Interest - Varying Time Units

Video lesson3 worked examples

Theory

Now that we are comfortable using multipliers for a single percentage change, we can upgrade our approach to handle multiple time periods (compound interest) using a simple formula.

1. The Compound Interest Formula

To calculate the final balance of an investment or a loan over multiple time periods, we use the formula:

\text{New Balance} = \text{Original} \times (\text{multiplier})^n

$n$ represents the number of time periods.
Because the multiplier is raised to a power, the numbers will grow (or decay) exponentially.

2. The Golden Rule of Time Units

In exam questions, the interest rate will be given for a specific time period (e.g., per month, per year, per quarter).

Crucial Rule:

You must always change your time period ( $n$ ) to perfectly match the time period of the interest rate. You cannot change the multiplier to match the time!

If the interest rate is per month, but the question asks about 2 years, you must use $n = 24$ .

3. Common Time Periods

You must be confident converting between the following periods:

Per annum (p.a.) / Per year: 1 time period per year.
Per half-year: 2 time periods per year (every 6 months).
Per quarter: 4 time periods per year (every 3 months).
Per month: 12 time periods per year.

Worked examples

Example 1

Example 1: Years to Months Conversion

Aidan deposits £4,500 into a new savings account. The account offers an effective rate of interest of 0.35% per month.

Calculate the exact balance of Aidan's account after exactly 2 years.

Find the multiplier: $100\% + 0.35\% = 100.35\%$ . The multiplier is 1.0035.
Match the time period: The interest is applied monthly, so we must convert 2 years into months. $n = 2 \times 12 = 24$ months.
Apply the formula:

$\text{New Balance} = \text{\pounds}4500 \times 1.0035^{24}$

$\text{New Balance} = \text{\pounds}4500 \times 1.08745... = \text{\pounds}4,893.53$ .

Example 2

Example 2: Working with Quarters

A small business takes out a commercial loan of £18,000 to purchase new equipment. The lender charges an effective rate of interest of 2.4% per quarter.

If the business makes no repayments, calculate the total amount of interest added to the loan after 3 years.

Find the multiplier: $100\% + 2.4\% = 102.4\%$ . The multiplier is 1.024.
Match the time period: There are 4 quarters in a year. $n = 3 \text{ years} \times 4 = 12$ quarters.
Apply the formula:

$\text{New Balance} = \text{\pounds}18000 \times 1.024^{12}$

$\text{New Balance} = \text{\pounds}18000 \times 1.32922... = \text{\pounds}23,926.11$ .

Calculate the interest only:

$\text{Interest} = \text{Final Balance} - \text{Original Amount}$

$\text{Interest} = \text{\pounds}23,926.11 - \text{\pounds}18,000 = \text{\pounds}5,926.11$ .

Example 3

Example 3: Half-Years and Odd Months

Clara invests £6,250 in a long-term bond that offers an effective rate of interest of 1.85% per half-year.

Calculate the balance of Clara's investment after exactly 4 years and 6 months.

Find the multiplier: $100\% + 1.85\% = 101.85\%$ . The multiplier is 1.0185.
Match the time period: A half-year is 6 months. 4 years and 6 months is exactly 4.5 years. Since there are 2 half-years in a single year: $n = 4.5 \times 2 = 9$ half-years.
Apply the formula:

$\text{New Balance} = \text{\pounds}6250 \times 1.0185^9$

$\text{New Balance} = \text{\pounds}6250 \times 1.17935... = \text{\pounds}7,370.97$ .

Ready to practise?

5 Interest with Varying Time Units questions with answers and video solutions