Percentages, Similarity and Growth

A percentage is a fraction with denominator 100. The multiplier method is the most efficient way to calculate percentage changes.

Percentage increase: multiply by $(1+\frac{p}{100})$

Percentage decrease: multiply by $(1-\frac{p}{100})$

Change	Multiplier
20% increase	$\times 1.2$
15% decrease	$\times 0.85$
7.5% increase	$\times 1.075$

Worked example — a jacket costs £64. It is reduced by 35%. Find the sale price.

$64\times 0.65=\pounds 41.60$ ✓

One quantity as a percentage of another:

$\frac{15}{60}\times 100=25\%$ (15 is 25% of 60)

Original value problems (reverse percentage): divide by the multiplier.

Worked example — after a 20% increase, the price is £96. Find the original price.

$96÷1.2=\pounds 80$ ✓

Worked example — a population of 8500 is 85% of the original. Find the original.

$8500÷0.85=10000$ ✓

Simple interest — calculated on the original principal only:

$\text{Interest}=\frac{P\times r\times t}{100}$

where $P$ = principal, $r$ = annual rate (%), $t$ = time (years).

Worked example — find the simple interest on £500 at 4% per year for 3 years.

$\text{Interest}=\frac{500\times 4\times 3}{100}=\pounds 60$. Total = £560. ✓

Compound interest — calculated on the accumulating total each period:

$A=P{\left(1+\frac{r}{100}\right)}^t$

Worked example — find the value of £1000 invested at 3% compound interest for 5 years.

$A=1000\times 1.03^5=1000\times 1.1593=\pounds 1159.27$ (to the nearest penny) ✓

Comparing: compound interest grows faster than simple interest for $t>1$ year, because the interest earns interest.

Percentage change:

$\%\text{ change}=\frac{\text{change}}{\text{original}}\times 100$

Worked example — a price rises from £80 to £94. Find the percentage increase.

$\frac{94-80}{80}\times 100=\frac{14}{80}\times 100=17.5\%$ ✓

Percentages over 100%: if one value is more than twice another, the percentage is over 100%.

$\frac{250}{100}\times 100=250\%$ — the new value is 250% of the original.

A value 150% of the original represents a 50% increase. $150\%=1.5\times \text{original}$.

Similar shapes have the same angles and proportional corresponding sides. The ratio of corresponding lengths is the scale factor $k$.

• Length scale factor: $k$
• Area scale factor: $k^2$
• Volume scale factor: $k^3$

Worked example — two similar cylinders. The smaller has height 4 cm and volume 60 cm³. The larger has height 10 cm. Find the larger cylinder's volume.

Scale factor for length: $k=\frac{10}{4}=2.5$

Volume scale factor: $2.5^3=15.625$

Larger volume: $60\times 15.625=937.5$ cm³ ✓

Trigonometric ratios as proportions (R12 underlined): in similar right-angled triangles, the sine, cosine, and tangent ratios are equal regardless of size — they depend only on the angle.

(Extra context — this explains why trig ratios are constants for a given angle rather than depending on the specific triangle's side lengths.)

Compound growth/decay applies a percentage multiplier repeatedly over time.

$A=P\times m^t\text{where }m=1+\frac{r}{100}\text{ (growth) or }1-\frac{r}{100}\text{ (decay)}$

Worked example — population growth:

A town of 12 000 grows at 2.5% per year. Find the population after 10 years.

$A=12000\times 1.025^{10}=12000\times 1.2801\approx 15361$ ✓

Worked example — depreciation (decay):

A car worth £18 000 loses 18% of its value each year. Find the value after 4 years.

$A=18000\times 0.82^4=18000\times 0.4521\approx \pounds 8138$ ✓

(Extra context — Higher: iterative processes in R16H link to A20 iteration; a recurrence relation like $V_{n+1}=0.82V_n$ describes the same depreciation sequence term by term.)

1. Percentage decrease — multiplying by the percentage, not the multiplier

A 15% decrease is NOT $\times 0.15$. The multiplier is $1-0.15=0.85$. Multiplying by 0.15 gives the decrease itself, not the new value.

2. Reverse percentage — subtracting the percentage from the given value

If £60 is the price after a 20% reduction, the original is NOT $\pounds 60+20\%\text{ of }\pounds 60=\pounds 72$. Divide by the multiplier: $\pounds 60÷0.8=\pounds 75$.

3. Similarity — using length scale factor for areas

If the length scale factor is 3, the area scale factor is $3^2=9$. Multiplying the area by 3 instead of 9 is a very common error.

4. Compound interest — applying simple interest formula

Compound interest is recalculated on the new total each period; simple interest uses the original principal throughout. The compound formula $A=P(1+r/100{)}^t$ does this automatically.

Mistake	Correction
"Original before 25% increase: $80-25\%=80-20=60$"	Subtracting 25% of the new value is wrong; divide by the multiplier: $80÷1.25=64$ ✓
"If length scale = 4, area of similar shape = $4\times$ area"	Area scale $=4^2=16$; new area $=16\times$ original area
"3% compound interest on £200 for 2 years: $200\times 0.03\times 2=\pounds 12$"	Compound: $200\times 1.03^2=\pounds 212.18$; the simple interest answer is £12 growth but compound interest gives £12.18 growth