Proportionality, percentage and scale

Masika sews uniforms in Bukavu. If $6$ m of fabric cost $27\,000$ FC, how much do $10$ m cost? How much is a $15\,\%$ discount on a uniform? And on a map, how many kilometres do $7.5$ cm stand for? These three questions use the same idea: proportionality. In this lesson you learn to use ratios, proportions, percentages and scales.

Ratios and proportions

The ratio of $a$ to $b$ ($b\ne0$) is the quotient $\dfrac{a}{b}$.

A proportion is an equality of two ratios: $\dfrac{a}{b}=\dfrac{c}{d}$. The terms $a$ and $d$ are the extremes, $b$ and $c$ the means. Key property:
$$\frac{a}{b}=\frac{c}{d}\iff a\times d=b\times c$$
(the product of the extremes equals the product of the means).

Two quantities are proportional if you always go from one to the other by multiplying by the same number $k$, the proportionality coefficient. To find an unknown value, use the rule of three: first compute the value for one unit.

Percentages

A percentage is a ratio with denominator $100$: $p\,\%=\dfrac{p}{100}$.

• Taking $p\,\%$ of a quantity $A$: $\dfrac{p}{100}\times A$. Example: $15\,\%$ of $32\,000$ FC is $0.15\times32\,000=4\,800$ FC.
• Increasing $A$ by $p\,\%$: new value $A\times\left(1+\dfrac{p}{100}\right)$.
• Decreasing $A$ by $p\,\%$: new value $A\times\left(1-\dfrac{p}{100}\right)$.
• Finding a percentage: $\dfrac{\text{part}}{\text{whole}}\times100$.

Scale

On a map or a plan, the scale is the ratio
$$\text{scale}=\frac{\text{distance on the map}}{\text{real distance}}$$
with both distances expressed in the same unit.

A scale of $\dfrac{1}{50\,000}$ means that $1$ cm on the map stands for $50\,000$ cm, that is $500$ m, in reality.

• Real distance $=$ map distance $\times$ $50\,000$.
• Map distance $=$ real distance $\div$ $50\,000$.