Reading a fraction solution means giving each equality one of four labels: E for equivalence, O for operation, R for reduction and C for check. In 5/6 − 1/3 = 5/6 − 2/6 = 3/6 = 1/2 there are three equalities, one E, one O and one R, and the C is written separately. If the final result is wrong, the first equality that fails its condition is the one to fix; the ones before it still hold.
| Label | What must hold | Sample line |
|---|---|---|
| E, equivalence | Numerator and denominator are multiplied by the same nonzero number | 3/7 = 9/21 |
| O, operation | In addition and subtraction, the pieces are the same size; in products and quotients, the rule for that operation is applied | 4/9 + 2/9 = 6/9 |
| R, reduction | Numerator and denominator are divided by the same factor | 6/9 = 2/3 |
| C, check | Another operation recovers a piece of data from the problem | 6/9 − 2/9 = 4/9 |
The table is the standard for the whole post: every line of a solution falls in one row, and every row has its own condition.
A complete solution, line by line
Worked example: 5/6 − 1/3
A pitcher holds 5/6 of a quart of juice and 1/3 of a quart is poured out. How much juice is left?
- E, equivalence. The least common multiple of 6 and 3 is 6. Multiply 1/3 by 2/2: 1/3 = 2/6. The value is the same and only the writing changes.
- O, operation. Subtract pieces of the same size: 5/6 − 2/6 = (5 − 2)/6 = 3/6. The denominator stays because you are still counting sixths.
- R, reduction. Since 3 and 6 both divide by 3, 3/6 = 1/2.
The result is 1/2 quart.
C, check. 1/2 + 2/6 = 3/6 + 2/6 = 5/6, which is the starting amount.
Reduction is a separate step from the subtraction. The result 3/6 was not wrong, just not reduced. In 4/9 + 2/9 = 6/9 you also have to reduce, while in 2/8 + 1/8 = 3/8 there is no R row, because 3 and 8 share no factor. To find the common denominator you can use the least common denominator, and to decide whether a fraction is already reduced, simplify fractions.
Where the first equality fails
A solution can have a wrong final result and almost every line right. Mark each equality with its label and stop at the first one that does not hold.
Adding numerators is valid when the pieces are equal, as in 2/7 + 3/7 = 5/7. What decides whether the rule can be applied is the condition in the O row, not how the symbols look.
The sign and the mixed number
A minus sign in front of a mixed number applies to the whole number. −2 1/4 means −(2 + 1/4) = −9/4, while −2 + 1/4 = −7/4. If a solution jumps from one to the other without saying so, the failure is in that E, even if the other lines look right. With a negative fraction, the sign is kept in every equivalence: −3/5 = −6/10, never 6/10.
The other things to watch are few. No denominator can be zero and, in a division, neither can the divisor. If the problem has units, they are written on every line: 7/10 qt + 1/5 qt = 9/10 qt is a sum of quarts, while 7/10 qt + 1/5 ft is not a quantity you can add.
Exact equality or approximation
A solution can also swap an exact value for a rounded one without warning. The decimal of 1/9 does not end: 1/9 = 0.(1), with the repeating digits in parentheses. If you write it as 0.11, the correct symbol is ≈, and the difference is 1/9 − 11/100 = 1/900. The error grows as steps are chained: 9 × 0.11 = 0.99, while 9 × 1/9 = 1.
That is why, to verify a chain of equalities, you use exact fractions and leave rounding for the end. If you need to convert, the fraction to decimal tool marks the repeating part.
Exercise: find the first false line
First false line
In each solution, give the label of the first line that does not keep the value and write it correctly.
- The solution says 3/7 + 1/5 gives 4/12.
- The solution says 4/5 × 3/8 gives 12/13.
- The solution goes from −1/5 + 3/10 to 2/10 + 3/10 and arrives at 5/10.
- In 7/8 ÷ 3/4, the solution goes to 8/7 × 3/4 and arrives at 6/7.
Show the answers
- The O fails. Both the numerators and the denominators were added. The correct sum is 3/7 + 1/5 = 15/35 + 7/35 = 22/35.
- The O fails. 5 × 8 = 40, not 13. The correct product is 4/5 × 3/8 = 12/40 = 3/10.
- The E fails, the first equality. −1/5 = −2/10, not 2/10. The correct sum is −2/10 + 3/10 = 1/10.
- The O fails (the quotient rule). The dividend was flipped, not the divisor. The correct quotient is 7/8 ÷ 3/4 = 7/8 × 4/3 = 28/24 = 7/6.
After each correction, ask for at least one of these checks: the inverse operation, a size estimate or substitution into the original problem.
Reading, verifying and producing a solution
This post is about reading a solution that already exists. For the opposite job, confirming that a result is right with an independent method, there is the guide to verifying a fraction result. If you want to see the most repeated mistakes and why they happen, look at mistakes with fractions. To produce the steps of an addition, go to the add fractions calculator, and the fraction calculator shows the full chain of an operation. If you are helping a child with this exercise, how to help with fractions suggests questions to ask.
Sources
- OpenStax, Prealgebra 2e, chapter 4, section 5, Add and Subtract Fractions with Different Denominators: you need a common denominator to add or subtract fractions, but not to multiply or divide them (consulted October 17, 2026).