## Rounding Help

*This is the third in a series of posts which have been written for a site aimed at our A level students. The first two (fractions and indices) have led to people giving me great feedback on things I could improve and have enabled me to ensure the site, when launched, will be in great shape to help our learners. The third installment is on rounding, a topic I’ve seen really bright students get wrong and one I’ve known senior teachers to teach incorrectly. (It’s a pandemic!) Please get in touch if I’ve omitted anything, made an error, or just lack clarity anywhere.*

It seems ridiculous to even contemplate a help page for A level students which is based on rounding, it’s something that really should have been learned before now. Yet every year we see students who struggle to round correctly and lose daft marks through rounding errors. We hope this page will help you minimise the rounding errors and stop losing silly marks.

**Don’t round until the end**

One way people can go wrong with rounding is to round early. This can lead to an answer being far enough away from the correct answer to lose marks, even though the maths is correct all the way through. Rather than round, use the “ans” button on your calculator, or write the whole display down each time. This may still technically be rounded, but the full display should have sufficient accuracy to ensure your final answer is correct. When you are rounding at the end of a question, it is always best to write the full display on one line, then the rounded answer on the next.

**Exact Answers **

Before we go into rounding, I feel I should mention exact answers. This is a related area where people lose marks. The exact answer isn’t “all the digits of your calculator display”, this is still rounded. If a question asks for an exact answer it will want it in surd form, or in terms if a known irrational constant such as pi.

**Decimal places**

These are easy, this is the number of digits after the decimal point. 3.547 has 3 decimal places (3dp), 7.4147 has 4dp

To round to a specific number of decimal places you need to pay attention to the digit that falls one place after. If you are asked to round 4.735 to 2dp it’s the same as saying “to the nearest hundredth”. You look at the 3rd decimal place, the midpoint between 4.73 and 4.74 is 4.735, so if it’s that or higher we round up, if it’s lower we round down.

You only look at the next digit. If you were asked to round 4.4444444445 to 1 dp you would round it to 4.4 as 4.4444444445 is lower than 4.45 YOU DO NOT ROUND FROM THE RIGHT. That would lead to every 4 becoming a 5 and the answer being wrong. I know it’s obvious, but I’ve known A grade A level candidates mess up here!

**Significant figures **

For some reason these are like kryptonite to some students. Even some students who are generally fine with rounding. Significant figures are about the numbers which hold most significance. In the number 78315 the 7 is the most significant number, as it holds the highest place value. This is the number we use as our first significant figure. We then round the same as for decimal places, use the next digit to see if we are closest to the number we have, or if we need to round up. So 78315 would round to 80000 to 1sf 78000 to 2sf and 78300 to 3sf

Likewise for decimal numbers. In 0.00537 the 5 holds the highest place value, so is most significant. This would round to 0.005 to 1sf and 0.0054 to 2sf. An issue some have is when a 0 appears after a non zero digit. Ie in 0.05071, I this case we still start from the most significant, so it’s 0.05 to 1sf and 0.051 to 2sf etc.

A problem some have is when it’s something like 4.745 and you’re rounding to significant figures. People get confused and round to decimal places, when actually the first significant figure falls before the decimal point. So in this case we’d get 4.7 (2sf).

**What should I round to?**

Usually exams will specify what to round to, in which case you should round to that. If it’s not specified then as long as you have used exact values then an exact answer should be fine, if you have used a rounded value then round to whatever that was rounded to. For example, if you are working with gravity in M1 you take g to equal 9.8 which is the constant rounded to two significant figures, so round your answer to that.