Rounding & Estimation

Identify the digit in the required place value and circle the number to the right
- This number will determine whether to round up or round off
- e.g. To round 1294 to the nearest 100 you would find the 2 digit and then use the 9 to decide how to round $1294$
Identify the two options that the number could round to
- e.g. the two nearest 100's to 1294 are 1200 and 1300
- Be careful if your digit is a 9 and the next number up will affect the higher place values
  - e.g. the nearest 2 decimal places to 3.497 are 3.49 and 3.50
If the circled number is 5 or more then you round to the bigger number
If the circled number is less than 5 then you round to the smaller number
You then put a zero in any following place values before the decimal
- If you are rounding to nearest decimal places then make sure you leave your answer with the required amount of decimal places - do not put unnecessary zeros
  - e.g. 1297 to the nearest 100 is 1300
  - e.g. 3.497 to two decimal places (nearest 100th) is 3.50 (exactly two decimal places in answer)

Rounding to significant figures is the same as rounding to place value
- You just need to identify the relevant place value
Find the first significant figure
- Find the biggest place value that has a non-zero digit
  - The first significant figure of 3097 is 3
  - The first significant figure of 0.006207 is 6
Start with this number and count along to the right
- You do count the following zeros
  - e.g. 0 is the second significant figure of 3097
  - e.g. 9 is the third significant figure of 3097
Use the normal rules for rounding
- Circle the number to the right
- Use this to determine whether the given significant figure rounds up or rounds off

Truncation is essentially the same as rounding down to a given degree of accuracy
It is usually used in situations where a whole number of items is needed
- For example if the question is about the maximum number of people that can fit in a taxi and the answer is 4.6, the answer would be truncated to 4, rather than rounded to 5

We round numbers to something sensible before calculating
GENERAL RULE:
- Round numbers to 1 significant figure
  - 7.8 ➝ 8
  - 18 ➝ 20
  - 3.65 × 10^-4 ➝ 4 × 10^-4
  - 1080 ➝ 1000
EXCEPTIONS:
- It can be more sensible (or easier) to round to something convenient
  - 16.2 ➝ 15
  - 9.1 ➝ 10
  - 1180 ➝ 1200
It wouldn’t usually make sense to round a number to zero

For addition
- If you round both numbers up then you will overestimate
- If you round both numbers down then you will underestimate
For multiplication
- If you round both numbers up then you will overestimate
- If you round both numbers down then you will underestimate
Subtraction and division are more complicated
You need to consider the effects of rounding each number
- For subtraction a - b
  - Increasing a and/or decreasing b will increase the answer so you will overestimate
  - Decreasing a and/or increasing b will decrease the answer so you will underestimate
  - If both numbers are increased or both are decreased then you can not easily tell if it is an underestimate or underestimate
- For division a ÷ b
  - Increasing a and/or decreasing b will increase the answer so you will overestimate
  - Decreasing a and/or increasing b will decrease the answer so you will underestimate
  - If both numbers are increased or both are decreased then you can not easily tell if it is an underestimate or underestimate

When rounding or truncating to a degree of accuracy greater than ten, remember to add the necessary zeros to ensure the place value of the number is not changed
- For example to truncate the number 5760 to the nearest hundred, the answer should be 5700, not 57

Calculate an estimate for $\frac{17.3 \times 3.81}{11.5} .$ State, with a reason, whether the estimate is an overestimate or an underestimate.

Round each number to 1 significant figure.

17.3 → 20
3.81 → 4
11.5 → 10

Perform the calculation with the rounded numbers.

$\frac{20 \times 4}{10} = \frac{80}{10} = 8$

An estimate is 8.
This is an overestimate as the numerator was rounded up and the denominator was rounded down.

Rounding & Estimation (AQA GCSE Maths)