Percentages, Profit & Loss

Section 1 of 5

Finding a percentage of an amount

"Per cent" means "out of $100$". So $15\%$ means $\dfrac{15}{100} = 0.15$.

Decimal method: change the percent to a decimal, then multiply.
$r\%$ of $X = \dfrac{r}{100} \times X$

(i) Find $15\%$ of $\text{€}650$.

$15\% = 0.15$

$0.15 \times 650$

$= \text{€}97.50$

(ii) Find $2.3\%$ of $\text{€}97.51$.

$2.3\% = 0.023$

$0.023 \times 97.51 = 2.242\,73\ldots$

$\approx \text{€}2.24$ (nearest cent)

Try this: find $25\%$ of $\text{€}80$.

$0.25 \times 80 = \text{€}20$

$\text{€}20$

Try this: find $7\%$ of $\text{€}450$.

$0.07 \times 450 = \text{€}31.50$

$\text{€}31.50$

Try this: find $12.5\%$ of $\text{€}240$.

$0.125 \times 240 = \text{€}30$

$\text{€}30$

Try this: find $0.5\%$ of $\text{€}10{,}000$.

$0.005 \times 10{,}000 = \text{€}50$

$\text{€}50$

Section 2 of 5

Percentage increase and decrease

For an increase or decrease, you can find the percent of the original and add/subtract — but the multiplier method is faster.

Multiplier method:
Increase by $r\%$: multiply by $(1 + \dfrac{r}{100})$. e.g. $+14\%$ → $\times 1.14$.
Decrease by $r\%$: multiply by $(1 - \dfrac{r}{100})$. e.g. $-15\%$ → $\times 0.85$.

(i) Increase $\text{€}350$ by $14\%$.

Multiplier: $1 + 0.14 = 1.14$

$350 \times 1.14$

$= \text{€}399$

Check the other way: $0.14 \times 350 = 49$. Then $350 + 49 = 399$ ✓

(ii) Decrease $\text{€}250{,}000$ by $15\%$.

Multiplier: $1 - 0.15 = 0.85$

$250{,}000 \times 0.85$

$= \text{€}212{,}500$

Try this: increase $\text{€}500$ by $8\%$.

$500 \times 1.08 = \text{€}540$

$\text{€}540$

Try this: decrease $\text{€}1{,}500$ by $20\%$.

$1{,}500 \times 0.80 = \text{€}1{,}200$

$\text{€}1{,}200$

Try this: a $\text{€}45$ item is reduced by $35\%$ in a sale. Find the sale price.

$45 \times 0.65 = \text{€}29.25$

$\text{€}29.25$

Try this: VAT at $23\%$ is added to a net price of $\text{€}80$. Find the price including VAT.

$80 \times 1.23 = \text{€}98.40$

$\text{€}98.40$

Section 3 of 5

Comparing two quantities as percentages

To compare two test scores, success rates, or any two fractions, convert each to a percentage. The bigger percentage wins.

$\dfrac{\text{part}}{\text{whole}} \times 100\%$

(i) Ann got $8$ out of $18$ right. John got $251$ out of $653$ right. Who did better?

Ann: $\dfrac{8}{18} \times 100 = 44.4\ldots\%$

John: $\dfrac{251}{653} \times 100 = 38.4\ldots\%$

$44.4\% > 38.4\%$

Ann did better.

Try this: in a class test, Liam scored $42$ out of $60$ and Aoife scored $58$ out of $80$. Who did better?

Liam: $\dfrac{42}{60} \times 100 = 70\%$
Aoife: $\dfrac{58}{80} \times 100 = 72.5\%$

Aoife (by $2.5$ percentage points)

Try this: a shop sells $87$ out of $300$ items on Monday, $134$ out of $500$ on Tuesday. Which day had the better sell-through rate?

Mon: $\dfrac{87}{300} \times 100 = 29\%$
Tue: $\dfrac{134}{500} \times 100 = 26.8\%$

Monday ($29\%$ vs $26.8\%$)

Try this: free-throw rates — Player A made $18$ of $25$, Player B made $34$ of $50$. Who is more accurate?

