Ratios (Edexcel GCSE Maths): Revision Notes
Ratios
Introduction to ratios
Ratios are a way of comparing quantities and showing the relationship between different amounts. While they can seem tricky at first, understanding the key techniques will make working with ratios much clearer. A ratio shows how much of one thing there is compared to another, and we can manipulate ratios in several useful ways.
Ratios are fundamental to many areas of mathematics and everyday life, from cooking recipes to map scales. Mastering ratio techniques will help you solve a wide variety of problems.
Reducing ratios to their simplest form
When you have a ratio, you often need to simplify it to make it easier to work with. The simplest form of a ratio is when you can't divide both numbers by anything else - similar to how you simplify fractions.
To reduce a ratio to its simplest form, you need to find a common factor that divides into all parts of the ratio, then divide each part by that same number.
Worked Example: Simplifying 15:18
Step 1: Find the common factor Both 15 and 18 can be divided by 3
Step 2: Divide both sides by the common factor 15 ÷ 3 = 5 18 ÷ 3 = 6
Therefore, 15:18 = 5:6 in simplest form
Calculator tip: If you're allowed to use a calculator, you can enter ratios as fractions using the fraction button. For instance, entering 8/12 as a fraction will automatically reduce to 2/3, which you can then write as the ratio 2:3.
Handling more challenging cases
Working with decimals or fractions
When your ratio contains decimals or fractions, you need to eliminate these before simplifying. This is a crucial step that many students forget.
For decimals: Multiply both sides by 10 (or 100, 1000, etc.) to remove the decimal points, then simplify the resulting whole numbers.
For fractions: Put the fractions over a common denominator, then multiply both sides by that denominator to eliminate the fractions completely.
Always convert decimals and fractions to whole numbers first, then simplify. Never try to simplify ratios while they still contain decimals or fractions.
Converting mixed units
When a ratio has different units (like millimetres and centimetres), you must convert everything to the same unit before simplifying. Always convert to the smaller unit to avoid decimals.
Worked Example: Converting 24 mm : 7.2 cm
Step 1: Convert to the same unit (millimetres) 7.2 cm = 7.2 × 10 = 72 mm
Step 2: Write the ratio with the same units 24 mm : 72 mm
Step 3: Simplify by dividing by the common factor (24) 24 ÷ 24 = 1 72 ÷ 24 = 3
Therefore, 24 mm : 7.2 cm = 1:3
Getting ratios into 1 form
Sometimes the most useful way to express a ratio is in the form 1
, where one side equals 1. This makes it very clear what the proportional relationship is.To convert to 1
form, simply divide both sides of the ratio by the first number.Worked Example: Converting 3:56 to 1
formStep 1: Divide both sides by the first number (3) 3 ÷ 3 = 1 56 ÷ 3 = 18⅔
Therefore, 3:56 = 1:18⅔
This shows that for every 1 of the first quantity, you have 18⅔ of the second quantity.
Scaling up ratios
When you know the ratio between different parts and you know the actual amount of one part, you can find the amounts of the other parts by scaling up the ratio. The process involves multiplying both sides of the ratio by the same number to match the known quantity.
Worked Example: Mortar mixture
If mortar is made from sand and cement in the ratio 7:2, and you have 21 buckets of sand, how much cement do you need?
Step 1: Find the scaling factor 7 × ? = 21 ? = 21 ÷ 7 = 3
Step 2: Apply the same scaling factor to cement 2 × 3 = 6 buckets of cement
Therefore, you need 6 buckets of cement.
You can also work backwards from total quantities. If you know the total number of items and the ratio they're in, you can find the individual amounts.
The key principle is that both sides of the ratio must be multiplied by the same number to maintain the proportional relationship.
Proportional division
This is one of the most important applications of ratios. When you need to split a total amount according to a given ratio, you follow a three-step process:
Worked Example: Dividing £9100 in the ratio 2:4:7
Step 1: Add up the parts 2 + 4 + 7 = 13 parts total
Step 2: Divide to find one "part" £9100 ÷ 13 = £700 per part
Step 3: Multiply to find the amounts
- First person: 2 × £700 = £1400
- Second person: 4 × £700 = £2800
- Third person: 7 × £700 = £4900
Check: £1400 + £2800 + £4900 = £9100 ✓
This three-step method works for any proportional division problem, whether you're splitting money, resources, or any other quantity according to a given ratio. Always check your answer by adding up the parts.
Key Points to Remember:
- Always reduce ratios to their simplest form by dividing by common factors
- Convert mixed units to the same unit (preferably the smaller one) before simplifying
- For decimals and fractions, eliminate them first by multiplying, then simplify
- The 1 form is often the clearest way to show proportional relationships
- For proportional division, remember the three steps: add the parts, divide to find one part, multiply to find individual amounts
- Ratios show relationships between quantities and can be scaled up or down while maintaining the same proportional relationship