Percentages (Leaving Cert Mathematics): Revision Notes
Percentages
What are percentages?
Percentage is a way of expressing parts of a whole. The word per cent literally means "per hundred" or "out of every hundred". When we see the % symbol, we know it represents a percentage.
The concept of percentages is fundamental to many mathematical calculations. Understanding that "per cent" literally means "per hundred" helps you visualise what percentages represent in practical terms.
For example, 10% means 10 out of every 100, just like saying 10 parts out of 100 parts total.
Converting between percentages and fractions
Understanding how percentages and fractions relate to each other is essential for solving percentage problems effectively.
To convert a percentage to a fraction:
- Put the percentage number over 100
- Simplify the fraction if possible
For example:
- 20% =
- 60% =
To convert a fraction to a percentage:
- Multiply the fraction by 100 and add the % symbol
Converting Fractions to Percentages:
To convert to a percentage:
Step 1: Multiply the fraction by 100
Step 2: Simplify and add the % symbol
Converting between percentages and decimals
Percentages and decimals are closely connected, making conversions between them straightforward when you understand the pattern.
Converting decimals to percentages:
- Multiply by 100 and add the % sign
Examples:
- 0.23 = 23%
- 0.04 = 4%
- 0.54 = 54%
Converting percentages to decimals:
- Divide by 100 and remove the % sign
Examples:
- 38% = 0.38
- 8% = 0.08
- 3½% = 0.035
These conversion rules work because percentages are essentially decimals expressed "out of 100". This connection makes switching between the two formats logical and predictable.
Finding percentages of quantities
When you need to calculate a percentage of a given amount, convert the percentage to a decimal first, then multiply.
Method:
- Convert the percentage to a decimal (divide by 100)
- Multiply the quantity by this decimal
Finding 35% of €380:
Step 1: Convert the percentage to a decimal 35% = 35 ÷ 100 = 0.35
Step 2: Multiply the quantity by the decimal €380 × 0.35 = €133
Therefore, 35% of €380 is €133.
For percentage increases, remember that you need more than 100% of the original amount. To increase 450 by 5%, you need 105% of 450:
- 105% of 450 = 450 × 1.05 = 472.5
VAT calculations
VAT (Value Added Tax) calculations are common percentage problems you'll encounter in exams and real life.
Finding the original amount before VAT: When a price includes VAT, the total represents more than 100% of the original price.
VAT Calculation Problem:
If a bill for €57.60 includes VAT at 20%, find the amount before VAT was added.
Solution: Step 1: Identify what the total represents €57.60 represents 120% of the original bill (100% + 20% VAT)
Step 2: Find what 1% equals If 120% = €57.60, then 1% = €57.60 ÷ 120 = €0.48
Step 3: Calculate 100% Therefore, 100% = €0.48 × 100 = €48
The original bill before VAT was €48.
Profit and loss calculations
Profit and loss percentages are always calculated based on the cost price unless stated otherwise.
Always remember that profit and loss percentages use the cost price as the base for calculations. This is a common source of errors in examinations.
Key formulas:
- Percentage profit =
- Percentage loss =
Profit and Loss Problem:
A car dealer sells a car for €14,400 and loses 4% on the purchase price.
Part (i): Finding the cost price Step 1: Understand what the selling price represents €14,400 represents 96% of the cost price (100% - 4% loss)
Step 2: Find what 1% equals If 96% = €14,400, then 1% = €14,400 ÷ 96 = €150
Step 3: Calculate the cost price Therefore, 100% = €150 × 100 = €15,000 The dealer paid €15,000 for the car
Part (ii): Finding percentage profit at different selling price If the car had been sold for €17,250:
Step 1: Calculate the profit Profit = €17,250 - €15,000 = €2,250
Step 2: Apply the percentage profit formula Percentage profit =
Reverse percentage problems
These problems give you the result after a percentage change and ask you to find the original amount.
Method:
- Identify what percentage the given amount represents
- Find what 1% equals by dividing by that percentage
- Find 100% by multiplying by 100
This approach works for any reverse percentage problem, whether dealing with increases, decreases, VAT, or discounts. The key is correctly identifying what percentage your given amount represents.
Exam tips for percentages
Critical Exam Strategies:
- Always read the question carefully to identify whether you're finding a percentage OF something or finding the original amount
- For VAT problems, remember the final price includes the original price plus the VAT
- For profit and loss, use the cost price as your base unless told otherwise
- Convert percentages to decimals for calculations - it's usually easier
- Check your answers make sense - a 50% discount shouldn't give you a negative price!
Key Points to Remember:
- Percentage means "out of 100" - the % symbol always represents this concept
- Convert percentages to decimals by dividing by 100 for easier calculations
- VAT problems: the total price = original price + VAT, so represents more than 100%
- Profit and loss percentages are calculated using the cost price as the base
- Reverse percentage problems: work out what 1% equals, then find 100%