Population growth, depreciation & elections — notes, question types & practice · Percentage

01

One rate, many years

A town grows by $r\%$ every year. Each year's population is the previous year's times the chip $\dfrac{100+r}{100}$ .

\text{After } n \text{ years} = P \times \left(\frac{100+r}{100}\right)^n

A town of 50,000 grows $10\%$ a year. After 2 years: $50000 \times \dfrac{11}{10} \times \dfrac{11}{10} = 60500$ .

Rule: Growth per year compounds. Never use $P + P \times r \times n$ , which is simple interest thinking.

02

Different rates in different years

When each year has its own rate, multiply one chip per year in order.

A town of 40,000 grows $10\%$ one year and falls $5\%$ the next: $40000 \times \dfrac{11}{10} \times \dfrac{19}{20} = 41800$ .

Three rates work the same way. A value that rose $20\%$ , then $25\%$ , then $10\%$ is now $66,000$ ; before those years it was $66000 \div \dfrac{33}{20} = 40000$ .

Note: $+10\%$ then $-5\%$ is not $+5\%$ . The chips multiply, they do not cancel.

03

Travelling back in time

To find an earlier value, divide by the chips.

A town has 16,000 people now and grows $25\%$ yearly. Two years ago: $16000 \div \dfrac{5}{4} \div \dfrac{5}{4} = 16000 \times \dfrac{16}{25} = 10240$ .

Tip: Dividing by $\dfrac{5}{4}$ twice is multiplying by $\left(\dfrac{4}{5}\right)^2$ . Reverse the chip, then square it.

04

Depreciation compounds too

A machine loses value every year. That is a fall applied to the falling value, so it compounds.

A machine worth Rs 12,500 loses $10\%$ a year. After 2 years: $12500 \times \left(\dfrac{9}{10}\right)^2 = 10125$ .

Running backwards: a machine is worth 8,100 after two years of $10\%$ yearly loss. It cost $8100 \times \left(\dfrac{10}{9}\right)^2 = 10000$ .

Watch: Each year's fall is taken on that year's value, never on the original price.

A rise of $25\%$ followed by a fall of $20\%$ does come back: $\dfrac{5}{4} \times \dfrac{4}{5} = 1$ . But $+25\%$ then $-25\%$ leaves $\dfrac{5}{4} \times \dfrac{3}{4} = \dfrac{15}{16}$ , a loss.

05

Elections: valid votes first

Votes polled $=$ valid votes $+$ invalid votes. Candidate percents are always on valid votes.

Worked: $20\%$ of votes polled were invalid. The winner got $60\%$ of valid votes and beat the only rival by 1,200 votes.

Two candidates, so the rival got $40\%$ of valid votes.
Margin $= 20\%$ of valid votes $= 1200$ , so valid $= 6000$ .
Polled $=$ valid $\times \dfrac{100}{80} = 6000 \times \dfrac{5}{4} = 7500$ .

Careful: "Votes polled" is not "valid votes". Remove the invalid share first.

06

Voters who did not turn up

The electoral roll lists everyone. Votes cast $=$ roll $-$ those who did not vote.

Worked: $20\%$ of the roll did not vote. The winner got $55\%$ of votes cast and won by 1,200 votes.

Margin $= 55\% - 45\% = 10\%$ of votes cast $= 1200$ , so cast $= 12000$ .
Cast is $80\%$ of the roll, so roll $= 12000 \times \dfrac{5}{4} = 15000$ .

Example: 27,000 votes were polled in another ward and $10\%$ were invalid. Valid votes $= 27000 - 2700 = 24300$ .

07

Question types you will see

Each type: how to recognise it, the method step by step, and one question to try.

Type 1very common2 practice Q

Population after n years

How to spot it:

A present population and a yearly growth rate are given; the future population is asked.

P\left(1+\frac{r}{100}\right)^n

Method

Write the chip for the yearly rate.
Raise it to the number of years.
Multiply by the present population.
Keep the answer exact with fractions.

Why it works:

The same chip applies every year, so it compounds to a power.

Try this

The population of a town is 6,400 and it grows by 25% every year. What will it be after 2 years?

Show solution

Chip $= \dfrac{5}{4}$ ; power for 2 years $= \dfrac{25}{16}$ .
$6400 \times \dfrac{25}{16} = 400 \times 25 = 10000$ .

Answer

10,000

Type 2common2 practice Q

Population or value n years ago

How to spot it:

The present value and a yearly rate are given; an earlier value is asked.

\text{past} = \frac{\text{now}}{\left(1+\frac{r}{100}\right)^n}

Method

Write the chip for the yearly rate.
Divide the present value by the chip, once per year.
Check by growing your answer forward.

Why it works:

Growth moved the number forward by chips, so division walks it back.

Try this

A town's population grows by 25% every year. It is 16,000 now. What was it 2 years ago?

Show solution

Chip $= \dfrac{5}{4}$ ; divide twice, same as $\\times \dfrac{16}{25}$ .
$16000 \times \dfrac{16}{25} = 640 \times 16 = 10240$ .

Answer

10,240

Type 3common3 practice Q

Different rates in different years

How to spot it:

Each year has its own rise or fall: 'increased by 5% in the first year and decreased by 10% in the second'.

\text{final} = P \times \frac{100+a}{100} \times \frac{100-b}{100} \times \cdots

Method

Write one chip per year, in order.
Multiply all the chips.
Multiply by the starting value.
Going back in time, divide instead.

Why it works:

Multipliers handle mixed rises and falls in one line.

