Rationals and proportionality (VAT)

In a hardware shop in Matadi, the label on a bag of cement shows a price before tax (HT); at the till you pay the price including all taxes (TTC), which includes the value added tax (VAT, in French TVA). Computing VAT requires handling fractions and proportionality well. In this lesson you place fractions and decimals in $\mathbb{Q}$, reduce a fraction to lowest terms, operate on fractions and compute VAT.

The set $\mathbb{Q}$ and fractions in lowest terms

A rational number can be written $\dfrac{a}{b}$ with $a\in\mathbb{Z}$, $b\in\mathbb{Z}$ and $b\ne0$. We have $\mathbb{N}\subset\mathbb{Z}\subset\mathbb{D}\subset\mathbb{Q}$.

• $-7=\dfrac{-7}{1}$ and $0.25=\dfrac{1}{4}$ are rational numbers.
• $\dfrac{1}{3}=0.333\dots$ is rational but not a decimal: the division never ends.

A fraction is in lowest terms when its numerator and denominator have no common divisor other than $1$. To reduce it, divide both terms by their GCD (in French PGCD).

Operations on fractions

• Addition and subtraction: write with the same denominator (the LCM of the denominators), then add or subtract the numerators: $\dfrac{5}{6}-\dfrac{1}{4}=\dfrac{10}{12}-\dfrac{3}{12}=\dfrac{7}{12}$.
• Multiplication: $\dfrac{a}{b}\times\dfrac{c}{d}=\dfrac{a\times c}{b\times d}$. Simplify before multiplying when possible.
• Division: dividing by a non-zero fraction means multiplying by its reciprocal: $\dfrac{a}{b}\div\dfrac{c}{d}=\dfrac{a}{b}\times\dfrac{d}{c}$.
• The sign rules of $\mathbb{Z}$ apply: $\dfrac{-a}{b}=-\dfrac{a}{b}$.

VAT, a proportionality situation

In the DRC, the standard VAT rate is $16\%$. VAT is always computed on the price before tax:
$$\text{VAT}=\text{HT}\times\dfrac{16}{100}\qquad\text{TTC}=\text{HT}+\text{VAT}=\text{HT}\times1.16$$

So the TTC price is proportional to the HT price, with coefficient $1.16$.

To find the HT price from the TTC price, divide by $1.16$: $\text{HT}=\text{TTC}\div1.16$.