Year 11 General Consumer Arithmetic: Loans And Investments

Inflation

Q: How do you calculate the future price after inflation?

Multiply the current price by 1 plus the inflation rate as a decimal, raised to the power of the number of years. For example, $4.20 with 3 percent annual inflation for 5 years is 4.20 times 1.03 to the power 5.

Q: How do you find what a price was in the past?

Divide the current price by 1 plus the inflation rate as a decimal, raised to the power of the number of years. This is the same compound formula run backwards.

Q: What is purchasing power?

Purchasing power is how much an amount of money can actually buy. Inflation reduces purchasing power over time. To find the real value of cash after t years, divide it by 1 plus r, all raised to the power t.

Q: How do you calculate the average annual inflation rate from two prices?

Divide the later price by the earlier price, raise the result to the power of 1 divided by the number of years, then subtract 1. Multiply by 100 to get a percentage.

10 practice questions 2 video lessons Theory + worked examples

Theory

Inflation is the gradual rise in prices over time. Mathematically, it works exactly like compound interest applied to prices: $P_{t} = P_{0} (1 + r)^{t}$ . The same formula goes backwards in time by dividing, and shows how cash loses purchasing power when not invested.

Inflation is the gradual rise in prices over time. The inflation rate is the average annual percentage increase. Even if a product doesn't change, its price tag does.

Inflation works exactly like compound interest applied to prices. The future price after $t$ years is $P_{t} = P_{0} (1 + r)^{t}$ , where $P_{0}$ is today's price and $r$ is the annual inflation rate as a decimal.

Purchasing power is what your money can actually buy. Cash kept under the mattress loses real value because prices rise around it. The real value after inflation is found by dividing the cash amount by $(1 + r)^{t}$ ; the actual dollar amount is the nominal value.

Same formula, different framing: compound interest grows your money in an account; inflation grows the price of things you buy. The maths is identical.

Each year multiplies the price by 1.03

Cash shrinks in purchasing power

Future price after $t$ years of inflation:

P_{t} = P_{0} \times (1 + r)^{t}

P_{t} = P_{0} {(1 + r)}^{t}

Past price (going backwards in time):

P_{0} = \frac{P_{t}}{(1 + r)^{t}}

P_{0} = \frac{P_{t}}{{(1 + r)}^{t}}

Real value of cash after $t$ years (purchasing power in today's dollars):

real value = \frac{cash}{(1 + r)^{t}}

Finding the average annual inflation rate from a price change over $t$ years:

r = {(\frac{P_{t}}{P_{0}})}^{1 / t} - 1

r = {(\frac{P_{t}}{P_{0}})}^{1 / t} - 1

Symbols: P0 is the starting price, Pt is the price after t years, r is the annual inflation rate as a decimal (e.g. 3%→0.03).

How to solve an inflation problem

Convert the rate to a decimal ( $3 % \to 0.03$ ).
Identify the direction: forward in time → multiply by $(1 + r)^{t}$ ; backward → divide by $(1 + r)^{t}$ .
For purchasing power of cash, divide the cash amount by $(1 + r)^{t}$ to get the real value in today's dollars.
To find the average rate from two prices, use $r = (P_{t} / P_{0})^{1 / t} - 1$ and multiply by $100$ for a percentage.

Example 1 — Future price

A loaf of bread costs

$ 4.20

today. With inflation at

3 %

p.a., what will it cost in

5

years?

Solution

Multiply by $(1.03)^{5}$ .

$P_{5}$	$=$	$4.20 \times {1.03}^{5}$
$P_{5}$	$=$	$4.20 \times 1.15927$
$P_{5}$	$\approx$	$$ 4.87$

P_{5} \approx 4.87

The bread will cost about $$4.87$ .

Example 2 — Past price

A car costs

$ 32,000

today. With inflation averaging

2.5 %

p.a., what was it worth

4

years ago?

Solution

Divide by $(1.025)^{4}$ .

$P_{0}$	$=$	$\frac{32,000}{{1.025}^{4}}$
$P_{0}$	$=$	$\frac{32,000}{1.10381}$
$P_{0}$	$\approx$	$$ 28,990.42$

P_{0} \approx 28990.42

The car was worth about $$28{,}990.42$ four years ago.

Example 3 — Purchasing power

Sara hides

$ 2,000

in cash. After

6

years of

4 %

average annual inflation, what is its real value (in today's dollars)?

Solution

Divide by $(1.04)^{6}$ to get the real value.

$real$	$=$	$\frac{2,000}{{1.04}^{6}}$
$real$	$=$	$\frac{2,000}{1.26532}$
$real$	$\approx$	$$ 1,580.63$

real value \approx 1580.63

The cash will be worth only about $$1{,}580.63$ in today's purchasing power.

Example 4 — Find the rate

A house was

$ 650,000

five years ago and is now worth

$ 780,000

. Find the average annual inflation rate.

Solution

Use the ratio raised to $1 / t$ , then subtract $1$ .

$r$	$=$	${(\frac{780,000}{650,000})}^{1 / 5} - 1$
$r$	$=$	${1.2}^{0.2} - 1$
$r$	$\approx$	$1.03714 - 1$
$r$	$\approx$	$0.03714 = 3.71 %$ p.a.

r \approx 0.0371

The average annual inflation rate is about $3.71\%$ .

Common pitfalls

Treating inflation as simple growth. Inflation compounds — using

P \times (1 + r t)

instead of

P \times (1 + r)^{t}

under-estimates the future price.

Forgetting direction. Multiply by

(1 + r)^{t}

for future prices, divide for past prices and purchasing power. Doing the wrong operation flips the answer.

Mixing rate forms.

r

is a decimal in the formula —

3 %

becomes

0.03

. Using

3

gives a price that triples each year.

Confusing nominal and real value. If a question asks "what is its real value in today's dollars?", you must divide by

(1 + r)^{t}

. The cash amount itself doesn't shrink — its purchasing power does.

Frequently asked questions

What is inflation?

Inflation is the gradual rise in prices over time. The inflation rate is the average annual percentage by which prices increase.

How do you calculate the future price after inflation?

Multiply the current price by $(1 + r)^{t}$ . For $$ 4.20$ at $3 %$ annual inflation over $5$ years: $4.20 \times {1.03}^{5} \approx $ 4.87$ .

How do you find what a price was in the past?

Divide the current price by $(1 + r)^{t}$ . The same compound formula, run backwards.

What is purchasing power?

Purchasing power is what a sum of money can actually buy. To find the real value of cash after $t$ years, divide by $(1 + r)^{t}$ .

What is the difference between real and nominal value?

Nominal value is the actual dollar amount. Real value adjusts for inflation and shows what the money is worth in today's purchasing power.

How do you calculate the average annual inflation rate from two prices?

Divide the later price by the earlier price, raise to the power $1 / t$ , then subtract $1$ . Multiply by $100$ for the percentage.

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