### GCSE Maths

Numbers
Algebra
Geometry and Measures
Probability
Statistics

# Rounding and Estimation

The approximation symbol (≈) is used when you are indicating that a value is close to, but not exactly equal to, another value. It is typically used where:

• You have rounded a number to a specific decimal place or a whole number. For example, if you calculate a value as 3.14159 and round it to two decimal places, you would write it as π ≈ 3.14.
• You are using an estimated value or an average value to make a calculation. For example, if you estimate that the average speed of a car is around 55 mph, you would write it as v ≈ 55 mph.
• The exact value is not necessary or practical for the given context. In some cases, an approximate value is sufficient for understanding or solving a problem, and using the exact value would be cumbersome or unnecessary.
• You are working with irrational numbers, such as the square root of a non-perfect square, and you need to express the value as a decimal. For example, you might write √2 ≈ 1.414.

When faced with calculations involving awkward numbers, we can round these numbers to simplify the process. This is particularly useful for estimating results when exact calculations aren’t necessary. Keep in mind that the estimations we’ll look at are not perfect, but they provide a quick idea of the result without resorting to a calculator.

Let’s look at some examples involving estimation.

## Examples

Example 1:

Estimate the value of .

We will round up 36.012 to a number, as the square root is easy to find. Instead of using 36.012, we will use 36 and . So, 6 will be used in place of .

Example 2:

Estimate the value of: We take as and 2.967 as 3. We will also take 3.008 as 3. We then have:   A calculator gives 1.008644 to six decimal places. Our estimate of 1 is not bad, and all the calculations involved were much easier than if we used the original values.

Example 3:

Estimate, to two significant figures, the value of We round 27.0413 to 27, 0.995 to 1, and 1.97 to 2. We then have:    A calculator gives a value of 3.943, to 3 decimal places.

The percentage error involved is: The estimation in this example has a percentage error of 1.45%.

Example 4:

By writing each number correct to one significant figure, estimate the value of To one significant figure: and .

So, our calculation gives:   Example 5:

Estimate, to one significant figure, the value of We shall use , , and in place of , , and respectively. We then have    Example 6:

Estimate the value of: We approximate the numbers to get:    Example 7:

Estimate We approximate the numbers and get:  