Number 4. Rounding and Approximations Mr F’s Maths Notes Number 4. Rounding and Approximations

4. Rounding and Approximations A Question… Imagine you are walking along the street and someone stops you and asks you to do this nice little sum in your head in 30 seconds… If you are anything like my students, I can imagine what you might say, and I can’t write it here…other than say...eek!... But, with a little knowledge about rounding and approximations, you should be able to tell that person that the answer is about 2, and then ask them to kindly leave you alone 1. Rounding Now, there are lots of degrees of accuracy you will need to know how to round to, but the way to tackle any question you could ever possibly be asked is always the same: 1. Circle the last digit you need – what I will call the Key Digit 2. Look at the unwanted digit to the right to it – if it is 5 or above add one on to your Key Digit, if it is less than five, leave your Key Digit alone. 3. Be very careful of the dreaded number 9…

(a) decimal places The most common degree of accuracy you are asked to round to is a number of decimal places. Because mathematicians are lazy, this is normally shortened down to dp. e.g. 5.96 (2dp) means that the answer was probably really long, but when rounded to two decimal places, it was 5.96 The thing you need to remember, and the thing that sounds really obvious, is that if the question asks for two decimal places, you must give two, no more, no less! Example 1 Example 2 Round 5.639 to 1dp Round 12.0482 to 2dp 5 . 6 3 9 1 2 . 0 4 8 2 1. We start by putting a ring around our Key Digit. Now the question has asked for 1 decimal place, so our key digit is the 6, as it occupies the 1st decimal place 1. This time the Key Digit is in the 2nd decimal place, which makes it the 4 2. The unwanted digit to the right of it is an 8, which is definitely 5 or above, so we must add one onto our Key Digit 2. Next we look at the digit to the right to it – the unwanted number 3. It is less than 5, so we leave the key digit alone. 3. So, to two decimal places, our answer is: 3. So, to one decimal place, our answer is: 12.05 5.6

Example 3 Example 4 Round 25.72037 to 3dp Round 3.7952 to 2dp 2 5 . 7 2 0 3 7 3 . 7 9 5 2 1. This time the Key Digit is in the 3rd decimal place, which makes it the 0 1. This time the Key Digit is in the 2nd decimal place, which makes it the 9 2. The unwanted digit to the right of it is 3, which is definitely less than 5, so just leave our Key Digit alone 2. The unwanted digit to the right of it is a 5, which is 5 or above, so we must add one onto our Key Digit 3. So, to three decimal places, our answer is: But: if we add one to our key digit, we get 10! So, we must add one to the next digit as well, which is the 7 25.720 Be careful: Some silly people will put 25.72 down as the answer thinking that the 0 makes no difference. But it does! The question has asked for 3dp, so give them 3dp! 3. So, to two decimal places, our answer is: 3.80 (b) nearest whole, 10, 100, 1000 etc These are the nicest types of rounding questions, and so long as you have your brain switched on, you shouldn’t get too many of them wrong. But don’t get cocky, as you can easily make mistakes! Remember: the size of your rounded number should be a similar size to the number in the question, and you must use zeros to help you with this

Example 1 Example 2 Round 3.825 to the nearest whole number Round 32,825.2 to the nearest hundred 3 . 8 2 5 3 2 8 2 5 . 2 1. Our Key Digit is always the degree of accuracy the question asks for, which in this case is whole numbers, so we need the 3. 1. We want the nearest hundred, so stick the ring around the digit in the hundreds column, which is the 8. 2. The unwanted digit to the right of it is 8, which is definitely more than 5, so we add one to our Key Digit. 2. The unwanted digit to the right of it is a 2, which is less than 5, so we leave our Key Digit alone. 3. So, to the nearest whole number, our answer is: 3. So, to the nearest hundred, our answer is: 4 32,800 Example 3 Example 4 Round 4,365,901 to the nearest thousand Round 3,999 to the nearest ten 4 3 6 5 9 0 1 3 9 9 9 1. We want the nearest thousand, so our Key Digit must be the number that represents the thousands which is the 5 1. We want the nearest ten, so the Key Digit must be the 9 in the tens column 2. The unwanted digit to the right of it is a 9, so we add one on, but we then need to add one on the next 9, and then the 3! 2. The unwanted digit to the right of it is 9, which is definitely more than 5, so we add one to our Key Digit. 3. So, to the nearest ten, our answer is: 3. So, to the nearest whole number, our answer is: 4,000 4,365,000

(c) significant figures In higher level SATs, and especially in GCSE and A Level, the nasty examiners are obsessed with Significant Figures. Again, there is a lazy way of writing this, which is sf or sig fig. Crucial: The first significant figure is always the first non-zero number you come across. The second significant figure is the number to the right of that, and so on… Remember: the size of your rounded number should be a similar size to the number in the question, and you must use zeros to help you with this. Example 1 Example 2 Round 28.53 to 1 sig fig Round 5,322 to 2 sig figs 2 8 . 5 3 5 3 2 2 1. The Key Digit has the be the first significant figure, which must be the 2, as it is the first non-zero number 1. The Key Digit is in the place of the 2nd significant figure, which is the 3 2. The unwanted digit to the right of it is 2, which is definitely less than 5, so we leave our Key Digit alone 2. Now we carry on as normal looking to the number to the right, which is an 8, so we add one on. 3. So, keeping the size of the answer the same as the question with a zero, to 1 sig fig the answer must be: 3. So, again using zeros to help us, to two sig figs, our answer is: 5300 30

Example 3 Example 4 Round 0.027 to 1 sig fig Round 305,216 to 3 sig figs 0 . 0 2 7 3 0 5 2 1 6 1. Our first significant figure is the first non-zero number, which means it’s the 2 1. The 1st sig fig is the 3, the 2nd is the 0 (it is after the 3, so it’s significant), so the Key Digit is the 5 2. The unwanted digit to the right of it is 7, so we add one to our Key Digit. 2. The unwanted digit to the right of it is a 2, so we leave our Key Digit alone. 3. No need for extra zeros here, so to the 1 significant figure our answer is: 3. We need some zeros to make our answer the correct size, so to 3 sig figs:: 0.03 305,200 Example 5 Example 6 Round 4.0004 to 2 sig figs Round 0.089722 to 2 sig figs 4 . 0 0 0 4 0 . 0 8 9 7 2 2 1. The 1st sig fig is the 4, and so the 2nd is the 0 (it is after the 4, so it’s significant). 1. Our 1st non zero number is the 8, so the Key Digit must be the 9. 2. The unwanted digit to the right of it is a 7, so we add one on, but that gives us 10, so we must add one to our 8 as well. 2. The unwanted digit to the right of it is 0, which is definitely less than 5, so we leave our Key Digit alone 3. Keeping our answer the right size, we have: 3. So, to 2 sig figs, our answer is: 4.0 0.090

2. Approximations Right, now that we are experts at rounding, we can use our skills to find approximate answers (or estimates) to horrible looking questions like the one at the start: Now, you would need to be a bit of a freak to do this in your head, but if you were to round each number to the nearest whole number, or 1 significant figure then you get: And if you now use our BIDMAS skills, you should be able to say: The actual answer on the calculator is pretty close: 1.98521838… So, if we want to sound clever, we can say that: Where the funny sign means: “approximately equal to” So, always look out for ways to use your rounding skills to turn tricky looking sums into pretty easy ones!

Good luck with your revision!