Estimation, Approximation and Rounding
Sometimes, you may find it helpful to know the approximate answer to a calculation.
You may be in a shop and want to know broadly what you’re going to have to pay.
You may need to know roughly how much money you need to meet a couple of bills.
You may also want to know roughly what the right answer to a more complicated calculation is likely to be, to check that your detailed work is correct.
Whatever your precise need, you want to know how to estimate or approximate the right answer.
One very simple form of estimation is rounding. Rounding is often the key skill you need to quickly estimate a number. This is where you make a long number simpler by ‘rounding’, or expressing in terms of the nearest unit, ten, hundred, tenth, or a certain number of decimal places.
For example, 1,654 to the nearest thousand is 2,000. To the nearest 100 it is 1,700. To the nearest ten it is 1,650.
The way it works is straightforward: you look at the number one place to the right of the level that you are rounding to and see whether it is closer to 0 or 10.
In practice, this means that if you’ve been asked to round to the nearest 10, you look at the units. If you are rounding to three decimal places, you look at the fourth decimal place (the fourth number to the right of the decimal point) and so on. If that number is 5 or over, you round up to the next number, and if it is 4 or under, you round down.
Round Up or Round Down?
We round numbers to reduce their number of digits while keeping the result as close to the original number as possible.
Numbers that are less than 5 get rounded down.
Numbers that are 5 or higher get rounded up.
Rounding to one decimal place:
- 1.47 rounds to 1.5
- 1.42 rounds to 1.4
- 1.4535412 rounds to 1.5
Rounding: Worked Examples
Express 156 to the nearest 10
In this example you look at the tens and units. The hundreds will not change. You need to decide whether 56 will be rounded up to 60 or down to 50.
Looking at the units, you know that 6 is more than 5, so you round up.
The answer is 160.
Express 0.4563948 to three decimal places.
As you're working to three decimal places, the answer will start 0.45 and you need to determine the third number after the decimal point
To work out whether the third number is 6 or 7, you need to look at the fourth number, which is 3. As 3 is less than 5, you round down.
The answer therefore is 0.456.
You can use the technique of rounding to start estimating the answer to more complex problems.
Estimating can be considered as ‘slightly better than an educated guess’. If a guess is totally random, an educated guess might be a bit closer.
Estimation, or approximation, should give you an answer which is broadly correct, say to the nearest 10 or 100, if you are working with bigger numbers.
Probably the simplest way to estimate is to round all the numbers that you are working with to the nearest 10 (or 100, if you are working in thousands at the time) and then do the necessary calculation.
For example, if you are estimating how much you will have to pay, first round each amount up or down to the nearest unit of currency, pound, dollar, euro etc. or even to the nearest 10 units (£10, $10, €10), and then add your rounded amounts together.
Many stores like to give prices ending in .09 and especially 0.99. The reason for this is that a shirt that costs 24.99 'sounds' cheaper than one that costs 25.00. When shopping for numerous items it can be useful to keep a running tally, an estimate of the total cost, by rounding items to the nearest currency unit, £, $, € etc.
If you are trying to work out how much carpet you will need, round the length of each wall up to the nearest metre or half-metre if the calculation remains simple, and multiply them together to get the area.
If you are relying on your calculation to make sure that you have enough of something, whether money or carpet, always round up. That way you will always over-estimate. Even engineers take this approach when thinking about the design of a structure before doing a detailed specification. It’s better to have a component that’s a bit stronger than it needs to be than one that is too weak.
You want to buy carpet for two rooms. The first is 3.2m by 2.7m. The second is smaller, 1.16m by 2.5m. How much carpet do you need to buy to be sure of having enough for both rooms?
The first room is approximately 3m by 3m, which is 9m2.
The second is just over 1m by 2.5m. Strictly speaking, you would round this to 1m by 2.5m, or 2.5m2.
In total, then, that’s 11.5m2. It’s hard to buy carpet in anything except whole m2, so you’ll need to round up to 12m2. In each case, you have rounded up one of the numbers by more than you have rounded the other one down, so you’re probably fine.
A quick check with a calculator will, indeed, confirm that you need exactly 11.54m2. 12m2 will be plenty.
You’ve decided to add another room to the carpet buying. The last room is 3.9m by 2.2m. How much carpet do I need for all three rooms?
3.9m is rounded up to 4m. 2.2m rounds down to 2m.
2 × 4 is 8m2, which gives a total, for all three rooms of 20m2.
However, in rounding down to 2m, you have taken out 0.2m. In rounding up to 4m, you have only added 0.1m.
You may not order quite enough carpet although you might get away with it because you rounded up to 12m2 for the first two rooms.
However, to be absolutely sure, you probably want to round 2.2m up, to 2.5m.
Multiply 2.5 by 4 to get 10m2. This means you need 22m2 of carpet for all three rooms.
A quick check with a calculator will confirm that 20m2 is not quite enough: You need 20.9m2 exactly.
Need a refresher on how to calculate area? See our page Calculating Area for help.
Estimated Time of Arrival (ETA)
Estimated time of arrival is used frequently used when travelling. Trains, buses, planes, ships and in-car satellite navigation (sat-nav) all use ETA.
The ETA is based on distance and speed of travel, it is ‘estimated’ because it cannot take into account changes to speed during the journey. Your flight may arrive early because of favourable tail winds. Your road trip may take longer than expected because of traffic.
The ETA is usually calculated by a computer and can change during your trip. As you near your destination, more data becomes available so the estimated time that you will arrive becomes more accurate.
A Special Case: Estimating for Work
You will almost certainly come across ‘estimates’ for work to be done, whether from a builder, plumber, mechanic or other tradesperson.
In this case, the tradesperson concerned has probably estimated how much time they are likely to take to do the work, multiplied it by their hourly or daily rate, and perhaps added additional charges for materials or a call-out.
They may also have added a ‘contingency’ for extra work needed, which is likely to be 10 or 20%, and will mean that you are not unpleasantly surprised by the bill if they find something unexpected that needs fixing.
An ‘estimate’ is not legally binding. It is just what it says: an estimate.
However, a ‘quote’ or ‘quotation’ for work done is legally binding on cost, provided that the work done is what was quoted for. However, if you have asked for extra work: ‘just add that bit’ or ‘do that while you’re here’, don’t be surprised if the bill is larger than you were expecting.
A Useful Skill
You may be wondering why you’d ever use estimation when you have a calculator on your phone.
The ability to estimate will mean that you will know if the answer you get from the calculator is not right, and do it again.