If you added 2/5 + 1/4 and got 13/20, subtract one of the addends. Since 13/20 − 1/4 = 2/5, you recover the other addend and the result is confirmed. This check helps catch a calculation error, and an estimate of the sign and size completes it. Repeating the same calculation can repeat the same mistake, so it pays to take a different route. If the check fails, the guide to common fraction mistakes explains where the error usually comes from.
| Operation | Check | What it catches |
|---|---|---|
| Addition | Subtract one addend from the result | An error when finding the common denominator |
| Subtraction | Add the subtracted number to the difference | A wrong sign or a miscalculated common denominator |
| Multiplication | Divide the product by a nonzero factor | A product that was simplified incorrectly |
| Division | Multiply the quotient by the divisor | A fraction flipped by mistake |
| Equivalence | Compare the cross products | Two different values that look equal |
For division, the quickest check is to multiply the quotient by the divisor, because it does not require finding a common denominator.
Undo a multiplication or a division
For a division, multiply the quotient by the divisor and you should get the dividend back. With 3/4 ÷ 2/3 = 3/4 × 3/2 = 9/8, the check is 9/8 × 2/3 = 18/24 = 3/4. It matches, so the quotient is correct. This check requires a divisor other than zero.
For a multiplication, such as 3/7 × 2/9 = 6/63 = 2/21, divide the product by one of the factors. So 2/21 ÷ 3/7 = 2/21 × 7/3 = 14/63 = 2/9. If a factor is zero you cannot divide by it, and it is enough to check that the product is 0.
Worked example: a sum with an inverse check
You added these two fractions and want to know whether the result is correct.
- Add. The common denominator is 20. With it, 2/5 = 8/20 and 1/4 = 5/20, so the sum is 13/20.
- Pick the inverse check. Subtract one of the addends.
- Use the same denominator. 13/20 − 5/20 = 8/20.
- Simplify and compare. 8/20 = 2/5, which is the other addend.
The sum is correct, and the subtraction confirms it.
By size, the result is larger than each addend and smaller than 1, as expected for two small fractions.
Compare with cross products
To find out whether a/b and c/d have the same value, with nonzero denominators, compare a × d with b × c. For 6/9 and 10/15 you get 6 × 15 = 90 and 9 × 10 = 90, so they are equivalent. For 6/9 and 11/15 you get 90 and 99, so they are not. The products are whole numbers, so nothing is rounded.
This check confirms the value, but it does not tell you whether the fraction is in lowest terms. If you are asked for the simplest form, check separately that the greatest common factor (GCF, also called the greatest common divisor, GCD) of the numerator and denominator is 1. To compare sizes, the compare fractions calculator applies the same method, and the equivalent fractions tool generates pairs such as 6/9 and 10/15.
Estimate the sign and size before you calculate
An estimate does not prove that a result is exact, but it rules out impossible answers in seconds. Compare with 0, 1/2 and 1. A sum of positive fractions is larger than each addend, and a subtraction where the first number is smaller than the second gives a negative number. With 2/9 − 5/6, the result must be negative, and the calculation gives −11/18. The inverse check confirms it: −11/18 + 5/6 = 2/9.
If you work with decimals, do not compare rounded values. The decimal 0.14 looks close to 1/7, but 7 × 0.14 = 0.98, not 1. The exact difference is 1/7 − 14/100 = 1/350. To decide whether two values are equal, go back to the fraction or to cross products.
What a check does not prove
A check confirms that the equation you wrote is true. It does not guarantee that the problem was set up correctly: you can add two quantities correctly that should not have been added, or use a number that was copied wrongly. After the check, reread the question and the units. To see how to read each line of a solution, the guide to interpreting solution steps separates what changes from what keeps the same value.
Practice
- Calculate 7/8 − 1/5 and check it with the inverse operation.
- Calculate 5/9 × 3/10 and check it by dividing.
- Are 7/12 and 21/36 equivalent? Use cross products.
- Someone claims that 5/8 + 1/5 = 3/11. Which estimate rules it out?
Show the answers
- 27/40. Check: 27/40 + 8/40 = 35/40 = 7/8.
- 1/6, because 15/90 = 1/6. Check: 1/6 ÷ 5/9 = 1/6 × 9/5 = 9/30 = 3/10.
- Yes, because 7 × 36 = 252 and 12 × 21 = 252.
- 3/11 is less than 5/8, and a sum of positive numbers cannot be less than an addend. The correct result is 33/40.
For practice with automatic checking, fraction practice checks your answer, and the guide to common fraction mistakes explains why the most common calculations go wrong.
Sources
- Wikipedia, article Division (mathematics): states that in the rational and real numbers division is the inverse operation of multiplication, and that division by zero is not defined (accessed October 17, 2026).