Percentage Change, Compound Interest, Depreciation & Reverse Percentages
Fast, accurate percentage calculations in GCSE Maths rely on converting percentage increases and decreases into single decimal multipliers: adding r% multiplies by (1 + r/100), and subtracting r% multiplies by (1 - r/100). Repeated percentage changes, such as bank compound interest, population growth, or car value depreciation, are modelled using the exponential formula Total = Initial x (Multiplier)ⁿ. Examiners regularly test reverse percentages (finding the original value before a percentage change was applied by dividing by the multiplier) and multi-year compound growth problems.
- Decimal Multipliers
- To increase a value by r%, multiply by (1 + r/100). To decrease a value by r%, multiply by (1 - r/100). For example, increasing £240 by 17.5% is 240 x 1.175 = £282; decreasing £240 by 8% is 240 x 0.92 = £220.80.
- Percentage Change (Profit / Loss / Error)
- Percentage Change = ((Final Value: Original Value) / Original Value) x 100. For percentage profit or loss, divide the profit or loss by the original cost price and multiply by 100.
- Compound Interest & Exponential Growth
- When a quantity N₀ increases by r% per time period for n time periods: Final Amount N = N₀ x (1 + r/100)ⁿ. To find the interest earned alone, remember to subtract the initial principal N₀ from the final amount N at the end.
Quadratic Parabola Plotter, Roots, Turning Point & Quadratic Formula
Adjust coefficients a, b, and c in y = ax² + bx + c to see how the parabola, roots (x-intercepts), and turning point move on the graph.
y = x² - 2x - 3
(-2)² − 4(1)(-3) = 16 → 2 real roots (crosses x-axis twice)x = 1.00b in brackets when squaring: (-2)² = 4 (always positive!). Typing -2² into a calculator without brackets gives -4 and loses the accuracy mark.Key Concepts & Exam Definitions (6)
- Decimal Multipliers
- To increase a value by r%, multiply by (1 + r/100). To decrease a value by r%, multiply by (1 - r/100). For example, increasing £240 by 17.5% is 240 x 1.175 = £282; decreasing £240 by 8% is 240 x 0.92 = £220.80.
- Percentage Change (Profit / Loss / Error)
- Percentage Change = ((Final Value: Original Value) / Original Value) x 100. For percentage profit or loss, divide the profit or loss by the original cost price and multiply by 100.
- Compound Interest & Exponential Growth
- When a quantity N₀ increases by r% per time period for n time periods: Final Amount N = N₀ x (1 + r/100)ⁿ. To find the interest earned alone, remember to subtract the initial principal N₀ from the final amount N at the end.
- Compound Depreciation & Exponential Decay
- When an asset (like a car or machinery) or a radioactive/biological population N₀ decreases by r% per time period for n time periods: Final Value N = N₀ x (1 - r/100)ⁿ.
- Reverse Percentages (Finding the Original Amount)
- When given the final amount after an r% increase or decrease: set up the equation Original x Multiplier = Final Amount, and solve by dividing: Original = Final Amount ÷ Multiplier. For example, if a coat costs £68 in a '15% off' sale, the multiplier is 0.85, so Original Price = £68 ÷ 0.85 = £80.
- Finding an Unknown Percentage Rate r or Time n (Higher)
- 1. If given N₀, N, and n, and asked for the annual percentage rate r%: substitute into N₀ x (m)ⁿ = N, divide by N₀ to find mⁿ = N / N₀, take the n-th root m = ⁿ√(N / N₀), and convert the multiplier m into a percentage change. 2. If asked for the number of years n needed to exceed a target value: substitute trial integer values of n = 1, 2, 3, 4. systematically and show both the year just below and the year just above the target.
Quick-Fire Flashcards (6)
Click any card to check the answer, then mark it as Known to track your progress.
Write down the single decimal multiplier for: (a) a 4% increase, (b) a 12.5% decrease, and (c) a 0.6% increase.
A house was bought for £250,000 and sold 3 years later for £295,000. Calculate the percentage profit.
State the compound interest formula for investing £P at r% compound interest per year for n years.
After a 12% pay rise, Sarah's new annual salary is £31,360. Calculate her salary before the pay rise.
A price is increased by 20% and then decreased by 20%. Is the final price equal to the starting price?
A car bought for £20,000 depreciates to £12,800 after 2 years at a constant annual percentage rate r%. How do you find r?
Exam-Style Questions & Mark Schemes (4)
Write or think through your answer first, then click Check Mark Scheme to see the exact points examiners award marks for.
