Basic Operations & Estimation

Video coming soon3 worked examples

Theory

The Golden Rule: candidates frequently lose valuable marks in Paper 1 (the non-calculator paper) by not demonstrating solid basic number skills. It is essential to maintain and practise these non-calculator arithmetic skills to secure your marks.

1. Decimals: Addition & Subtraction

You are expected to add and subtract numbers with up to two or three decimal places.

The Rule: You must line up the decimal points directly below one another in your working.
Trailing Zeroes: It is highly recommended to add "trailing zeroes" to the end of your numbers so that they all have the exact same number of decimal places before you begin the calculation (e.g., change 34.7 to 34.70).

2. Multiplying and Dividing by Multiples of 10, 100, and 1000

To multiply or divide a decimal by a multiple of 10, 100, or 1000, you must split the calculation into two easier steps:

Step A: Multiply or divide by the single digit.
Step B: Move the digits left or right by multiplying or dividing by 10, 100, or 1000.

Example: To calculate $3.5 \times 400$ , you calculate $(3.5 \times 4) \times 100$ .

3. Order of Operations (BODMAS/BIDMAS)

Calculations must be performed in a specific mathematical order:

Brackets
Order / Indices (Powers)
Divide & Multiply (Left to right)
Add & Subtract (Left to right)

Operations within brackets must always be dealt with first, followed by multiplication and division, before you attempt any addition or subtraction.

4. Estimation & Checking Answers

You are expected to use your knowledge of rounding and estimation to check if your final answer makes logical sense in the context of the question. Always stop and ask yourself: "Does this answer seem sensible?"

⚠️ Common Examiner Traps

The Left-to-Right BODMAS Trap: calculations are frequently set that look tempting to read simply from left to right (e.g., $27.2 - 4.6 \times 3 + 4.7$ ). Candidates lose marks by ignoring the rule that multiplication must happen before addition and subtraction.
The Decimal Alignment Trap: When subtracting a decimal from a whole number (e.g., $30 - 14.26$ ), candidates often forget to write the whole number as 30.00 to allow them to borrow correctly.
The Unrealistic Answer Trap: Candidates often calculate an answer but fail to check if it is logically possible (e.g., calculating that a person worked 150 hours in a single day). Always estimate your answer first to ensure your final calculation is realistic.
The Money Formatting Trap: You will lose marks if you fail to write amounts of money to exactly two decimal places (e.g., writing £4.5 instead of £4.50).

Worked examples

Example 1

A plumber is fitting a new pipe. He starts with a copper pipe that is exactly 5 metres long. He cuts off two pieces: one piece measuring 1.35 metres and another measuring 2.8 metres. Calculate the length of the copper pipe he has left. (2 marks)

Step 1: Add the two cut lengths together. Add a trailing zero to the 2.8 so both numbers have two decimal places.

1.35 + 2.80 = 4.15\text{ metres}

Step 2: Subtract the total cut pieces from the original 5 metres. Crucially, add a decimal point and two trailing zeroes to the 5 to allow for correct borrowing.

5.00 - 4.15 = 0.85\text{ metres}

Answer: $0.85\text{ m}$ .

Example 2

A local cafe sells 300 premium hot chocolates over the course of a weekend. If each hot chocolate requires 14.2 grams of cocoa powder, calculate the total weight of cocoa powder used. (2 marks)

Calculation: $14.2 \times 300$

Step A (Multiply by the single digit):

14.2 \times 3 = 42.6

Step B (Multiply by 100):

42.6 \times 100 = 4260\text{ grams}

Estimation Check: $15\text{ grams} \times 300 = 4500\text{ grams}$ , so 4260 grams is a highly sensible answer.

Example 3

A student is checking an invoice for school supplies. The invoice lists the following calculation to find the total cost in pounds: $12.50 + 4.20 \times 6$

The student claims the total cost is £100.20. Determine if the student is correct. Use your working to justify your answer. (2 marks)

Step 1: According to BODMAS, the multiplication must be calculated before the addition.

4.20 \times 6 = 25.20

Step 2: Substitute this back into the original sum and perform the addition.

12.50 + 25.20 = 37.70

Final Answer (Communication): The student is incorrect. They calculated the sum from left to right instead of using BODMAS. The correct total is £37.70. (Note the money format to two decimal places).

Ready to practise?

8 Basic Operations questions with answers and video solutions