Percentage Problems

Spec reference: Ratio, proportion and rates of change
Key idea: Solve real-life percentage problems including interest, profit, loss and VAT.

---

---

Simple interest

$$\text{Interest}=\frac{P\times R\times T}{100}$$

where $P$ = principal, $R$ = rate (%), $T$ = time (years).

Example

£2,000 is invested at 3% simple interest per year. Find the total after 4 years.

Interest per year $=2000\times 0.03=\pounds 60$

Total interest $=60\times 4=\pounds 240$

Total $=\pounds 2000+\pounds 240=\boxed{{\displaystyle \pounds 2,240}}$

---

Compound interest

$$A=P{(1+\frac{r}{100})}^n$$

where $A$ = final amount, $P$ = principal, $r$ = rate (%), $n$ = number of years.

Example

£3,000 is invested at 4% compound interest per year. Find the value after 3 years.

$$A=3000\times {1.04}^3=3000\times 1.124864=\boxed{{\displaystyle \pounds 3,374.59}}$$

---

Profit and loss

$$\text{Profit \%}=\frac{\text{Profit}}{\text{Cost price}}\times 100$$$$\text{Loss \%}=\frac{\text{Loss}}{\text{Cost price}}\times 100$$

Example

A phone bought for £400 is sold for £460. Find the percentage profit.

Profit $=\pounds 60$

$$\text{Profit \%}=\frac{60}{400}\times 100=\boxed{{\displaystyle 15\mathrm{\%}}}$$

---

VAT

VAT is currently 20% in the UK. To find the price including VAT, multiply by 1.20.

To find the price excluding VAT, divide by 1.20.

Example

A laptop costs £840 including 20% VAT. Find the price before VAT.

$$\text{Price before VAT}=\frac{840}{1.20}=\boxed{{\displaystyle \pounds 700}}$$

---

Exam tips

Watch out for

• Simple interest: calculate on the original amount each year
• Compound interest: each year's interest is added before the next year's interest is calculated
• Always use the cost price (not selling price) as the denominator in profit/loss %

---

Test Yourself

Question 1 of 5

£2000 at 5% simple interest, 3 years. Total interest?