Ratio, proportion and rates of change: set up, solve and interpret the answers in growth and decay problems, including compound interest and work with general iterative processes (Higher tier)

The idea

Higher tier only. For compound interest, the amount after n periods is A = P(1 + r)ⁿ, where r is the decimal rate per period. For example, £2000 at 5% per year for 3 years gives 2000 × 1.05³ = £2315.25. Decay uses a multiplier below 1, such as 0.85 for a 15% fall. An iteration turns one value into the next, written xₙ₊₁ in terms of xₙ. Start from x₀, work out x₁, feed that answer back in to get x₂, and carry on; the calculator answer key does the feeding back. With xₙ₊₁ = √(3xₙ + 4) and x₀ = 2, x₁ ≈ 3.162, x₂ ≈ 3.672 and x₃ ≈ 3.875, closing in on 4, the positive solution of x² = 3x + 4. Growth questions can run backwards, asking for the rate or the number of periods, so try whole-number powers until the target is passed.

What earns the marks

• A correct compound interest multiplier, such as 1.05 for 5% growth or 0.85 for 15% decay, used with the correct power.
• The correct final amount after n periods, with the calculation shown rather than just an answer.
• A correct first iteration, with the starting value substituted into the rule.
• Two further terms that follow correctly from your own previous answer, even where the first term was wrong.
• The value the terms settle on, given to the number of decimal places asked for.
• A correct multiplier raised to a trial power in a growth or decay problem, and one for naming the period at which the target is first passed.

Common mistakes and how to put them right

• Adding simple interest each year instead of multiplying by the same factor each period, so the total drifts below the compound answer.

• Rounding each term to two decimal places and feeding the rounded value back in, so later terms drift away from the true value.

• Starting again from the first value each time instead of using the term just found.

• Naming the last period before a target is passed when the question asked for the first period at which the amount is above it.

Exam tips

• For compound interest, write the multiplier and the power before reaching for the calculator, so the method is visible even if the arithmetic slips.
• Enter the starting value, press equals, then key the iteration rule using the answer button; each further press gives the next term.
• Keep every digit on the calculator between steps and round only when you write a term down.
• For growth with an unknown number of periods, work out the total for a few whole numbers of periods and pick the first that passes the target.