Dividing a Quantity in a Ratio
This topic covers how to split a total quantity into proportional shares based on a given ratio, and how to determine the ratio when the shares are known. It's a foundational skill for problems involving mixtures, scaling, and distribution.
Part of the ESAT Mathematics 1 syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.
Key points
- To divide a quantity in a ratio (e.g., a:b:c), first sum the ratio's components (a+b+c) to find the total number of parts.
- Calculate the value of a single part by dividing the total quantity by the total number of parts.
- Find the size of each share by multiplying the single part's value by its corresponding number in the ratio.
- When expressing a division as a ratio, all quantities must be converted to the same units before comparison.
- Ratios should always be simplified to their simplest form by dividing all parts by their highest common factor.
Formulae
Value of one part = Total Quantity / (Sum of ratio parts) Use this as the first step when dividing a total quantity according to a known ratio.
Share Size = (Relevant ratio part) × (Value of one part) Use this to calculate the actual value of each share after finding the value of a single part.
Definitions
- Ratio
- A comparison of the relative sizes of two or more quantities, written with a colon, e.g., 3:2.
- Part
- One of the numbers in a ratio, representing a proportional share of the whole.
Worked example
An industrial solvent is a mixture of water, acetone, and ethanol in the ratio 9:4:7 by volume. If you have 3.6 litres of the solvent, what volume of acetone does it contain, in millilitres?
- 1
First, convert the total volume to the target unit (millilitres) to avoid confusion later:
3.6 L = 3600 mL - 2
Calculate the total number of parts in the ratio:
9 + 4 + 7 = 20 parts - 3
Determine the volume of a single part:
3600 mL / 20 parts = 180 mL per part.
- 4
Acetone corresponds to 4 parts in the ratio.
Calculate its volume:
4 parts × 180 mL/part = 720 mL.
Answer: 720 mL
Common mistakes
- ×Basic arithmetic errors during division or multiplication are the most frequent mistake. Double-check your calculations, as there is no calculator safety net.
- ×Forgetting to convert all quantities to the same units before forming a ratio (e.g., mixing litres and millilitres, or metres and centimetres).
- ×Mixing up which value corresponds to which part of the ratio after calculating the share sizes.
No-calculator tips
- ✓When dividing, simplify the fraction by cancelling zeros or common factors first. For example, 3600 / 20 is the same as 360 / 2.
- ✓Use distributive multiplication for tricky numbers. For instance, to calculate 4 × 180, think of it as (4 × 100) + (4 × 80) = 400 + 320 = 720.
- ✓When checking your answer, quickly estimate if the proportions make sense. In the example, acetone (4 parts) should be less than half the volume of water (9 parts), and 720 mL is indeed less than half of 1620 mL (9 × 180).