## Approximation and Estimation Definition

An **approximation** is a value that is **close** to the **true** **value** but **not** **quite** **equal** to it. Note that the symbol for "approximately" is$\approx $.

We could approximate pi by saying that$\mathrm{\pi}\approx 3.14$. In actual fact, pi is an irrational number that never ends, so $3.14$ is certainly not an exact value. However, it is a very good approximation.

**Estimation** is a process where we either guess or roughly calculate something. Our objective is to obtain a value that is as close to the true value as possible.

The definition of pi is the ratio of a circle's circumference and diameter. Thus, if we get a piece of string to measure the circumference of a circle, and divide it by its diameter, we can estimate pi. Suppose, we measure a diameter of a circle as 10 cm, and a circumference of 31.4 cm, we could say$\mathrm{\pi}\approx \frac{31.4}{10}=3.14$. Therefore, we have just estimated pi to be 3.14.

### Rounding

For this topic, it is also important that we know how to round numbers. Let's quickly recap before going any further...

The process of **rounding** a number means to make the number something that it is close to that is easier to work with.

**Round the number 3728 to the nearest 10, 100, and 1000. **

**Solution:**

When rounding to the nearest 10, we need to look at the digits from the 10s column onwards. In this case, we have 28. Now, we must ask ourselves, is 28 closer to 20 or 30? The answer is 30, since 28 is only 2 away from 30, whereas it is 8 away from 20. Thus, 28 rounds to 30 and so 3728 rounds up to 2730.

When rounding to the nearest 100, we look at the digits from the 100s column onwards. In this case, we have 728. Now, we must ask ourselves, is 728 closer to 700 or 800? Clearly, it is closer to 700 and so in this case, 3728 rounds down to 3700.

When rounding to the nearest 1000, we look at the digits from the 1000s column onwards. This is 3728. Now, we must ask ourselves, is 3728 closer to 3000 or 4000. In this case, it is closer to 4000 so we round up to 4000 when rounding to the nearest 1000.

Now that we have defined some key terms, we will now look at some examples using approximation and estimation.

## Approximation and Estimation Examples

### How to Estimate

To estimate a calculation, first **round** all the numbers to something that is "easy" to work with. For example, it is hard to multiply 72 by 91, however, it is much easier to work out $70\times 90$and so we would round each of the numbers to the nearest 10 to form an estimate. This **rounding** process is an example of **approximating**. In other words, 72 is approximately 70 and 91 is approximately 90 so we use those numbers to work out our estimation. Sometimes, it is easier to round to the nearest whole number, hundred, or even decimal place. Choose something sensible that enables you to work out the estimation in your head.

### Examples

**Work out an estimate for $\frac{92.1\times 19.2}{98.1}$**

**Solution:**

This calculation is quite difficult to work out without using a calculator. However, if we round each number to the nearest 10, we obtain$\frac{90\times 20}{100}=\frac{1800}{100}=18$. Thus, we can say that the estimation for the calculation is 18.

We could go one step further and work out the **percentage error** between the estimated value and the real value. Using a calculator, we can find that$\frac{92.1\times 19.2}{98.1}=18.0256881$. If we subtract the estimated value from the real value, we get what is called the absolute error. In this case, the absolute error is 0.0256881 (which is promising as it shows that our estimated value is close to the real value due to such a small absolute error).

Now, if we divide the absolute error by the actual value, and then multiply by 100, we get the percentage error. If we do this, we get the percentage error as$\frac{0.0256881}{18.0256881}\times 100\%=0.1425\%$. Since this number is small, we can see that we have a good estimation as we have such a small percentage error.

**I buy 32 packets of crisps for a party. Each packet costs 21p. Estimate the total cost of the crisps.**

**Solution:**

The total cost is 32 lots of 21p. Thus, we need to calculate$32\times 21$ to work out the cost (in pence). We can round both numbers to the nearest 10 to get$30\times 20$ which is far easier to work out. We get$20\times 30=600p=\pounds 6$. Thus, we can say that the total cost of the 32 packets of crisps is about £6. The true value of$32\times 21=672p=\pounds 6.72$and so we can see that our estimated value is close to the actual value.

If we were to go one step further and work out the percentage error, we would need to subtract the estimated value from the true value , divide by the actual value and then multiply by the true value as follows:

$\frac{6.72-6}{6.72}\times 100\%=10.7\%$. Thus, the percentage error of our estimate is 10.7%.

**Estimate the cost of 123 paper plates that cost 11p each and 157 napkins that cost 9p each **

**Solution:**

The total cost (in pence) of the paper plates is going to be$123\times 11$and the total cost of the napkins is$157\times 9$. Thus the total cost of both the paper plates and napkins is$123\times 11+157\times 9$. We can approximate the numbers in this calculation and instead work out $120\times 10+160\times 10=1200+1600=2800p=\pounds 28$. Thus, an estimate for the total cost is £28.

We could work out the percentage error by working out the true value of the cost. In this case, it is $123\times 11+157\times 9=2766p=\pounds 27.66$. Thus, the percentage error is$\frac{28-27.66}{27.66}\times 100\%=1.23\%$.

