Profit, Loss & Error — Mathslive.ie

Section 1 of 3

One Number as a Percentage of Another

First over second

To write one number as a percentage of another: $\dfrac{\text{first}}{\text{second}} \times \dfrac{100}{1}$. Percentages are handy for comparing two results.

Worked example — a mark

Write $4$ as a percentage of $5$; and find $63$ out of $120$ as a percentage.

$\tfrac{4}{5} \times 100 = 80\%$

$\tfrac{63}{120} \times 100 = 52.5\% \approx 52\%$

$80\%$ and $52.5\%$

You try

Mary got $73$ out of $135$; Ann got $89$ out of $162$. Who did better?

Turn each into a percentage and compare.

Mary: $\tfrac{73}{135} \times 100 = 54.07\%$

Ann: $\tfrac{89}{162} \times 100 = 54.93\%$

Ann did better ($54.93\% > 54.07\%$)

Section 2 of 3

Profit & Loss

The formulas

$\text{Profit} = \text{Selling price} - \text{Cost price}$. Then $\%\text{ profit} = \dfrac{\text{profit}}{\text{cost}} \times 100$; similarly $\%\text{ loss} = \dfrac{\text{loss}}{\text{cost}} \times 100$. Always over the cost price.

Worked example — percentage profit

A car costs €2500 and is sold for €3000. Find the percentage profit.

Profit $= 3000 - 2500 = 500$

$\%\text{ profit} = \tfrac{500}{2500} \times 100$

$20\%$ profit

Worked example — a house

A house costs €400,000 and sells for €420,000. Find the percentage profit.

Profit $= 420000 - 400000 = 20000$

$\tfrac{20000}{400000} \times 100$

$5\%$ profit

You try

An item costing €750 is sold for €680. Find the percentage loss.

Loss = cost − sell, then divide by the cost.

Loss $= 750 - 680 = 70$

$\%\text{ loss} = \tfrac{70}{750} \times 100$

$9.3\%$ loss

Section 3 of 3

Percentage Error

Error over real

When a number is rounded, $\text{error} = |\text{real} - \text{rounded}|$ and $\%\text{ error} = \dfrac{\text{error}}{\text{real}} \times 100$ — always over the real value.

Worked example — rounding to the nearest whole

$5.2$ is rounded to $5$. Find the percentage error.

Error $= 5.2 - 5 = 0.2$

$\%\text{ error} = \tfrac{0.2}{5.2} \times 100 = 3.84$

$3.8\%$

Worked example — a big number

$16{,}529$ is rounded to $17{,}000$. Find the percentage error.

Error $= |16529 - 17000| = 471$

$\tfrac{471}{16529} \times 100 = 2.84$

$2.8\%$

You try

$42{,}521$ is rounded to $43{,}000$. Find the percentage error.

Error $= 43000 - 42521$, then divide by the real value $42521$.

Error $= 43000 - 42521 = 479$

$\tfrac{479}{42521} \times 100$

$1.1\%$

That’s Part 3.

Writing one number as a percentage of another (and comparing), profit & loss over the cost, and percentage error over the real value. Next: compound interest, income tax, and foreign exchange.