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Rounding: worked examples and method
Foundation · Grade 1
This page sets out the method for rounding in the order you would use it, then walks through the full lesson explanation. For three complete worked examples with the slips that cost marks, see the Rounding topic page.
Key facts
Look at the digit after the one you are rounding to: 5 or more rounds up, 4 or less leaves it alone.
Decimal places are counted after the decimal point: 14.29 has 2 d.p.
Significant figures start from the first non-zero digit: in 0.004072 the first significant figure is the 4.
Zeros that hold a place must stay: 649 500 to the nearest 1000 is 650 000, not 650.
Estimate means round each number to 1 s.f., then calculate. Use the ≈ sign.
The method, step by step
Method: rounding to a column, d.p. or s.f.
When you see 'round to …', find the last digit you will keep, then look at the next digit.
Underline the last digit you are keeping (the tens digit, the 2nd decimal place, the 2nd significant figure…).
Look only at the digit immediately to its right.
If that digit is 5, 6, 7, 8 or 9, add 1 to the underlined digit. If it is 0 to 4, leave the underlined digit as it is.
For whole numbers, replace every digit after the underlined one with 0. For decimals, just cut the rest off.
Check the size is right: the rounded number should be close to the original.
Method: estimating a calculation
When you see 'estimate' or 'work out an approximate value', round every number to 1 significant figure.
Round each number in the calculation to 1 s.f.
Rewrite the calculation with the rounded numbers.
Work it out without a calculator.
Write the ≈ sign and your estimate. Do not round the final answer again.
How the lesson explains it
Find the neighbours
To round to the nearest 10, find the two tens the number sits between.
74 sits between 70 and 80.
The decider digit
Look at the digit just after the column you are rounding to.
5 or more rounds up. 4 or less stays down.
Halfway rounds up
A number ending in 5 is exactly halfway to the next ten. The usual rule rounds it up.
65 to the nearest 10 is 70, not 60.
Nearest hundred
Look at the tens digit. 349 has tens digit 4, so it rounds down to 300.
350 has tens digit 5, so it rounds up to 400.
One decimal place
To round to 1 decimal place, the decider is the second decimal digit.
3.46 to 1 decimal place looks at 6, so it rounds up to 3.5.
Rounding is not the exact value
A rounded number is an approximation. 3.5 is not equal to 3.46.
Say “about” when you use a rounded figure in a sentence.
Significant figures
Significant figures count from the first digit that is not zero. In 4 726 the 4 is the first significant figure. To round to 2 significant figures, keep 2 digits and look at the next one: 4 726 becomes 4 700.
Keep the zeros that hold the place value: 4 726 to 1 significant figure is 4 000, not 4.
Zeros at the start do not count
In 0.04072 the first significant figure is the 4, because the zeros before it only show the size. To 2 significant figures it is 0.041.
Zeros after the first non-zero digit do count, so 0.0407 has 3 significant figures.
Nearest 10, 100 and 1000
Find the two multiples the number sits between, then use the decider. For the nearest 10, the decider is the units digit. 765 sits between 760 and 770. The units digit is 5, and a decider of 5 rounds up, so 765 to the nearest 10 is 770. 764 has units digit 4, so it rounds down to 760.
For the nearest 100, the decider is the tens digit. 349 has tens digit 4, so it rounds to 300. 350 has tens digit 5, so it rounds to 400. For the nearest 1000, the decider is the hundreds digit. 6482 sits between 6000 and 7000. The hundreds digit is 4, so it rounds to 6000. 45628 has hundreds digit 6, so it rounds to 46000.
Rounding up can change several digits when the kept digit is 9. 3960 to the nearest 100 looks at the tens digit 6, so the hundreds digit 9 rounds up. That carries into the thousands: 3960 becomes 4000. Write the zeros that fill the columns you have cleared. An answer of 40 has dropped the size of the number.
Method: 1. Find the two multiples of 10, 100 or 1000 that the number sits between. 2. Look at the next digit, the decider. 3. 5 or more rounds up, and 4 or less stays down.
Decimal places
To round to 2 decimal places, the answer will have two digits after the point, and the decider is the third decimal digit. 3.846 has third decimal digit 6, so the second decimal digit 4 rounds up to 5. The value to 2 decimal places is 3.85. It is not 3.84, and it is not 3.8, which would be 1 decimal place.
To round to the nearest whole number, the decider is the first decimal digit. 3.846 has tenths digit 8, so the nearest whole number is 4. The same number can give different answers under different instructions. Read the instruction before you look at a digit. 'Two decimal places' and 'the nearest ten' are not the same request.
If the digit you round up is 9, the 9 becomes 0 and you carry. 2.497 to 2 decimal places looks at 7, so 2.50 is the rounded value, written with the zero if two places were asked for. 1.96 to 1 decimal place is 2.0. Keep the place that the question asked you to show.
