2.3 Percentages
Finding a Percentage of a Number
To find a percentage of a number, convert the percentage to a decimal and multiply.
You must be wary with percentages since sometimes you have to find percentages of percentages.
To find a percentage of a number, convert the percentage to a decimal and multiply.
⚡ The Method:
To find $x\%$ of a number:
$$\text{Result} = \text{Number} \times \frac{x}{100}$$
$$\text{Result} = \text{Number} \times \frac{x}{100}$$
Example 1: Find 73% of 54
Step 1: Convert 73% to a decimal
$$73\% = \frac{73}{100} = 0.73$$
Step 2: Multiply
$$0.73 \times 54 = 39.42$$
$$73\% = \frac{73}{100} = 0.73$$
Step 2: Multiply
$$0.73 \times 54 = 39.42$$
Example 2: Find 15% of £240
Multi-Step Percentage Problems
Step 1: Convert 15% to a decimal
$$15\% = 0.15$$
Step 2: Multiply
$$0.15 \times 240 = £36$$
$$15\% = 0.15$$
Step 2: Multiply
$$0.15 \times 240 = £36$$
You must be wary with percentages since sometimes you have to find percentages of percentages.
Example: In a safari, 60% of lions are male. Of the males, 25% are cubs. What percentage of all lions are fully grown males?
Step 1: Identify what we need
• 60% of all lions are male
• 25% of males are cubs, so 75% of males are fully grown
Step 2: Calculate percentage of fully grown males
$$0.60 \times 0.75 = 0.45 = 45\%$$
Answer: 45% of all lions are fully grown males
• 60% of all lions are male
• 25% of males are cubs, so 75% of males are fully grown
Step 2: Calculate percentage of fully grown males
$$0.60 \times 0.75 = 0.45 = 45\%$$
Answer: 45% of all lions are fully grown males
💡 Shortcuts you might want to memorise:
• 50% = divide by 2
• 25% = divide by 4
• 10% = divide by 10 (move decimal one place left)
• 1% = divide by 100 (move decimal two places left)
• 5% = find 10% then halve it
• 15% = find 10% + 5%
• 20% = find 10% then double it
• 25% = divide by 4
• 10% = divide by 10 (move decimal one place left)
• 1% = divide by 100 (move decimal two places left)
• 5% = find 10% then halve it
• 15% = find 10% + 5%
• 20% = find 10% then double it
🧮 % of a Number
Find % of
Find % of
🎯 Percentage Practice
Real Life Uses:
• Shopping: Calculating discounts and sale prices
• Taxes: Calculating VAT or income tax
• Grades: Converting marks to percentages
• Shopping: Calculating discounts and sale prices
• Taxes: Calculating VAT or income tax
• Grades: Converting marks to percentages
What is Simple Interest?
Simple interest is calculated only on the original amount. The interest earned each year stays the same.
Simple interest is calculated only on the original amount. The interest earned each year stays the same.
⚡ Simple Interest Formula:
$$E = I \times r \times t$$
Where:
• $E$ = Interest earned
• $I$ = Initial amount
• $r$ = Interest rate (as a decimal)
• $t$ = Time
• $E$ = Interest earned
• $I$ = Initial amount
• $r$ = Interest rate (as a decimal)
• $t$ = Time
Example: You have £1000 in a bank account paying 2.5% simple interest per year.
a) How much interest do you earn each year?
Step 1: Convert percentage to decimal
$$2.5\% = 0.025$$
Step 2: Calculate yearly interest
$$E = £1000 \times 0.025 = £25 \text{ per year}$$
Step 1: Convert percentage to decimal
$$2.5\% = 0.025$$
Step 2: Calculate yearly interest
$$E = £1000 \times 0.025 = £25 \text{ per year}$$
Example (continued):
b) How much interest after 5 years?
$$E = £1000 \times 0.025 \times 5 = £125$$
c) Total amount after 5 years?
$$\text{Total} = £1000 + £125 = £1125$$
d) What is the percentage increase?
$$\frac{£125}{£1000} \times 100 = 12.5\%$$
(Or simply: $5 \times 2.5\% = 12.5\%$)
$$E = £1000 \times 0.025 \times 5 = £125$$
c) Total amount after 5 years?
$$\text{Total} = £1000 + £125 = £1125$$
d) What is the percentage increase?
$$\frac{£125}{£1000} \times 100 = 12.5\%$$
(Or simply: $5 \times 2.5\% = 12.5\%$)
🧮 Simple Interest Calculator
Initial (£):
Interest Rate (%):
Time (years):
Real Life Uses:
• Savings accounts: Some basic accounts use simple interest
• Loans: Some short-term loans use simple interest
• Bonds: Government bonds often pay simple interest
• Hire purchase: Some payment plans use simple interest
• Savings accounts: Some basic accounts use simple interest
• Loans: Some short-term loans use simple interest
• Bonds: Government bonds often pay simple interest
• Hire purchase: Some payment plans use simple interest
What is Compound Interest?
