Direct and inverse proportionIB MYP Maths Standard: Revision notes

Two quantities are in direct proportion if one is multiplied by a number and the other is multiplied by the same number. Doubling $x$ doubles $y$. We write $y\propto x$, or $y=kx$, where $k$ is the constant of proportionality. Find $k$ from one pair of values: if $y=18$ when $x=4$, then $k=\frac{18}{4}=4.5$, so $y=4.5x$. Then use the equation: when $x=10$, $y=45$; when $y=63$, $x=63\div4.5=14$. The ratio $\frac{y}{x}$ is always the same.

Two quantities are in inverse proportion if one increases as the other decreases so that their product stays the same. Doubling $q$ halves $p$. We write $p=\frac{k}{q}$, so $k=pq$. If $p=6$ when $q=5$, then $k=30$ and $p=\frac{30}{q}$. When $p=4$, $q=\frac{30}{4}=7.5$. When $q=12$, $p=2.5$.

The graph of direct proportion $y=kx$ is a straight line through the origin; the gradient is $k$. A larger $k$ gives a steeper line. The graph of inverse proportion $y=\frac{k}{x}$ (for $x>0$) is a curve that falls from left to right and never touches either axis, because $y$ can never be $0$ and $x$ can never be $0$. To sketch it, mark a point such as $(5,6)$ for $y=\frac{30}{x}$, and also $(10,3)$ and $(2,15)$, then draw a smooth curve through them, getting close to the axes.

The unitary method finds the value of one unit first. USD $68.50$ for QAR $250$ gives $68.50\div250=0.274$ USD for one riyal, so QAR $1800$ gives $0.274\times1800=\text{USD }493.20$. This is the same as finding $k$. Going the other way, QAR $=\text{USD}\div0.274$, so USD $150$ costs $150\div0.274=547.4\ldots\approx547$ riyals. Ratios also work: $1800:250=7.2:1$, so multiply $68.50$ by $7.2$.

Currency: exchange at a fixed rate is direct proportion. Recipes and scaling: ingredients scale directly with the number of people ($9$ kg of rice for $60$ guests is $0.15$ kg per guest). Time and workers: the time taken falls as more workers are used, so $t=\frac{k}{c}$; with $5$ cooks taking $8$ hours, $k=40$, and $8$ cooks take $5$ hours. Always match the right model to the situation by asking: does more of one mean more or less of the other?

Proportion models are only valid in a sensible range. The inverse model predicts that $40$ cooks need $\frac{40}{40}=1$ hour, but in a real kitchen cooks share limited space and equipment, and some tasks cannot be shared, so the real time would be longer. When you answer a justify or comment question, say what the model gives, then give one realistic reason why it may not hold.