- Q1
Liam invests £6,000 in a savings account for 4 years. The account pays compound interest at a rate of 3.5% per year for the first 2 years, and 4.2% per year for the final 2 years. Calculate the total amount of interest Liam has earned at the end of the 4 years, to the nearest penny. [4 marks]
Click “Check Mark Scheme” to compare your answer, 1. Identify the decimal multipliers: 3.5% increase = 1.035; 4.2% increase = 1.042 [1 mark]. 2. Set up the combined compound calculation for the 4 years: Final Balance = £6,000 x (1.035)² x (1.042)² [1 mark]. 3. Calculate the final balance: £6,000 x 1.071225 x 1.085764 = £6,978.597. = £6,978.60 [1 mark]. 4. Subtract the initial £6,000 principal to find total interest earned: £6,978.60 - £6,000 = £978.60 [1 mark].
- Q2
In a winter sale, all prices in a furniture shop are reduced by 15%. On the final day of the sale, the sale prices are reduced by a further 10%. A sofa costs £612 on the final day of the sale. Calculate the original price of the sofa before the sale began. [4 marks]
Click “Check Mark Scheme” to compare your answer, 1. Decimal multiplier for the first 15% reduction = 1 - 0.15 = 0.85; decimal multiplier for the further 10% reduction = 1 - 0.10 = 0.90 [1 mark]. 2. Set up the equation linking the original price P to the final price: P x 0.85 x 0.90 = £612 [1 mark]. 3. Simplify the combined multiplier: P x 0.765 = £612 [1 mark]. 4. Divide by 0.765 to find the original price: P = £612 ÷ 0.765 = £800 [1 mark].
- Q3
A company buys a new delivery van for £32,000. The value of the van depreciates by x% each year. At the end of 3 years, the value of the van is £16,384. Work out the value of x. (Higher Tier) [4 marks]
Click “Check Mark Scheme” to compare your answer, 1. Let m be the annual decimal multiplier, and set up the compound depreciation equation for 3 years: 32,000 x m³ = 16,384 [1 mark]. 2. Divide both sides by 32,000: m³ = 16,384 / 32,000 = 0.512 [1 mark]. 3. Take the cube root of both sides to find the multiplier m: m = ³√0.512 = 0.80 [1 mark]. 4. Convert the multiplier m = 0.80 (80%) into the percentage decrease x: 100% - 80% = 20%, so x = 20 [1 mark].
- Q4
A radioactive substance decays by 18% every hour. Its initial mass is 500 g. Calculate the minimum number of complete hours needed for the mass of the substance to fall below 200 g. [3 marks]
Click “Check Mark Scheme” to compare your answer, 1. The decimal multiplier for an 18% decrease is 1 - 0.18 = 0.82, so Mass after n hours = 500 x (0.82)ⁿ [1 mark]. 2. Test integer values of n systematically: for n = 4 hours, Mass = 500 x (0.82)⁴ = 226.06 g (still above 200 g) [1 mark]. 3. For n = 5 hours, Mass = 500 x (0.82)⁵ = 185.37 g (below 200 g); therefore, 5 complete hours are needed [1 mark].
Common Exam Questions (4)
How do you find the decimal multiplier for a percentage increase or decrease?
Always start from 100%: 1. For a percentage increase of r%, add r% to 100% and divide by 100 (e.g., +15% -> 100% + 15% = 115% = 1.15; +3.5% -> 103.5% = 1.035). 2. For a percentage decrease (or depreciation) of r%, subtract r% from 100% and divide by 100 (e.g., -15% -> 100% - 15% = 85% = 0.85; -6% -> 94% = 0.94).
How do you spot a 'reverse percentage' question in the exam, and what is the golden rule?
A reverse percentage question gives you the new price/value after a percentage increase or discount has already happened, and asks you to find the original price/value (in the past). Golden rule: Never calculate the percentage of the new price. Instead, write the equation: Original x Multiplier = New, and divide the new value by the decimal multiplier: Original = New ÷ Multiplier.
What is the difference between simple interest and compound interest?
With simple interest, the exact same fixed cash amount of interest is added every year (based only on the starting principal), so if you earn 5% simple interest on £1,000, you get £50 every single year. With compound interest, the interest is added to the account balance at the end of each year, so in the next year you earn 'interest on your interest' using the formula Balance = P x (1 + r/100)ⁿ.
What is the formula for calculating percentage change (or percentage profit/loss)?
Use the formula: Percentage Change = ((Change in value) / (original value)) x 100, where Change = |New value: Original value|. Always divide by the original starting value, never the final value.
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