Compound interest is calculated on the original amount AND any accumulated interest. Your money grows faster because you earn "interest on interest".
Compound interest is calculated on the original amount AND any accumulated interest. Your money grows faster because you earn "interest on interest".
⚡ Compound Growth/Decay Formula:
$$N = N_0 \times (\text{multiplier})^n$$
Where:
• $N$ = Final amount
• $N_0$ = Initial amount
• Multiplier = $1 + \frac{\text{rate}}{100}$ for growth, or $1 - \frac{\text{rate}}{100}$ for decay
• $n$ = Number of years, days, etc.
• $N$ = Final amount
• $N_0$ = Initial amount
• Multiplier = $1 + \frac{\text{rate}}{100}$ for growth, or $1 - \frac{\text{rate}}{100}$ for decay
• $n$ = Number of years, days, etc.
💡 Finding the Multiplier:
Compound Growth Example
For Growth:
• 5% growth → multiplier = $1.05$
• 2.5% growth → multiplier = $1.025$
• 12% growth → multiplier = $1.12$
For Decay:
• 5% decay → multiplier = $0.95$
• 10% decay → multiplier = $0.90$
• 15% decay → multiplier = $0.85$
• 5% growth → multiplier = $1.05$
• 2.5% growth → multiplier = $1.025$
• 12% growth → multiplier = $1.12$
For Decay:
• 5% decay → multiplier = $0.95$
• 10% decay → multiplier = $0.90$
• 15% decay → multiplier = $0.85$
Example: £1000 at 2.5% compound interest per year
After 1 year:
$$N = £1000 \times (1.025)^1 = £1025$$
After 5 years:
$$N = £1000 \times (1.025)^5 = £1131.41$$
$$N = £1000 \times (1.025)^1 = £1025$$
After 5 years:
$$N = £1000 \times (1.025)^5 = £1131.41$$
📊 Comparing Simple vs Compound Interest (£1000 at 2.5%):
Compound Decay Example| Year | Simple | Compound | Difference |
|---|---|---|---|
| 1 | £1,025.00 | £1,025.00 | £0.00 |
| 2 | £1,050.00 | £1,050.63 | £0.63 |
| 5 | £1,125.00 | £1,131.41 | £6.41 |
| 10 | £1,250.00 | £1,280.08 | £30.08 |
Example: A car worth £15,000 loses 10% of its value each year
Other Applications
Using Simple Decay (for comparison):
After 5 years: $5 \times 10\% = 50\%$ lost
$$£15000 \times 0.5 = £7500$$
Using Compound Decay:
Multiplier = $1 - 0.10 = 0.90$
$$N = £15000 \times (0.9)^5 = £8857.35$$
With compound decay, you lose less over time because each year's loss is based on a smaller value!
After 5 years: $5 \times 10\% = 50\%$ lost
$$£15000 \times 0.5 = £7500$$
Using Compound Decay:
Multiplier = $1 - 0.10 = 0.90$
$$N = £15000 \times (0.9)^5 = £8857.35$$
With compound decay, you lose less over time because each year's loss is based on a smaller value!
Example: A bacteria population doubles every day. Starting with 5 bacteria, how many after one week?
Step 1: Identify the values
• $N_0 = 5$ (initial population)
• Multiplier = $2$ (doubles = 200% = ×2)
• $n = 7$ days
Step 2: Apply the formula
$$N = 5 \times 2^7 = 5 \times 128 = 640 \text{ bacteria}$$
• $N_0 = 5$ (initial population)
• Multiplier = $2$ (doubles = 200% = ×2)
• $n = 7$ days
Step 2: Apply the formula
$$N = 5 \times 2^7 = 5 \times 128 = 640 \text{ bacteria}$$
🧮 Compound Calculator
Initial Amount:
Rate (%):
Time Periods:
🎯 Find the Multiplier
🎯 Compound Interest Practice
Real Life Uses:
• Savings: Most savings accounts use compound interest
• Investments: Stocks, funds, and retirement accounts grow with compounding
• Mortgages: Home loans use compound interest
• Car depreciation: Vehicles lose value through compound decay
• Population growth: Humans, animals, bacteria populations
• Radioactive decay: Half-life calculations in physics
• Savings: Most savings accounts use compound interest
• Investments: Stocks, funds, and retirement accounts grow with compounding
• Mortgages: Home loans use compound interest
• Car depreciation: Vehicles lose value through compound decay
• Population growth: Humans, animals, bacteria populations
• Radioactive decay: Half-life calculations in physics