AQA GCSE Mathematics (8300) — Ratio, proportion and rates of change
Worked Example: Expressing and Simplifying Ratios
Question: Mix g of sand and kg of cement. Express the division of sand to cement as a ratio in its simplest form.
Step 1: Convert to the same units so they can be compared. kg g
Step 2: Write using colon notation.
Step 3: Perform ratio simplification by dividing both sides by the Highest Common Factor (HCF). The Highest Common Factor (HCF) of and is .
Result: This creates an equivalent ratio in its simplest form, where the parts are integers with no common factors other than .
Worked Example: The Additive Method
Question: Divide £450 between Anna and Ben in the ratio .
Step 1: Sum the parts of the ratio to find the total parts. total parts.
Step 2: Divide the total quantity by this sum to find the share value (the value of one single share). per share.
Step 3: Multiply the share value by the original ratio numbers. Anna: Ben:
Students often incorrectly divide the total quantity by the largest number in the ratio instead of summing all the numbers to find the total parts.
In a unit conversion question, always check units first! Writing for cm and m instead of will cost you accuracy marks.
Always label your ratios in your working (e.g., writing 'Anna : Ben' above the numbers) to avoid 'reversal errors' where you assign the correct final amounts to the wrong person.
Avoid using equals signs () between different equivalent ratios in your working (e.g., do not write ); AQA examiners prefer arrows or the phrase 'is equivalent to'.
In calculation questions, explicitly write down your method (e.g., showing or ) to secure M1 (method) marks even if your final answer is incorrect.
Ratio
A way of comparing one part of a whole to another part (part-to-part) or to the whole itself (part-to-whole).
Colon notation
The standard mathematical format for writing a ratio, using a colon between the numbers (e.g., a : b).
Ratio simplification
The process of reducing a ratio to its lowest integer values by dividing all parts by their highest common factor.
Highest Common Factor (HCF)
The largest integer that divides two or more numbers without leaving a remainder.
Equivalent ratio
Ratios that represent the same proportion, found by multiplying or dividing all parts by the same non-zero number.
Simplest form
A ratio where the parts are integers with no common factors other than 1.
Additive method
The specific sequence of adding ratio parts together to determine the total number of shares.
Unitary method
A mathematical process where the value of a single unit (one share) is found first to calculate the value of multiple units.
Total parts
The sum of all the numbers in a ratio, representing the total number of shares an amount is split into.
Share value
The calculated amount that one single part of a ratio is worth.
Put your knowledge into practice — try past paper questions for Mathematics
Ratio
A way of comparing one part of a whole to another part (part-to-part) or to the whole itself (part-to-whole).
Colon notation
The standard mathematical format for writing a ratio, using a colon between the numbers (e.g., a : b).
Ratio simplification
The process of reducing a ratio to its lowest integer values by dividing all parts by their highest common factor.
Highest Common Factor (HCF)
The largest integer that divides two or more numbers without leaving a remainder.
Equivalent ratio
Ratios that represent the same proportion, found by multiplying or dividing all parts by the same non-zero number.
Simplest form
A ratio where the parts are integers with no common factors other than 1.
Additive method
The specific sequence of adding ratio parts together to determine the total number of shares.
Unitary method
A mathematical process where the value of a single unit (one share) is found first to calculate the value of multiple units.
Total parts
The sum of all the numbers in a ratio, representing the total number of shares an amount is split into.
Share value
The calculated amount that one single part of a ratio is worth.
Step 4: Check that the final amounts sum to the original total.
