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Percentages – Repeated (Compound Changes)
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Foundational Skills
Understand simple vs compound interest
\[ \text{Simple: £200, Compound: £?} \]
Compare simple and compound interest for the same scenario.
Form the compound multiplier for growth
\[ 5\% \text{ for 3 yrs} \rightarrow 1.05^3 \]
Determine the compound multiplier for repeated percentage growth.
Form the compound multiplier for decay
\[ 15\% \text{ for 4 yrs} \rightarrow 0.85^4 \]
Determine the compound multiplier for repeated percentage decay.
Compound Growth
Compound growth over 2 years
\[ \text{£}500 \times 1.08^2 = \text{?} \]
Calculate the value after 2 years of compound growth.
Compound growth over 3+ years
\[ \text{£}800 \times 1.05^4 = \text{?} \]
Calculate the value after 3 or more years of compound growth.
Compound interest earned
\[ \text{Interest} = \text{Final} – \text{Principal} \]
Calculate the total compound interest earned (not the final value).
Value growth
\[ \text{Painting: £4000} \times 1.06^4 \]
Apply compound growth to asset appreciation.
Compound Decay
Compound decay over 2 years
\[ \text{£}12000 \times 0.85^2 = \text{?} \]
Calculate the value after 2 years of compound decay.
Compound decay over 3+ years
\[ \text{£}8000 \times 0.80^4 = \text{?} \]
Calculate the value after 3 or more years of compound decay.
Depreciation in context – vehicles
\[ \text{Car: £18000} \times 0.82^3 \]
Apply depreciation to vehicle values.
Depreciation in context – equipment
\[ \text{Machine: £25000} \times 0.88^5 \]
Apply depreciation to business equipment.
Radioactive decay / population decline
\[ 800\text{g} \times 0.95^6 = \text{?} \]
Apply compound decay to scientific contexts.
Overall Percentage Change
Overall % change from compound growth
\[ 8\% \text{ for 3 yrs} \neq 24\% \]
Find the total percentage change from repeated compound growth.
Overall % change from compound decay
\[ 15\% \text{ for 4 yrs} \neq 60\% \]
Find the total percentage change from repeated compound decay.
Calculate loss in value from depreciation
\[ \text{Loss} = \text{Initial} – \text{Final} \]
Find the total amount lost through depreciation.
Finding Unknowns
Find original value before compound growth
\[ \text{Original} = \text{Final} \div 1.06^3 \]
Work backwards to find the original value before compound growth.
Find original value before compound decay
\[ \text{Original} = \text{Final} \div 0.85^4 \]
Work backwards to find the original value before compound decay.
Find number of years by trial
\[ 1.1^n > 1.6 \rightarrow n = \text{?} \]
Find the number of time periods for a value to reach a target.
Find number of years for value to halve
\[ 0.8^n < 0.5 \rightarrow n = \text{?} \]
Find the number of years for depreciation to halve a value.
Find the percentage rate given values and time
\[ r = \sqrt[3]{\frac{2662}{2000}} – 1 \]
Work backwards to find the annual percentage rate.
Special Cases
Compare simple and compound interest
\[ \text{Compound} – \text{Simple} = \text{?} \]
Calculate the difference between compound and simple interest.
Compound changes with different rates
\[ 1.10 \times 1.15 = \text{?} \]
Find overall percentage change when rates differ each period.
Growth then decay
\[ 1.20 \times 0.90 = \text{?} \]
Find overall change when growth is followed by decay.
Find annual rate to achieve target growth
\[ \text{Double in 6 yrs} \rightarrow r = \text{?} \]
Find the rate required to achieve a specific growth target.
Timer (Optional)
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