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Higher Applications of Mathematics

Regular and irregular payments

Payment patterns, deposits, withdrawals and cash flow.

Formula referenceFinance mixed quiz

Formula reference

Check the key Finance formulae and when to use each one.

Mixed revision quiz

Review the whole Finance section with adaptive feedback.

Before you start

  • Apply compound interest for one period at a time.
  • Read whether payments happen at the start or end of a period.
  • Use a table or timeline for repeated cash flows.
  • Keep balances to two decimal places only when money is being reported.

Method chooser

Which method do I use?

Are you finding interest on the original amount only?

Use simple interest.

Is the amount increasing by a percentage repeatedly?

Use compound interest.

Are you finding what future money is worth today?

Use present value.

Are you finding what today's money will be worth later?

Use future value.

Are there regular payments, deposits or withdrawals?

Use a payment timeline.

Are you tracking a loan balance after repayments?

Use a repayment schedule.

Are you comparing insurance policies or risk?

Use expected value and excess comparisons.

Are you working with gross pay, deductions and take-home pay?

Use net pay.

Are you comparing prices over time with inflation?

Use inflation / real value.

Finance lesson

Key idea

  • Regular payments are equal amounts paid at equal intervals. Irregular payments can change in size or timing. Higher Applications questions often involve building a timeline rather than using one single formula.
  • Each deposit earns interest only for the time it remains in the account. Withdrawals reduce the balance from the date they are taken.
  1. List the starting balance, deposits and withdrawals in time order.
  2. For each period, apply interest to the balance that is present during that period.
  3. Add deposits and subtract withdrawals at the stated time.
  4. Repeat until the required date.
  5. Check whether the answer is just before or just after a payment.

Key formulae

  • Future value of a payment: value at end = payment x (1 + r)number of periods left
  • Balance update: new balance = old balance x (1 + period rate) + deposit − withdrawal
  • For repeated payments, a table or timeline is usually the safest method.

Worked examples

Worked example 1

Regular annual deposits

Kara deposits £600 at the end of each year into an account paying 4% per year. Find the value just after the third deposit.

  1. The first deposit earns interest for 2 years: 600(1.04)² = 648.96.
  2. The second deposit earns interest for 1 year: 600(1.04) = 624.00.
  3. The third deposit is just paid, so it is £600.00.
  4. Add the values: 648.96 + 624.00 + 600.00.

So: The value just after the third deposit is £1,872.96.

Worked example 2

Irregular payments and a withdrawal

An account starts with £1,200. It grows by 3% each year. At the end of year 1, £500 is added. At the end of year 2, £300 is withdrawn. Find the balance after year 3 interest.

  1. After year 1 interest: 1200 × 1.03 = 1236. Then add 500: 1736
  2. After year 2 interest: 1736 × 1.03 = 1788.08. Then withdraw 300: 1488.08
  3. After year 3 interest: 1488.08 × 1.03 = 1532.7224

Final step: The balance is £1,532.72.

Worked example 3

Monthly deposits with monthly interest

A saver pays £120 at the end of each month for 4 months. The monthly interest rate is 0.4%. Find the balance just after month 4's deposit.

  1. Month 1 deposit grows for 3 months: 120(1.004)³ = 121.44
  2. Month 2 deposit grows for 2 months: 120(1.004)² = 120.96
  3. Month 3 deposit grows for 1 month: 120(1.004) = 120.48
  4. Month 4 deposit is just paid: 120. Total = 482.88

So: The balance is £482.88.

Worked example 4

Changing deposits

A pupil saves £300 now, £450 after one year, and £200 after two years. Interest is 3.5% per year. Find the value after three years.

  1. £300 grows for 3 years: 300(1.035)³ = 332.62
  2. £450 grows for 2 years: 450(1.035)² = 482.05
  3. £200 grows for 1 year: 200(1.035) = 207.00
  4. Add the accumulated values.

Final step: Total value = £1,021.67

Watch out

  • Assuming every deposit earns interest for the same length of time.
  • Adding a deposit before interest when the question says it is paid at the end of the period.
  • Ignoring withdrawals on the timeline.
  • Using an annuity formula when payments are irregular.

Spreadsheet connection

Spreadsheet connection

Create a savings plan table showing deposits, withdrawals, interest and final balance.

Open spreadsheet skill

Next step

Move into practice

Use the method notes to choose the correct financial model, then try varied rates, time periods, tables and decision contexts.

Finance mixed quiz