---
title: "Bounds and error in rounded values"
description: "Use approximation through rounding to estimate answers and calculate possible resulting errors expressed using inequality notation a < x ≤ b; understand upper and lower bounds of rounded values"
canonical: https://lightmysky.com/learn/mathematics/bounds-and-error-in-rounded-values-mt_VVn1IXjkzn
source: https://lightmysky.com/learn/mathematics/bounds-and-error-in-rounded-values-mt_VVn1IXjkzn.md
retrieved: 2026-09-02
---

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# Bounds and error in rounded values

Use approximation through rounding to estimate answers and calculate possible resulting errors expressed using inequality notation a < x ≤ b; understand upper and lower bounds of rounded values

Subject: Mathematics · Area: Number Representation & Place Value · Ages 12 to 14
Page: https://lightmysky.com/learn/mathematics/bounds-and-error-in-rounded-values-mt_VVn1IXjkzn

## Ready when they can

- Estimate the answer to a calculation by rounding all values appropriately
- Calculate upper and lower bounds of a rounded measurement
- Express error intervals using inequality notation

## Lesson: Bounds and error in rounded values

Before you work out an exact answer, it helps to guess a rough one first. Round each number in the calculation to a nearby value that's easy to work with, then do the easier calculation. That rough answer is your estimate, and it tells you whether your exact answer is reasonable.

**Example.** Estimate 39 × 21. Round 39 up to 40, and round 21 down to 20. Now multiply the easier numbers: 40 × 20 = 800. The exact answer is 819, so 800 is a quick check that you're in the right area.

Rounding a measurement works a bit differently. If a length is 7 cm to the nearest cm, it wasn't measured exactly. It just rounded to 7. The real length could be a little more or a little less than 7 cm, somewhere within half a centimetre either side.

*(drawing: The real length can be anywhere from 65 mm up to (but not quite reaching) 75 mm.)*

The two ends aren't treated the same. The lower bound, 6.5 cm, is included, because 6.5 cm itself would round up to 7 cm. The upper bound, 7.5 cm, is not included, because 7.5 cm would round up to 8 cm instead. We write that error interval as 6.5 ≤ x < 7.5, where x is the real length.

**Recap.** Round first to estimate fast, and remember every rounded value hides a range: half a unit below up to, but not including, half a unit above.

## Practice

8 questions on this page, each with its working shown.

## Needs first

- [Decimal place value (age 11+)](https://lightmysky.com/learn/mathematics/decimal-place-value-age-11-mt_U9sme87C32)
- [Rounding Answers](https://lightmysky.com/learn/mathematics/rounding-answers-mt_QU5R7Aajy9)