**Estimate the value of $\frac{301\times 9.01}{0.499}$**

**Solution:**

If we round 301 to the nearest 10, we get 300. If we round 9.01 to the nearest 10, we get 10. Now, 0.499 is approximately 0.5, so let's round it to that. Thus, we have $\frac{300\times 10}{0.5}=\frac{3000}{0.5}=6000$. Thus, 6000 is an estimate.

The true value of $\frac{301\times 9.01}{0.499}=5434.88978$ and so we can see that our estimated value is relatively close. The absolute error of our estimate is $6000-5434.88978=565.11022$ and our percentage error is $\frac{565.11022}{5434.88978}\times 100\%\approx 10.4\%$. Thus, we can say that our estimation is out by approximately 10.4%.

**Estimate the value of $\frac{{(4.98)}^{2}}{0.482}$**

**Solution: **

4.98 is quite close to 5, and it is easy to square 5, so let's approximate 4.98 as 5. 0.482 is close to 0.5, and it is fairly easy to divide by one half. Thus, we have the estimate $\frac{{(4.98)}^{2}}{0.482}\approx \frac{{\left(5\right)}^{2}}{0.5}=\frac{25}{0.5}=50$. Thus, the estimation for this calculation is 50.

We could work out the percentage error by working out the true value of$\frac{{(4.98)}^{2}}{0.482}=51.45$.

Thus, the percentage error is $\frac{51.45-50}{51.45}\times 100\%=2.82\%$.

**Estimate the value of $\sqrt{\frac{51.3}{0.53}}$**

**Solution:**

51.3 is approximately 50, and 0.53 is approximately 0.5. Thus, we have the estimation $\sqrt{\frac{51.3}{0.53}}\approx \sqrt{\frac{50}{0.5}}=\sqrt{100}=10$. Thus, an estimate for the value is 10.

We could work out the percentage error by working out the true value of $\sqrt{\frac{51.3}{0.53}}=9.84$.

Thus, the percentage error is $\frac{10-9.84}{9.84}\times 100\%=1.63\%$.

## Difference Between Estimation and Approximation

You may be questioning what the actual difference between estimation and approximation actually is. They are both very similar concepts, so how do we determine whether something is an estimation or an approximation?

Estimation is the process of **roughly** obtaining a solution to something that we don't already know. For example, you may estimate that the number of sweets in a jar is around 30, but you do not know how many there are **exactly**. You may also be in a shop, and want to estimate how much everything is going to come to. By rounding the price of everything in your basket to the nearest pound, you can get an estimation, but you do not know the true value until you go through the bills to pay. By rounding, you are approximating the price of each item; you know the true price but you want to make the calculation simpler. Going back to the first example, if it turns out there are actually 32 sweets in the jar, and each of them cost 23p, you can estimate the cost of all 32 by approximating 32 to 30 and 23 to 25.

So, the main difference between estimation and approximation is that with estimation, you do not know the true value. On the contrary, with approximation, you know the true value but want to alter it slightly to turn it into something that is easier to work with.

## Importance of Estimation and Approximation

Being able to estimate and approximate is actually a very handy tool to have in everyday life. It enables us to quickly make approximate calculations in our heads rather than relying heavily on a calculator. It also makes us really good at that game of "guess the bill", where we guess the cost of a restaurant bill before it has come.

Mathematicians often use approximations, for example, when working out solutions to equations that are hard to solve. Later in the GCSE course, you may come across iterative techniques to approximate solutions of higher-order equations (I know, we have some interesting stuff coming).

Some other uses of estimation include trying to work out the value of something. For example, property evaluators estimate the value of a property by looking at various factors including the house size, access to local travel networks, access to schools, number of bedrooms, socio-economic status of the area, and more.

Estimations enable us to make predictions, and approximations make numbers easier to work with. Of course, there are times when our estimations are rubbish and deviate massively from the true value. There are also times when approximations are not sensible. However, in general, they provide us with a really useful tool way of working things out.

## Approximation and Estimation - Key takeaways

- An
**approximation**is a value that is**close**to the**true value**but**not quite equal**to it. **Estimation**is a process where we either guess or roughly calculate something.- To estimate a calculation, first round (approximately) all the numbers involved to something that is "easy" to work with. Then compute the calculation in your head.
- The
**difference**between estimation and approximation is that estimation is where we are trying to work out the true value by either guessing or using rounding techniques. An approximation is where we already know the true value, but take a value close to the true value so that it is easier to work with.

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##### Frequently Asked Questions about Approximation and Estimation

What is approximation in math?

An approximation is where we take a value close to the true value of something to make it easier to work with or explain.

What is an example of approximation and estimation?

We may approximate the cost of a television to be £135. In actual fact, the true value may be £133.99 but that is not a very easy number to work with. We may estimate the cost of two televisions to be 2x135=£270. This is an estimation.

What are the rules of approximation?

Choose a value that is close to the true value.

What is the meaning of estimation in maths?

Where we try to work out the value of something we don't already know using approximations.

What is the difference between approximation and estimation in mathematics?

With approximation, we know the true value but change it to make it easier to work with. With estimation, we do not know the true value but are trying to work out an answer as close to the true value as possible.

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