.3333...), so you need enough decimal places for accurate comparison. In practice, four to six decimal places are sufficient for most comparisons.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy