Iterative growth processes Worked Solutions — GCSE Maths

Start from 100 percent

An increase of $8\%$ leaves the original $100\%$ plus another $8\%$.

Convert the percentage to a decimal

Divide by $100$ to turn a percentage into a multiplier.

State the multiplier

Each step multiplies the quantity by $1.08$, so after $n$ steps the model is $A = P \times 1.08^{n}$.

Start from 100 percent

A decrease of $15\%$ leaves $85\%$ of what was there before.

Convert the percentage to a decimal

Divide by $100$ to get the multiplier.

State the multiplier

Each step multiplies by $0.85$, so the model is $A = P \times 0.85^{n}$.

Identify the multiplier and the number of steps

Doubling means the multiplier is $2$, and $3$ hours is $3$ steps.

Use the iterative model

Substitute $P = 400$, $r = 2$ and $n = 3$ into $A = P \times r^{n}$.

Evaluate

$2^{3} = 8$, so there are $3200$ bacteria after $3$ hours.

Count the half-lives

Each half-life lasts $2$ days, so $6$ days is $3$ half-lives.

Apply the halving multiplier three times

After one half-life exactly half remains, so $3$ half-lives multiply the mass by $\left(\tfrac{1}{2}\right)^{3} = \tfrac{1}{8}$.

State the mass

$8\text{ g}$ of the isotope remains after $6$ days.

Write the multiplier

A $10\%$ rise means multiplying by $1.1$ every year.

Apply the model for two steps

Two years means the multiplier is used twice, so raise it to the power $2$.

Work out the population

The population after $2$ years is $6050$.