Household Finance

• Step 1: Calculate the actual profit.

  • $\text{Profit} = SP - CP$
  • $\text{Profit} = 58.50 - 45.00 = £13.50$

• Step 2: Substitute into the percentage profit formula.

  • $\text{Percentage Profit} = \frac{13.50}{45} \times 100$

• Final Answer: $30\%$

Worked Example: Reverse Percentage (Finding Cost Price)

Question: A sofa is sold for £630.00 at a 10% loss. Calculate the cost price.

• Step 1: Find the multiplier for a 10% loss.

  • $100\% - 10\% = 90\%$
  • $\text{Multiplier} = 0.90$

• Step 2: Set up the formula and rearrange.

  • $\text{Selling Price} = \text{Cost Price} \times \text{Multiplier}$
  • $\text{Cost Price} = 630.00 \div 0.90$

• Final Answer: £700.00

Value Added Tax (VAT)

You might not realise it, but almost everything you buy has a hidden government tax attached to it. VAT (Value Added Tax) is added to goods and services in the UK, usually at a standard rate of 20% or a reduced rate of 5% for items like domestic energy.

The price before VAT is added is called the Net Price, and the total price including VAT is the Gross Price. To add 20% VAT, you multiply the net price by $1.2$. To remove it, you divide the gross price by $1.2$.

$$\text{Gross Price} = \text{Net Price} \times \left(1 + \frac{\text{VAT rate}}{100}\right)$$

Worked Example: Removing VAT

Question: A TV costs £480.00 including 20% VAT. Calculate the net price.

• Step 1: Identify the multiplier for 20% VAT.

  • $\text{Multiplier} = 1.20$

• Step 2: Substitute into the rearranged formula.

  • $\text{Net Price} = \text{Gross Price} \div \text{Multiplier}$
  • $\text{Net Price} = 480.00 \div 1.20$

• Final Answer: £400.00

Worked Example: Finding the VAT Amount Only (Non-Calculator)

Question: A receipt totals £126.00 including 20% VAT. Calculate the amount of VAT paid without a calculator.

• Step 1: Use the "one-sixth" rule (since $20/120 = 1/6$, the VAT is exactly one-sixth of the gross price).

  • $\text{VAT Amount} = \frac{1}{6} \times \text{Gross Price}$

• Step 2: Substitute and calculate.

  • $\text{VAT Amount} = 126.00 \div 6$

• Final Answer: £21.00

Simple and Compound Interest

If you leave your money in a bank, it grows, but the way it grows depends entirely on the type of interest applied. The original sum of money invested is the Principal ($P$), and the percentage charged or earned is the Interest Rate ($R$).

Simple Interest means you earn a fixed amount based only on the original principal. Compound Interest means interest is added to the new total each year, so you earn "interest on your interest." The same principles apply to decreasing values: Simple Depreciation removes a fixed percentage of the original value, while compound Depreciation uses a continuous Multiplier less than 1.

Simple Interest Formula:

$$I = \frac{P \times R \times T}{100}$$

Worked Example: Simple Interest

Question: Invest £250.00 at 4% simple interest for 6 years. Calculate the final balance.

• Step 1: Calculate the interest for one year.

  • $4\% \text{ of } 250.00 = £10.00$

• Step 2: Calculate total interest for 6 years.

  • $\text{Total Interest} = 10.00 \times 6 = £60.00$

• Step 3: Add interest to the principal to find the final balance.

  • $\text{Final Balance} = 250.00 + 60.00$

• Final Answer: £310.00

Compound Interest Formula (Not provided in exams):

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

Worked Example: Compound Interest Balance

Question: Invest £1,200.00 at 4% compound interest per year for 7 years. Calculate the final balance.

• Step 1: Identify the principal ($P$), the multiplier for a 4% increase, and the number of years ($n$).

  • $P = 1200.00$
  • $\text{Multiplier} = 1 + \frac{4}{100} = 1.04$
  • $n = 7$

• Step 2: Substitute into the compound interest formula.

  • $A = 1200.00 \times 1.04^7$

• Step 3: Calculate the final amount and round to 2 decimal places.

  • $A = 1579.118...$

• Final Answer: £1,579.12

Worked Example: Finding the Exact Compound Interest Rate

Question: Sasha invests £4,000.00. After 3 years, the account balance is £4,060.30. Calculate the annual compound interest rate, $r$.

• Step 1: Rearrange the formula to find the multiplier ($m$).

  • $m = \sqrt[n]{\frac{\text{Final Amount}}{\text{Principal}}}$

• Step 2: Substitute the values.

  • $m = \sqrt[3]{\frac{4060.30}{4000.00}}$

• Step 3: Calculate the multiplier.

  • $m = \sqrt[3]{1.015075} = 1.005$

• Step 4: Convert the multiplier back to a percentage rate.

  • $r = (1.005 - 1) \times 100$

• Final Answer: $0.50\%$

Income Tax: Net and Gross Income

Understanding how a salary is taxed explains why you never take home exactly what your employment contract states. Your Gross Income is your total earnings before any deductions, while your Net Income is your "take-home pay" after taxes.

In the UK, you do not pay tax on your entire salary. You are given a tax-free Personal Allowance. The income above this allowance is your Taxable Income, which is the only part subject to Income Tax. Deductions may also include National Insurance (NI).

$$\text{Taxable Income} = \text{Gross Income} - \text{Personal Allowance}$$

Worked Example: Marginal Tax Calculation

Question: An employee earns a gross income of £32,000.00. Their personal allowance is £12,570.00. They pay 20% tax on the rest of their earnings. Calculate their net income.

• Step 1: Calculate the Taxable Income.

  • $\text{Taxable Income} = 32{,}000.00 - 12{,}570.00 = £19{,}430.00$

• Step 2: Calculate the 20% Income Tax.

  • $\text{Tax} = 0.20 \times 19{,}430.00 = £3{,}886.00$

• Step 3: Subtract the tax from the Gross Income to find the Net Income.

  • $\text{Net Income} = 32{,}000.00 - 3{,}886.00$

• Final Answer: £28,114.00

Bank Statements and Balances

Think of a bank account like a vertical number line; money coming in moves you up, and money going out moves you down. A bank Statement tracks your running Balance, adding Credits (money in, like wages) and subtracting Debits (money out, like bills).

If your balance drops below zero, the account is Overdrawn due to an Overdraft agreement. On statements, this is shown with a minus sign (e.g., -£50.00) or the letters "OD". Adding a credit to a negative balance moves it closer to zero or back into positive numbers.

$$\text{New Balance} = \text{Previous Balance} + \text{Credits} - \text{Debits}$$

Worked Example: Moving from Overdrawn to Positive

Question: A bank account is overdrawn by £113.56. Wages of £1,460.55 are paid into the account. Calculate the new balance.

• Step 1: Set up the calculation using the previous negative balance.

  • $\text{New Balance} = \text{Previous Balance} + \text{Credits}$

• Step 2: Substitute the values.

  • $\text{New Balance} = -113.56 + 1460.55$

• Step 3: Calculate the final sum.

  • $\text{New Balance} = 1346.99$

• Final Answer: £1,346.99