The bounds and rounding explainer takes a rounded measurement and shows the interval of true values it could have come from. You can then combine two measurements by adding, subtracting, multiplying or dividing, and see the lowest and highest results that are possible.
It shows every step, so you can compare it with your own working rather than just copy an answer.
How do you use it?
- Enter the rounded value, for example 5.2.
- Choose Rounded to: the nearest unit (such as 0.1 or 10), a number of decimal places, or a number of significant figures.
- Enter the unit or number of places that matches. For nearest 0.1, type 0.1. For 1 decimal place, type 1.
- Pick the convention for halves: half up, or half away from zero.
- Add units if you like, such as cm. This only labels the answer.
- To combine measurements, choose an operation under “Combine with a second measurement?” and enter the second value and its precision.
- Press Show bounds. Press Reset to return to the sample.
How do you read the result?
The first line gives the rounding unit and the half-unit. The lower bound is the value minus the half-unit, and the upper bound is the value plus it. The interval follows, written as an inequality such as 5.15 ≤ x < 5.25.
A filled ≤ means that endpoint is included. A plain < means it is not. The note under the result explains the test: an endpoint is included only if a value exactly there would round to your number.
When two measurements are combined, a table lists every combination of endpoints. The smallest result is the lower bound and the largest is the upper bound of the answer.
Example walk-through
Start with the sample: 5.2 cm rounded to the nearest 0.1, half up. The tool gives 5.15 ≤ x < 5.25, because 5.15 would round up to 5.2 and 5.25 would round up to 5.3.
Now choose Subtract and enter a second value of 3.4, also to the nearest 0.1. Its interval is 3.35 ≤ y < 3.45. The four corners are:
- 5.15 − 3.35 = 1.8
- 5.15 − 3.45 = 1.7
- 5.25 − 3.35 = 1.9
- 5.25 − 3.45 = 1.8
The lower bound is 1.7 and the upper bound is 1.9. The smallest result uses the smallest first value and the largest second value. Because that corner uses the excluded end of the second interval, the tool writes 1.7 < result < 1.9.
Try Multiply for comparison. The bounds are 5.15 × 3.35 = 17.2525 and 5.25 × 3.45 = 18.1125.
What are the assumptions and limits?
- Every true value inside the interval is treated as possible.
- Endpoints can be open or closed, and negative values reverse which end is which. The tool tests the endpoints under the convention you chose, so read the ≤ and < signs.
- Dividing is refused when the second interval includes 0, because the answer would be undefined.
- The tool rejects a value that could not come from the stated rounding. For example, 5.23 is not a possible result of rounding to the nearest 0.1.
- Zero cannot be rounded to significant figures, so use decimal places or a unit.
Which lessons explain the ideas behind it?
- Bound a sum and a difference shows why the subtraction corner is not the obvious one.
- Bound a product with positive measurements covers the multiplication case in this tool.
- Precision bounds and measurement practice gives mixed questions to test yourself.
- My bounds answers use the wrong endpoints helps if your method is right but the endpoints go wrong.
The whole topic sits in precision and bounds. If you would like a teacher to check your reasoning on real questions, see online one-to-one Mathematics tuition. More tools are in the learning tools directory.