Find the effect of inflation on a price practice questions | General Mathematics (Year 11, VCE Units 1 & 2)

Prices and inflation

A price affected by inflation grows geometrically. With different yearly rates, use each year's multiplier in turn; with one average rate, use one multiplier raised to the number of years

P_n=P_0\left(1+\frac{r}{100}\right)^n

Build an increase multiplier An increase of r% changes a price by the multiplier 1+r/100. $\displaystyle 1+\frac{4}{100}=1.04$
Use each yearly rate When the rates differ, apply the multiplier for each year to the price then current $\displaystyle P_2=P_0\times1.02\times1.05$
Use a power for an average rate A single average annual rate over n years uses the same multiplier n times $\displaystyle P_{10}=P_0\left(1.03\right)^{10}$
Round at the end Keep the exact calculation through the chain and round only at the precision requested $\displaystyle \$1.02\times1.05\times1.06=\$1.135260\ldots\approx\$1.14$

Worked example

A takeaway meal costs $12.50 now. Assume its price rises by average inflation of 3% per year for 10 years. Find the price after 10 years to the nearest cent

P_{10}=\$12.50\left(1+\frac{3}{100}\right)^{10}

$\displaystyle R=1+\frac{3}{100}=1.03$The average annual inflation rate gives a multiplier of 1.03
$\displaystyle P_{10}=\$12.50\times1.03^{10}=\$16.798954\ldots$The same multiplier is applied for all 10 years, so it is raised to the tenth power
$\displaystyle P_{10}\approx\$16.80$Round the exact price to the nearest cent only after the full calculation

Common mistakes

Adding yearly percentages

✕$\displaystyle P_2=P_0\left(1+\frac{3+5}{100}\right)$✓$\displaystyle P_2=P_0\left(1+\frac{3}{100}\right)\left(1+\frac{5}{100}\right)$

Each year's rate has its own multiplier, and the second multiplier acts on the price after the first year

Rounding before the calculation ends

✕$\displaystyle \$1.02\times1.05\approx\$1.07,\quad\$1.07\times1.06\approx\$1.13$✓$\displaystyle \$1.02\times1.05\times1.06\approx\$1.14$

Carry the exact value through the whole chain and round only the final price

Using the rate as the multiplier

✕$\displaystyle P_n=P_0\left(0.03\right)^n$✓$\displaystyle P_n=P_0\left(1.03\right)^n$

An increase multiplier includes the original 100% as well as the additional rate

Difficulty levels

introApply two whole-number yearly inflation rates to a small starting price

standardUse sequential rates or an average rate over about 10 to 15 years

challengeCompare long-term average rates or interpret a price rise after a longer span

examChoose the price produced by applying different yearly inflation rates in sequence

Questions about this topic

Why are different yearly inflation rates multiplied in sequence?+

The second year's percentage applies to the price after the first year's change. Each year therefore has a multiplier that acts on the current price

When can one multiplier be raised to a power?+

Use a power when one average annual rate applies for the whole span. The exponent is the number of years

Why should intermediate prices not be rounded?+

Rounding an intermediate price changes the amount used in the next multiplication. Carrying the exact value and rounding once gives the intended final approximation

Can I use a calculator?+

A calculator is allowed unless a question is marked no calculator

Which year level and chapter is this?+

Year 11, VCE Units 1 & 2, General Mathematics, chapter 3: Sequences and finance

In this chapter