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Intro Compound Interest Suppose £100 is invested in a building society at a rate of 10% per annum. At the end of the year the interest paid would be £10 and if this is withdrawn, leaving the original £100 in the account for a further year at 10% then this type of interest is called simple interest. £33.10£30.00Total Interest gained £133.10£130Amount after 3years £121£120Amount after 2 years £110 Amount after 1 year Compound InterestSimple Interest The table below shows how £100 grows over a 3 year period if it is invested at a rate of 10% If the annual interest is left in the account at the end of each year then the interest is called compound interest.

Worked Example 1 £2000 is invested at 6% compound interest for 3 years. Find: (a) the amount in the account at the end of the period. and (b) the interest accrued. Amount after 1 year = 2000 + 6% of 2000 =2000 + 120 = £2120 Amount after 2 years = 2120 + 6% of 2120 =2120 + 127.20 = £2247.20 Amount after 3 years = 2247.20 + 6% of 2247.20 =2247.20 + 134.83 = £2382.03 Interest accrued = £2382.03 – £2000 = £382.03 Long Method Compound Interest Long Method

Question 1 £600 is invested at 5% compound interest for 3 years. Find: (a) the amount in the account at the end of the period. and (b) the interest accrued. Amount after 1 year = 600 + 5% of 600 =600 + 30 = £630 Amount after 2 years = 630 + 5% of 630 =630 + 31.5 = £661.50 Amount after 3 years = 661.50 + 5% of 661.50 =661.50 + 33.08 = £694.58 Interest accrued = £694.58 – £600 = £94.58 Long Method Compound Interest

Question 2 £5000 is invested at 8% compound interest for 4 years. Find: (a) the amount in the account at the end of the period (nearest £). and (b) the interest accrued (nearest £). Amount after 1 year = 5000 + 8% of 5000 =5000 + 400 = £5400 Amount after 2 years = 5400 + 8% of 5400 =5400 + 432 = £5832 Amount after 3 years = 5832 + 8% of 5832 =5832 + £466.56 = £6298.56 Interest accrued = £6802 – £5000 = £1802 Amount after 4 years = 6298.56 + 8% of 6298.56 =6298.56 + 503.88 = £6802 Long Method Compound Interest

Tricky Question £3000 is invested at 7% compound interest for 15 years. Find: (a) the amount in the account at the end of the period (nearest £). and (b) the interest accrued (nearest £). What problems would you have with the following question using this method? Long Method We can avoid 15 calculations by considering a different approach to the problem to devise a more efficient method. We will rework Example Question 1(a) using this improved method. Compound Interest

Multiplicative Remember that 6% means 6 100 = 0.06 At the end of each year the money grows to 106% of its value at the start of the year = 106 100 = 1.06 After 1 year the money has been multiplied by 1.06  2000 x 1.06 After 2 years the money is again multiplied by 1.06  (2000 x 1.06) x 1.06 After 3 years the money is again multiplied by 1.06  (2000 x 1.06 x 1.06) x 1.06 So after 3 years the money will have grown to £2000 x 1.06 3. Worked Example 1 £2000 is invested at 6% compound interest for 3 years. Find: (a) the amount in the account at the end of the period. Efficient Method (a) Money at end of 3 years = 2000 x 1.06 3 = £2382.03 Explanation of the Method If the term had been 7 years and the interest rate 8% then we would simply have calculated 2000 x 1.08 7. Compound Interest

Question 1 £600 is invested at 5% compound interest for 3 years. Find: (a) the amount in the account at the end of the period. and (b) the interest accrued. Amount after 1 year = 600 + 5% of 600 =600 + 30 = £630 Amount after 2 years = 630 + 5% of 630 =630 + 31.5 = £661.50 Amount after 3 years = 661.50 + 5% of 661.50 =661.50 + 33.08 = £694.58 Interest accrued = £694.58 – £600 = £94.58 Long Method Use this efficient method to confirm the answer to Question 1 (a) below Amount after 3 years = 600 x 1.05 3 = £694.58 Compound Interest

Question 2 £5000 is invested at 8% compound interest for 4 years. Find: (a) the amount in the account at the end of the period (nearest £) and (b) the interest accrued (nearest £) Amount after 1 year = 5000 + 8% of 5000 =5000 + 400 = £5400 Amount after 2 years = 5400 + 8% of 5400 =5400 + 432 = £5832 Amount after 3 years = 5832 + 8% of 5832 =5832 + £466.56 = £6298.56 Interest accrued = £6802 – £5000 = £1802 Amount after 4 years = 6298.56 + 8% of 6298.56 = 6298.56 + 503.88 = £6802 Long Method Use this efficient method to confirm the answer to Question 2 (a) below Amount after 4 years = 5000 x 1.08 4 = £6802 Compound Interest

Question 3 £8000 is invested at 7% compound interest for 6 years. Find: (a) the amount in the account at the end of the period (nearest £) and (b) the interest accrued (nearest £) Question 4 £1250 is invested at 9% compound interest for 10 years. Find: (a) the amount in the account at the end of the period (nearest £) and (b) the interest accrued (nearest £) (a) 8000 x 1.07 6 = £12,006 (b) 12,006 – 8000 = £4,006 (a) 1250 x 1.09 10 = £2959 (b) 2959 – 1250 = £1709 Compound Interest

Question 5 £3750 is invested at 4.5% compound interest for 8 years. Find the amount in the account at the end of the period (nearest £100) Question 6 £7500 is invested at 8.5% compound interest for 15 years. Find the amount in the account at the end of the period (nearest £100) Answer: 3750 x 1.045 8 = £5300 Answer: 7500 x 1.085 15 = £25 500 Compound Interest

Worksheets Q1. £600 is invested at 5% compound interest for 3 years. Find: (a) the amount in the account at the end of the period. and (b) the interest accrued. Q2. £5000 is invested at 8% compound interest for 4 years. Find: (a) the amount in the account at the end of the period (nearest £) and (b) the interest accrued (nearest £) Worked Example 1: £2000 is invested at 6% compound interest for 3 years. Find: (a) the amount in the account at the end of the period and (b) the interest accrued. Worksheet 1

Q3. £8000 is invested at 7% compound interest for 6 years. Find: (a) the amount in the account at the end of the period (nearest £) and (b) the interest accrued (nearest £) Q4. £1250 is invested at 9% compound interest for 10 years. Find: (a) the amount in the account at the end of the period (nearest £) and (b) the interest accrued (nearest £) Q5.£3750 is invested at 4.5% compound interest for 8 years. Find the amount in the account at the end of the period (nearest £100) Q6. £7500 is invested at 8.5% compound interest for 15 years. Find the amount in the account at the end of the period (nearest £100) Worksheet 2