The nearer tenth
A bag of apples weighs 3.47 kg, and its price label has room for only one decimal place. 3.47 lies between two tenths, 3.4 and 3.5. Rounding to one decimal place means choosing the one it is nearer to.
Halfway between 3.4 and 3.5 is 3.45. 3.47 is above the halfway mark: it is 7 hundredths past 3.4 and only 3 hundredths short of 3.5. So 3.5 is nearer, and to one decimal place 3.47 rounds up to 3.5.
3.47 is above the halfway mark of 3.45, so it is nearer to 3.5 and rounds up to 3.5.
Below the halfway mark
2.62 lies between 2.6 and 2.7, and the halfway mark is 2.65. 2.62 is below it, only 2 hundredths past 2.6, so it rounds down to 2.6. Rounding down leaves the tenths digit as it is and drops the digits after it.
2.62 is below the halfway mark of 2.65, so it is nearer to 2.6 and rounds down to 2.6.
One digit decides
The hundredths digit says which side of the halfway mark the number is on, so it decides on its own. To round to one decimal place, keep the tenths and look only at the hundredths digit. If it is 5, 6, 7, 8 or 9, round the tenths digit up. If it is 0, 1, 2, 3 or 4, leave the tenths digit as it is. Then drop every digit after the tenths.
A number exactly on the halfway mark, such as 3.45, is as near to 3.4 as to 3.5. It rounds up to 3.5, by the same rule that rounds 45 up to 50.
Two decimal places
Rounding to two decimal places keeps the hundredths, and the digit just after them, the thousandths digit, decides. 1.283 lies between 1.28 and 1.29, and the halfway mark is 1.285. Its thousandths digit is 3, so it is below the mark and rounds down to 1.28.
In the same way, 1.287 has a thousandths digit of 7 and rounds up to 1.29.
1.283 is below the halfway mark of 1.285, so to two decimal places it rounds down to 1.28.
When a 9 rounds up
Take care when the digit you keep is a 9. 4.97 lies between 4.9 and 5.0, and its hundredths digit is 7, so it rounds up. One more tenth after 4.9 makes 5 wholes, so 4.97 rounds up to 5.0.
Write the 0 in 5.0. The question asked for one decimal place, and the answer should show one.
4.97 is above the halfway mark of 4.95, so it rounds up to the next tenth, which is 5, written 5.0 to one decimal place.
Round, do not cut off
Cutting 3.47 short to 3.4 is not rounding: it drops the 7 and chooses 3.4, although 3.5 is nearer. And the answer must have the number of places asked for. 3 is 3.47 to the nearest whole number, not to one decimal place, and 3.47 left as it is has not been rounded at all.
Worked example: Smallest and Largest Value Before Rounding
Question The mass of a parcel is measured in kilograms to 2 decimal places. Rounded to 1 decimal place, the mass is 4.7 kg. (a) What is the smallest possible mass of the parcel? (b) What is the largest possible mass of the parcel?
1.Mark 4.7 on a number line in hundredths, with 4.6 and 4.8 on either side.
Each small mark is one hundredth. The numbers nearest to 4.7 round to 4.7. 2.A number rounds up to 4.7 when its hundredths digit is 5 or more. The smallest such number is 4.65.
(a) 4.65 is the smallest number that rounds up to 4.7. 3.(a) The smallest possible mass is 4.65 kg.
4.A number rounds down to 4.7 when its hundredths digit is 4 or less. The largest such number with 2 decimal places is 4.74.
From 4.65 to 4.74, every number rounds to 4.7. 5.(b) The largest possible mass is 4.74 kg, because 4.75 rounds up to 4.8.
(b) 4.74 is the largest. The next hundredth, 4.75, rounds up to 4.8.
Answer: (a) 4.65 kg; (b) 4.74 kg
Common mistakes
- Giving 4.75 kg as the largest mass. 4.75 rounds up to 4.8, so the largest mass with 2 decimal places is 4.74 kg.
- Giving 4.6 kg as the smallest mass. 4.6 rounds to 4.6. The smallest number that rounds up to 4.7 is 4.65.