Method: 1. Count the decimal places the question asks you to keep. 2. The next digit is the decider. 3. Write the rounded number with exactly that many decimal places.
A decider of 5 rounds up
A decider of exactly 5 rounds up. 85 to the nearest 10 looks at the units digit 5, so it rounds to 90, not 80. The same rule is used for hundreds. 250 to the nearest 100 looks at the tens digit 5, so it rounds to 300.
On a decimal, 5 in the decider column also rounds up. 14.5 to the nearest whole number looks at the tenths digit 5, so the answer is 15. 6.05 to 1 decimal place looks at the second decimal digit 5, so it rounds to 6.1. Do not leave the answer on the lower value just because nothing follows the 5.
Write the zeros that keep the size. 90 is not 9, and 300 is not 3. If rounding up turns a 9 into 0, carry into the next column. 195 to the nearest 10 has units digit 5, so the tens digit 9 rounds up and the answer is 200.
Method: 1. A decider of 5 rounds up. 2. Carry if the digit you round up is a 9. 3. Keep the zeros that hold the size of the number.
Round in one pass
Look at the decider once. 6.048 to 2 decimal places uses the third decimal digit 8. The hundredths digit 4 rounds up, so the value is 6.05. Do not then round 6.05 on to 6.1. The question asked for two decimal places, and 6.05 already has them.
14.76 to 1 decimal place uses the hundredths digit 6. The tenths digit 7 rounds up to 8, so the value is 14.8. Rounding to the nearest whole number first would give 15, which answers a different instruction. Match the column to the words in the question.
1765 to the nearest 100 uses the tens digit 6, so it rounds to 1800. The same number to the nearest 10 uses the units digit 5, so it rounds to 1770. One number can have two correct rounded values. Write which instruction you used.
Method: 1. Look at the decider once only. 2. Do not round the rounded answer a second time. 3. Match the column to the instruction in the question.
Estimating a calculation by rounding
An estimate uses simple rounded numbers so you can calculate in your head or on paper. If the question says "round each number to the nearest 10", then 42 × 27 becomes 40 × 30 = 1200. If it says "round to 1 significant figure", 7.8 × 31 ÷ 0.48 becomes 8 × 30 ÷ 0.5 = 480. Write the rounded numbers on the page, because the method mark is for them.
A common command is "show that the answer is wrong". Estimate and then compare: 462 ÷ 19.8 is about 460 ÷ 20 = 23, so an answer of 2.33 is a factor of ten too small. Be careful with division by a number smaller than one, which makes the answer larger. Never use a calculator to get an exact value and then call it an estimate, since the question asks for the estimate by rounding.
Method: 1. Round every number in the calculation using the rule in the question. 2. Write down the rounded calculation. 3. Work it out with easy numbers. 4. State whether the given answer is close to your estimate and say why.
What number could have been rounded
A rounded value hides the original. A crowd of 3500 to the nearest 100 could be any number from 3450 up to, but not including, 3550. Half a step either side: the step is 100, half is 50, so the smallest value is 3500 - 50 = 3450. The number 3450 itself would round up to 3500, so it is allowed. The value 3550 would round up to 3600, so it is not.
The same rule holds for decimals. A height of 1.64 m to 2 decimal places has a step of 0.01, half-step 0.005, so it lies from 1.635 up to just below 1.645. A question may ask "Could the height be 1.645 m?" and the answer is no, because 1.645 rounds up to 1.65. Always give a reason in words and quote the number the value would round to.
Method: 1. Work out the size of the rounding step, for example 100 or 0.01. 2. Halve the step to get the amount either side of the rounded value. 3. Subtract it for the minimum and add it for the upper bound. 4. Remember that the upper bound itself would round up, so it is not a possible value.
Tip: The minimum is allowed. The upper bound is not.
Which way to round in a real situation
In context, the rounding rule depends on the question. A bill shared between 7 friends, £86.50 ÷ 7 = 12.357ldots, is rounded to the nearest penny, £12.36. If the question is how many boxes you need to cover a floor, 17.3 ÷ 2.4 = 7.2, and you must buy 8 boxes, because 7 boxes would not cover the floor. That is rounding up, whatever the decider digit says.
If the question asks how many whole items you can make or fill, you round down. For example, 7.9 lengths of ribbon means only 7 whole pieces. Write the unrounded answer first so the method mark is safe, then write the answer in the form that makes sense and add a short sentence explaining why. The wording "give your answer to the nearest penny" or "to a sensible degree of accuracy" tells you what is needed.
Method: 1. Work out the exact answer with the calculator display. 2. Decide whether the story needs rounding up, rounding down or the nearest value. 3. Round to the number of places asked for, or to the penny for money. 4. Write the final answer with its unit.