A: $\dfrac{18}{25} \times 100 = 72\%$
B: $\dfrac{34}{50} \times 100 = 68\%$

Player A ($72\%$ vs $68\%$)

Section 4 of 5

Percentage error

When you round an amount, you introduce an error. The percentage error tells you how big that error is relative to the real value.

$\% \text{ error} = \dfrac{\text{error}}{\text{real value}} \times 100$
Where $\text{error} = |\text{real} - \text{rounded}|$ (always positive).

(i) $\text{€}2.23$ is rounded to $\text{€}2.20$. Find the percentage error.

Error $= 2.23 - 2.20 = 0.03$

$\% \text{ error} = \dfrac{0.03}{2.23} \times 100$

$= 1.345\ldots\%$

$\approx 1.35\%$

Use the real value as the denominator, not the rounded one.

Try this: $\text{€}5.74$ is rounded to $\text{€}5.70$. Find the percentage error.

Error $= 0.04$
$\dfrac{0.04}{5.74} \times 100 = 0.697\ldots\%$

$\approx 0.70\%$

Try this: a measurement of $48.6\text{ cm}$ is rounded to $49\text{ cm}$. Find the percentage error.

Error $= |48.6 - 49| = 0.4$
$\dfrac{0.4}{48.6} \times 100 = 0.823\ldots\%$

$\approx 0.82\%$

Try this: a population is reported as $52{,}000$ but the real figure is $51{,}347$. Find the percentage error.

Error $= |52{,}000 - 51{,}347| = 653$
$\dfrac{653}{51{,}347} \times 100 = 1.272\ldots\%$

$\approx 1.27\%$

Section 5 of 5

Mark-up vs margin

Both compare profit to a price as a percentage — but the denominator is different. Mixing them up is a classic exam mistake.

Mark-up (also called $\%$ profit / loss): $\dfrac{\text{profit}}{\text{cost price}} \times 100$
Margin: $\dfrac{\text{profit}}{\text{selling price}} \times 100$

Memory hook: Mark-up uses Cost ("how much it's marked up from cost"). Margin uses Sell ("the slice of the sale kept as profit").

(i) A coat costs $\text{€}120$ and is sold for $\text{€}150$. Find mark-up and margin.

Profit $= 150 - 120 = \text{€}30$

Mark-up $= \dfrac{30}{120} \times 100 = 25\%$

Mark-up $= 25\%$

Margin $= \dfrac{30}{150} \times 100 = 20\%$

Margin $= 20\%$

Notice: same profit, but different percentages. Margin is always smaller than mark-up (because the selling price is always larger than the cost price for a profitable sale).

Try this: a phone costs the retailer $\text{€}400$ and is sold for $\text{€}500$. Find (a) mark-up, (b) margin.

Profit $= 100$
Mark-up $= \dfrac{100}{400} \times 100 = 25\%$
Margin $= \dfrac{100}{500} \times 100 = 20\%$

Mark-up $25\%$, margin $20\%$

Try this: cost $\text{€}80$, sold $\text{€}100$. Mark-up and margin?

Profit $= 20$
Mark-up $= \dfrac{20}{80} \times 100 = 25\%$
Margin $= \dfrac{20}{100} \times 100 = 20\%$

Mark-up $25\%$, margin $20\%$

Try this: a desk costs a shop $\text{€}75$ and is sold for $\text{€}120$. Mark-up and margin?

Profit $= 45$
Mark-up $= \dfrac{45}{75} \times 100 = 60\%$
Margin $= \dfrac{45}{120} \times 100 = 37.5\%$

Mark-up $60\%$, margin $37.5\%$

Try this (reverse): an item is sold for $\text{€}200$ at a margin of $25\%$. Find the cost price.

Margin $25\%$ of sell $= 0.25 \times 200 = \text{€}50$ profit
Cost $= 200 - 50 = \text{€}150$

Cost $= \text{€}150$

End of lesson

Percentages — HL · Mathslive.ie