21
20
0.75
0.714286
0.035714
1
0.035714
28
21
20
21
20
0.75
0.714286
0.035714
1
0.035714
28
21
20
The Comparing Fractions Calculator determines which of two fractions is larger, smaller, or whether they are equal. It uses the cross-multiplication method for comparison and also displays the decimal equivalents of both fractions, making it easy to see the relationship between them at a glance.
Comparing fractions is a fundamental mathematical skill that many people find challenging, especially when the fractions have different denominators. While it is easy to see that 3/5 is greater than 2/5 (same denominator, compare numerators), it is much harder to compare 3/7 and 5/12 without a systematic method. This calculator provides two reliable comparison methods simultaneously: cross-multiplication and decimal conversion.
The cross-multiplication method works as follows: to compare a/b and c/d (with positive denominators), compute the cross products a×d and c×b. If a×d > c×b, then a/b > c/d. If a×d < c×b, then a/b < c/d. If they are equal, the fractions are equal. This works because multiplying both sides of a/b vs. c/d by the positive value b×d preserves the inequality and transforms it into a×d vs. c×b.
Understanding fraction comparison is critical in many real-world situations. When shopping, you might need to compare 3/4 pound of cheese at one price versus 5/8 pound at another to determine the better deal. In cooking, knowing whether 2/3 cup is more or less than 3/4 cup determines if you have enough of an ingredient. In probability, comparing the likelihood of different events requires fraction comparison. In medicine, dosing ratios must be compared precisely to ensure patient safety.
There are several strategies for comparing fractions: (1) Common denominator — convert both fractions to the same denominator, then compare numerators; (2) Cross-multiplication — the method used by this calculator; (3) Decimal conversion — divide numerator by denominator for each fraction; (4) Benchmark comparison — compare each fraction to a benchmark like 1/2. Each method has advantages depending on the context.
The cross-multiplication method is particularly elegant because it avoids the need to find a common denominator. It is efficient, always works for fractions with positive denominators, and requires only two multiplication operations. The decimal method provides intuitive confirmation but may be less precise for repeating decimals.
This calculator also shows the difference between the two fractions (Fraction 1 minus Fraction 2) and a comparison indicator: 1 if the first fraction is greater, −1 if it is less, and 0 if they are equal. The difference tells you not just which fraction is larger, but by how much.
The tool supports positive denominators from 1 to 9999 and numerators from −9999 to 9999, covering a vast range of fractions including negative values and improper fractions.
The calculator uses two parallel methods:
Cross-Multiplication: Compare $$a \times d$$ with $$c \times b$$
If $$a \times d > c \times b$$, then $$\frac{a}{b} > \frac{c}{d}$$
If $$a \times d < c \times b$$, then $$\frac{a}{b} < \frac{c}{d}$$
If $$a \times d = c \times b$$, then $$\frac{a}{b} = \frac{c}{d}$$
Decimal Conversion: $$\text{Decimal}_1 = \frac{a}{b}, \quad \text{Decimal}_2 = \frac{c}{d}$$
The difference is: $$\frac{a}{b} - \frac{c}{d}$$
Cross Product 1 (a×d) and Cross Product 2 (c×b) are compared: whichever is larger corresponds to the larger fraction. The Decimal Values provide intuitive understanding. The Difference shows how far apart the fractions are. The Comparison result of 1 means Fraction 1 is greater, −1 means it is less, and 0 means they are equal.
Inputs
Results
Cross products: 3×7=21 vs 5×4=20. Since 21>20, 3/4 > 5/7
Inputs
Results
Cross products: 2×6=12 vs 4×3=12. Equal! 2/3 = 4/6 (equivalent fractions)
Use cross-multiplication: compare a×d with c×b. Alternatively, convert both fractions to decimals by dividing numerator by denominator, or find a common denominator and compare the numerators.
To compare a/b with c/d, compute a×d and c×b. The fraction whose numerator contributes to the larger cross product is the larger fraction. This works because it is equivalent to comparing fractions with a common denominator of b×d.
Yes. Cross-multiplying: 3×3=9 vs 2×4=8. Since 9>8, 3/4 > 2/3. As decimals: 0.75 > 0.6667.
Two fractions a/b and c/d are equivalent if a×d = c×b. For example, 2/3 and 4/6 are equivalent because 2×6 = 4×3 = 12. Equivalently, their decimal values are identical.
Yes. Negative fractions are less than positive fractions. When comparing two negative fractions, the one closer to zero is larger. For example, -1/4 > -1/2 because -0.25 > -0.5.
When denominators are equal, simply compare the numerators. The fraction with the larger numerator is larger. Cross-multiplication still works but is unnecessary in this case.
Cross-multiplication gives exact results and works for all fractions. Decimals are intuitive but may involve rounding for repeating decimals (like 1/3 = 0.333...). For precise comparison, cross-multiplication is preferred.
Convert all fractions to decimals, or find a common denominator for all fractions, then compare numerators. You can also use this calculator repeatedly for pairwise comparisons.
The comparison output is 1 if the first fraction is greater than the second, -1 if it is less, and 0 if they are equal. This is a standard mathematical sign convention.
Comparing fractions is essential in ordering data, making decisions about quantities, solving inequalities, understanding probability, and many real-world applications from cooking to finance to science.
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
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