Adding denominators, flipping the wrong fraction, losing the sign of a mixed number, taking a rounded decimal for an equality and believing that more parts make a larger fraction are five different mistakes. With 3/8 + 1/10 = 19/40 the result is greater than each addend, and the 4/18 you get by adding denominators is ruled out before the calculation is finished. Each mistake has a worked example here and a check you can do without a calculator. If you want the explanation with bars, it is in why denominators are not added.
Most of these mistakes show up before you finish if you compare the result with what should happen: two positive addends give something greater than each one, and dividing by a number less than one enlarges the dividend. The table collects the five signals and the fix for each.
| Mistake | How it looks | Fix | Check |
|---|---|---|---|
| Adding denominators | Numerators and denominators are both added | Match denominators before adding | The sum must exceed each addend |
| Flipping the wrong fraction | The dividend is flipped | Flip only the divisor | Multiply the quotient by the divisor |
| Losing the sign of a mixed number | The sign affects only the whole part | Apply the sign to the whole mixed number | The mixed number sits to the left of its whole part |
| Rounded decimal taken as equal | A cut-off decimal is written with = | Write ≈ or keep the fraction | Turn the decimal into a fraction |
| More parts, larger fraction | Only the denominator is compared | Compare with a common denominator | More parts of the same whole are smaller |
Each row is developed below, with the condition under which the correct rule works, and the worked example comes first.
Worked example: 3/8 + 1/10
A common result is 4/18. Find the mistake and correct the sum.
- Spot the signal. The sum of two positives must exceed each addend. Since 4/18 < 3/8, that result cannot be the sum.
- Find the common denominator. The least common multiple of 8 and 10 is 40.
- Convert each fraction. 3/8 = 15/40 and 1/10 = 4/40.
- Add the numerators. 15/40 + 4/40 = 19/40.
Result: 3/8 + 1/10 = 19/40.
Check. 19/40 − 4/40 = 15/40, which matches the first addend.
Mistake 1: adding the denominators
An eighth and a sixth are parts of different sizes, so they cannot be counted together until both are expressed in the same unit. The denominator tells you what the part is called, like “feet” or “inches” in a measurement. It is kept when the parts are already equal and changed when they are not. Two sevenths plus three sevenths is five sevenths, because parts of the same size are being counted. The English Wikipedia article on fractions says unlike fractions must first be converted to like quantities, with the smallest possible denominator given by the least common multiple of the denominators (consulted October 17, 2026).
The full procedure is in add fractions, and the guide why denominators are not added explains it with bars of the same whole.
Mistake 2: flipping the wrong fraction when dividing
When dividing, the dividend stays as it is and only the divisor is flipped. For 7/8 ÷ 3/4, the reciprocal of 3/4 is 4/3, so 7/8 × 4/3 = 28/24 = 7/6. If you flip the first term, you get 8/7 × 3/4 = 6/7, which answers a different question. If you flip neither, 7/8 × 3/4 = 21/32.
The check uses the inverse operation, and 7/6 × 3/4 = 21/24 = 7/8 gives back the starting dividend. Another quick signal is size. The divisor is less than one, so the quotient has to be greater than the dividend, and 7/6 is. That signal only holds with a positive divisor less than one. With 5/4, greater than one, the quotient 7/8 × 4/5 = 7/10 is less than the dividend.
| Case | Calculation | Result |
|---|---|---|
| Dividend zero | 0 ÷ 5/8 | 0 |
| Divisor zero | 5/6 ÷ 0 | Undefined |
Zero divided by a nonzero fraction gives zero, because nothing is left to share. Dividing by zero has no result, since zero has no reciprocal. When you divide by zero, the division calculator warns of the error, and the guide why dividing by a fraction can increase the result explains the size signal with a model.
Mistake 3: losing the sign of a mixed number
The sign of a negative mixed number applies to the whole number. So −4 1/6 means −(4 + 1/6), which as an improper fraction is −25/6. To convert it, first calculate the magnitude, 4 × 6 + 1 = 25, and put the sign at the end. The typical mistake is to subtract the fraction part, 4 × 6 − 1 = 23, and get −23/6.
The two values are a third apart on the line. With units it works the same. For a temperature, −1 1/4 degrees = −5/4 degrees, while −1 + 1/4 = −3/4. The guide to negative fractions covers signs in each operation, and reading and writing fractions explains mixed notation.
Mistake 4: taking a rounded decimal for an equality
The fraction 2/9 has a repeating decimal expansion, 0.(2), with twos that never end. The decimal 0.222 is exactly 222/1000 = 111/500, and the difference 2/9 − 111/500 = 1/4500 is not zero. Write 2/9 ≈ 0.222 (error 1/4500) when you show the rounding. Multiplying by nine makes it clear, because 9 × 0.222 = 1.998, which is not 2, while 9 × 2/9 = 2.
Not all decimals are approximations. An eighth is exactly 0.125, because 1/8 = 125/1000, and 1/4 = 0.25. To check a decimal equality, turn the decimal into a fraction and reduce. If it does not match the original fraction, keep the fraction in the calculations where precision matters. The fraction to decimal tool shows when the expansion ends and when it repeats. The English Wikipedia article on fractions notes that 1/3 cannot be written exactly as a decimal with a finite number of digits.
Mistake 5: believing a larger denominator gives a larger fraction
With the same numerator, more parts in the whole means smaller parts. That is why 1/8 < 1/5, since an eighth is what you get from sharing a whole among eight, and each piece is smaller than one of five. The numerator counts how many parts you take, so 5/8 > 1/5 because five eighths exceed one fifth.
For 1/8 and 1/5, the check is 1/5 = 8/40 and 1/8 = 5/40, and 5 < 8. When both numbers change, looking at just one is not enough. With 4/7 and 5/8, put both over a common denominator. So 4/7 = 32/56 and 5/8 = 35/56, and 32 < 35, so 4/7 < 5/8. There you compare the numerators. The compare fractions tool does that conversion. To choose which check to apply to each operation, see verifying the result of fractions.
Check and correct
- Check that 2/5 + 1/4 ≠ 3/9 and correct it.
- Check that 4/9 ÷ 2/3 ≠ 8/27 and correct it.
- The conversion −3 1/2 = −5/2 is wrong. Correct it and check.
Show the answers
- 8/20 + 5/20 = 13/20. Check: 13/20 − 1/4 = 13/20 − 5/20 = 8/20 = 2/5.
- Flip the divisor: 4/9 × 3/2 = 12/18 = 2/3. Check: 2/3 × 2/3 = 4/9.
- −(3 + 1/2) = −7/2. Check: −7/2 = −3.5, to the left of −3.
Sources
The English Wikipedia article Fraction explains that, to add fractions with different denominators, you first convert them to like quantities and the smallest possible denominator is the least common multiple of the denominators. It also says that 1/3 cannot be written exactly as a decimal with a finite number of digits, which is the basis of mistake 4 (consulted October 17, 2026).