Equivalent Fractions Guide
Make and recognize fractions that name the same amount.
Quick Answer
a/b = (a × k)/(b × k), b ≠ 0 and k ≠ 0When to Use This
Equivalent fractions name the same amount using a different number of equal-sized parts. This is the basis of common-denominator arithmetic. The scale factor changes both the count of parts and the size assigned to each part.
Step-by-Step Method
- Start with a fraction whose denominator is nonzero.
- Choose a nonzero factor for both numerator and denominator.
- Multiply both components by that factor.
- Verify with cross products or by reducing back to the starting fraction.
Calculator Tips and Limits
- For ordinary positive-denominator examples, choose a positive whole-number multiplier.
- The equivalence tool is useful for generating matching forms; use the fraction calculator when you also need arithmetic.
Worked Examples
Build twentieths
Multiply 3/5 by 4/4 to obtain 12/20. Since 4/4 equals 1, the value stays unchanged. Cross products verify it: 3 × 20 = 60 and 5 × 12 = 60.
Reject a near-looking pair
2/3 and 3/4 are not equal. Their cross products are 2 × 4 = 8 and 3 × 3 = 9. Similar-looking numerators and denominators do not establish equivalence.
Common Mistakes
Adding the same number to both parts
Adding 1 turns 3/5 into 4/6, which is 2/3 rather than 3/5. Scaling works; arbitrary addition does not.
Multiplying by zero
Multiplying both components by zero produces 0/0, an undefined expression. The scale factor must be nonzero.
FAQ
How do I reach a specified denominator?
Divide the target denominator by the original denominator to find the factor, then apply it to the numerator too.
Can equivalent fractions have negative values?
Yes. -3/5 and -12/20 are equivalent. Moving one minus sign between numerator and denominator also preserves the negative value.
References and corrections
OpenStax: Prealgebra 2e — further reading. The worked examples on this page are CalcSolver explanations. Read our calculation and correction method, or report an error with the input and expected result.