Enter the value, the step and the tie rule. There are three common uses:
- To a whole number: with step 1, 7/4 gives 2 and 13/10 gives 1.
- To a fixed step: with step 1/8, 0.6 gives 5/8, because 0.6 × 8 = 4.8 and the nearest integer is 5.
- With a denominator limit: that is a different search, handled by approximate a decimal as a fraction.
Worked example: 2.695 inches to the nearest sixteenth
A part measures 2.695 inches and the ruler is marked in sixteenths. Which mark do you read?
- Divide by the step. 2.695 ÷ 1/16 = 2.695 × 16 = 43.12.
- Round the quotient. The integer nearest to 43.12 is 43.
- Multiply by the step. 43 × 1/16 = 43/16, which is 2 11/16.
- Find the error. 43/16 − 2.695 = −3/400.
Result: 2 11/16 inches, which is 3/400 inch below the measured value.
Check. Since 43/16 = 2.6875 and 2.695 − 2.6875 = 0.0075 = 3/400, the error is below half the step, 1/32.
2.695 × 16 = 43.12 → 43; 43/16 = 2 11/16 in
If you then need that measurement in millimeters, fractional inches to mm converts it, and the guide on how to read fractions on a ruler helps you find the mark on the ruler itself.
How the nearest multiple is found
Each step divides the number line into equal intervals, and the value falls inside one of them. The two ends of that interval are the candidates, and the closer one wins. So it is enough to divide the value by the step and check whether the fractional part of the quotient is more than one half.
With v the value and p the step, write the quotient v/p as a whole number plus a remainder. If the fractional part of the quotient is less than 1/2, keep the whole number; if it is more, go up by one; if it is exactly 1/2, there is a tie. The calculations are exact, not approximate, so a true tie is never confused with a value that is only nearly halfway. The sign of the value is kept: −2.695 gives −43/16.
Which tie rule to choose
A tie happens when the value is exactly halfway between two multiples. For example, 5/16 with step 1/8 is the same distance from both neighboring multiples. The table shows what each rule returns.
| Rule | 5/16 (2.5 steps) | 7/16 (3.5 steps) |
|---|---|---|
| Away from zero (default) | 3/8 | 1/2 |
| Toward zero | 1/4 | 3/8 |
| Choose the even multiple | 1/4 | 1/2 |
Away from zero and toward zero depend on the sign; the even multiple picks the candidate whose number of steps k is even. For 5/16 the candidates are 2 and 3 steps, and 2 wins; for 7/16 they are 3 and 4, and 4 wins. That is why the third row matches the second in one column and the first in the other. With negative values the sign rules mirror each other: −5/16 gives −3/8 away from zero and −1/4 toward zero. Choosing the even candidate when two are equally close is a long-established way to avoid a bias in one direction; English Wikipedia describes it as round half to even, a tie-breaking rule without positive or negative bias. If your work requires a specific rule, select it before you calculate.
Signed error and absolute error
The signed error is rounded − original. For the tie above with the default rule, 3/8 − 5/16 = 1/16: positive, because the result ended up above the value. With the negative value it is −1/16, and the absolute error is the same. The sign tells you which way the value moved; the absolute error tells you how far.
If the value is already a multiple of the step, the error is 0 and the tie rule plays no part. To see the decimal of any result, use fraction to decimal.
The same value with finer steps
A finer step lowers the error bound, which is half the step, although the actual error for a particular value can go up or down. The table rounds 0.83 = 83/100 with five steps.
| Step | 0.83 ÷ step | Nearest multiple | Signed error | Bound |
|---|---|---|---|---|
| 1/2 | 1.66 | 1 | 17/100 | 1/4 |
| 1/3 | 2.49 | 2/3 | −49/300 | 1/6 |
| 1/5 | 4.15 | 4/5 | −3/100 | 1/10 |
| 1/10 | 8.3 | 4/5 | −3/100 | 1/20 |
| 1/20 | 16.6 | 17/20 | 1/50 | 1/40 |
Going from fifths to tenths improves nothing here: 0.83 lands on the same multiple, 4/5, and the error does not change. What always goes down is the bound, not the error of each value.
When to use another tool
Rounding changes the value on purpose. If you need the exact fraction of a decimal, use decimal to fraction, where 0.3125 is exactly 5/16. If the result will be used to make or measure something, keep the error the calculator shows as well.
Sources
English Wikipedia, article Rounding, section “Rounding half to even”, which describes choosing the even candidate as a tie-breaking rule without positive or negative bias (accessed October 17, 2026).
Frequently asked questions
How do I know whether my rule asks for the nearest value or rounding up?
This calculator always rounds to the nearest multiple and uses your chosen rule only for ties. If the instructions say "round up" or "ceiling", the result can differ: 2.01 with step 1 gives 2 here, but 3 when rounding up. Check which of the two your standard asks for before you calculate.
Which way does 0.5 round with step 1?
It depends on the tie rule. Away from zero gives 1; toward zero gives 0; the even multiple also gives 0, because 0 is even. With −0.5 the first two rules give −1 and 0. If your standard says nothing, note which rule you used so someone else can repeat the calculation.
Can I round a mixed number?
Yes, but convert it to an improper fraction first. 2 1/3 is 7/3, and with step 1/8 the quotient is 56/3, 18.666…, so the result is 19/8, which is 2 3/8. The signed error is 19/8 − 7/3 = 1/24, within the bound of 1/16.