Try this

A town had 80,000 people. The population rose by 5% in one year and fell by 10% the next. What was the population after the two years?

Show solution

Chips: $\dfrac{21}{20}$ then $\dfrac{9}{10}$ .
$80000 \times \dfrac{21}{20} = 84000$ .
$84000 \times \dfrac{9}{10} = 75600$ .

Answer

75,600

Type 4very common2 practice Q

Election with invalid votes and a margin

How to spot it:

Some per cent of votes polled is invalid; the winner's share of valid votes and the winning margin are given.

\text{margin} = (\text{winner}\% - \text{rival}\%) \times \text{valid votes}

Method

Find the rival's share of valid votes (two candidates).
Margin per cent of valid votes is the difference of shares.
From the margin votes, find valid votes.
Scale up to votes polled using the invalid per cent.

Why it works:

Candidate shares live on valid votes, so the margin fixes the valid count first.

Try this

In an election, 20% of the votes polled were invalid. The winner secured 60% of the valid votes and beat the only other candidate by 1,200 votes. How many votes were polled?

Show solution

Rival $= 40\%$ of valid; margin $= 20\%$ of valid $= 1200$ .
Valid votes $= 6000$ .
Valid is $80\%$ of polled: $6000 \times \dfrac{5}{4} = 7500$ .

Answer

7,500

Type 5occasional

Election with voters who stayed home

How to spot it:

A per cent of the electorate did not vote; the rest elect the winner.

Method

Votes cast = roll minus the absent share.
From the margin, find the votes cast.
Scale cast up to the full roll.

Why it works:

Turnout links the roll to the cast votes, and the margin links cast votes to the winner.

Try this

In an election, 20% of the voters on the electoral roll did not vote. The winner received 55% of the votes cast and won by 1,200 votes. How many voters were on the roll?

Show solution

Margin $= 55\% - 45\% = 10\%$ of cast $= 1200$ .
Cast $= 12000$ .
Cast is $80\%$ of the roll: $12000 \times \dfrac{5}{4} = 15000$ .

Answer

15,000

08

Formula sheet

Growth for n years

P\left(1 + \frac{r}{100}\right)^n

r% added every year, compounding.

Depreciation for n years

P\left(1 - \frac{r}{100}\right)^n

r% of value lost every year.

Value n years ago

\frac{\text{now}}{\left(1 \pm \frac{r}{100}\right)^n}

Divide by the chip power to go back.

Election margin

\text{margin votes} = (\text{winner}\% - \text{rival}\%) \times \text{valid votes}

Percents on valid votes only.

09

Shortcuts that save time

⚡ Chip powers beat formulas

Write one multiplier per year and multiply. Squares like $\left(\dfrac{5}{4}\right)^2 = \dfrac{25}{16}$ are worth remembering.

Example

A town of 6,400 people grows by 25% every year. What is its population after 2 years?

Show solution

Chip for $25\%$ growth: $\dfrac{5}{4}$ .
$6400 \times \dfrac{25}{16} = 400 \times 25 = 10000$ .

Answer

10,000

⚡ Remove invalid votes first

In elections, bring every statement onto valid votes before touching the margin.

Example

In an election, 20% of the votes polled were invalid. The winner secured 60% of the valid votes and beat the only other candidate by 1,200 votes. How many votes were polled?

Show solution

Margin $= 60\% - 40\% = 20\%$ of valid votes.
$20\%$ of valid $= 1200 \Rightarrow$ valid $= 6000$ .
Valid $= 80\%$ of polled, so polled $= 6000 \times \dfrac{5}{4} = 7500$ .

Answer

7,500 votes

⚡ Roll, cast, valid: walk the chain

Election numbers form a chain: roll, votes cast, valid votes, then candidates. Convert one link at a time with chips.

Example

In an election, 20% of the voters on the roll did not vote. The winner got 55% of the votes cast and won by 1,200 votes. How many voters were on the roll?

Show solution

Margin $= 10\%$ of cast $= 1200 \Rightarrow$ cast $= 12000$ .
Cast $= 80\%$ of roll.
Roll $= 12000 \times \dfrac{5}{4} = 15000$ .

Answer

15,000 voters

10

Mistakes to avoid

Where most students lose marks on this subtopic.

Mistake 01

Using simple growth $P + Prn$ for population.

Population compounds: $P \times \left(1 + \dfrac{r}{100}\right)^n$ .

Mistake 02

Taking every year's depreciation on the original value.

Each year falls on that year's value, so multiply chips.

Mistake 03

Applying the chip $n-1$ times for $n$ years.

Count on fingers: 2 years means the chip twice.

Mistake 04

Computing candidate percents on total votes polled.

Remove invalid votes first; percents are on valid votes.

Mistake 05

Taking the winner's margin as a percent of the roll.

The margin is the difference of two percents of valid or cast votes.

Mistake 06

Multiplying by the growth chip to find a past value.

Going back divides by the chip, or multiplies by its inverse.

11

Quick revision

Read this the night before the exam.

Growth: $P \times \left(\dfrac{100+r}{100}\right)^n$ .
Depreciation: $P \times \left(\dfrac{100-r}{100}\right)^n$ .
Past value: divide by the chip power.
Different rates: one chip per year, multiplied in order.
Roll $\to$ cast $\to$ valid votes: one chip at a time.
Margin $=$ (winner% $-$ rival%) of valid votes.

12

Practice: 12 questions

Sets of 10, mixed across the question types above. Every answer has a step-by-step explanation.

Topic test · 12 questions

Suggested time 9 min · wrong answers go to your mistake notebook automatically.