Rounding and Estimating
Theory
Rounding swaps a number for a simpler one that is close to it. Look at the digit just to the right of the place you keep: if it is \(5\) or more, round up; if it is less than \(5\), round down. An estimate uses rounded numbers to give a quick answer and to check a calculation. This Year 7 maths page shows both.
Place value. Each place is worth ten times the place to its right. In \(748\,215\) the \(7\) is in the hundred thousands place and the \(4\) is in the ten thousands place, worth \(40\,000\).
To round to the nearest 10, 100 or 1000: find the lower and upper multiples either side, and the halfway point. The deciding digit is the digit just to the right of the place you keep. If it is \(5\) or more (\(5, 6, 7, 8, 9\)), round up. If it is less than \(5\) (\(0, 1, 2, 3, 4\)), round down. Write \(\approx\) (“is approximately equal to”): \(4637 \approx 4600\).
A kept 9 carries. \(49\,961\) to the nearest hundred: \(9\) hundreds plus \(1\) more is \(10\) hundreds, one more thousand, so \(49\,961 \approx 50\,000\).
Leading digit. For a quick estimate keep only the first digit and let the next digit decide: \(7814 \approx 8000\). This is rounding to one significant figure.
Counting in hundreds. From \(24\,300\) to \(25\,100\) is \(800\), which is \(8\) hundreds.
Estimating. Round each number first, then calculate. Compare the estimate with the exact answer; if they are far apart, check the working.
The rounding rule:
| Deciding digit | What to do |
|---|---|
| \(0, 1, 2, 3, 4\) | round down (keep the digit) |
| \(5, 6, 7, 8, 9\) | round up (add 1 to the kept digit) |
| Round to the nearest | Deciding digit |
|---|---|
| ten | ones |
| hundred | tens |
| thousand | hundreds |
How to round a whole number
- Find the place you are rounding to.
- Look at the deciding digit just to its right.
- Round up if it is \(5\) or more, down if it is less than \(5\).
- Replace every digit after the kept place with \(0\).
How to estimate
- Round each number (to the leading digit, or to a number that is easy to work with).
- Calculate with the rounded numbers.
- Compare with the exact answer.
The hundreds digit is \(6\). The digit to its right is \(9\), so round up: \(5691 \approx 5700\).
The thousands digit is \(5\). The digit to its right is \(4\), so round down: \(35\,491 \approx 35\,000\).
(i) hundred
(ii) thousand
(i) Between \(58\,600\) and \(58\,700\); the tens digit \(3 < 5\): \(58\,600\)
(ii) The hundreds digit \(6 \ge 5\): \(59\,000\)
(ii) A whole number rounds to \(800\) to the nearest hundred. What are the smallest and largest numbers it could be?
(i) The tens digit is \(7\), so the \(9\) hundreds round up to \(10\) hundreds: \(39\,900 + 100 = 40\,000\).
(ii) The halfway points are \(750\) and \(850\): smallest \(750\) (rounds up), largest \(849\) (rounds down).
| \(700 + 500 + 400 + 50\) | \(=\) | \(1650\) |
| \(720 + 456 + 391 + 45\) | \(=\) | \(1612\) |
Estimate \(\approx 1650\); the exact answer is \(1612\).
| \(50 \times 100 \div 5\) | \(=\) | \(1000\) |
\(\approx 1000\). (The exact answer is \(1170\).)
(i) \(52 \times 29 = 1508\)
(ii) \(4158 \div 7\) gives \(64\)
| \(50 \times 30\) | \(=\) | \(1500\) |
| \(4200 \div 7\) | \(=\) | \(600\) |
(i) Reasonable: \(1508\) is close to \(1500\).
(ii) Not reasonable: \(64\) is far from \(600\). (The exact answer is \(594\).)
(i) Estimate the total, rounding each to the nearest hundred.
(ii) Find the exact total.
| \(400 + 400 + 700\) | \(=\) | \(1500\) |
| \(386 + 412 + 709\) | \(=\) | \(1507\) |
(i) About \(1500\) km
(ii) \(1507\) km: the estimate is only \(7\) km out.
Common pitfalls
Frequently asked questions
How do you round to the nearest hundred?
Look at the tens digit. \(5\) or more: round up. Less than \(5\): round down.
What does the symbol ≈ mean?
Is approximately equal to.
What is rounding to the leading digit?
Keeping only the first digit: \(6382 \approx 6000\). It is also called one significant figure.
Why do we estimate?
To get a quick answer and to check that a calculation is reasonable.
What happens when a 9 rounds up?
It becomes \(10\) and carries into the next place: \(49\,961 \approx 50\,000\) to the nearest hundred.