Direct variation in the real world

We have used direct proportion a lot already. This subtopic looks at it as a concept: what does it mean to say "$y$ varies directly with $x$"? When can we trust it? And what does the relationship look like on a graph?

Concept

Definition. Two quantities $x$ and $y$ are in direct variation (or direct proportion) if $y = kx$ for some constant $k > 0$ called the constant of proportionality.

Equivalent ways of saying the same thing:

• $\dfrac{y}{x} = k$ (a fixed value).
• Doubling $x$ doubles $y$; halving $x$ halves $y$.
• Whenever $x_1, y_1$ and $x_2, y_2$ are corresponding pairs, $\dfrac{y_1}{x_1} = \dfrac{y_2}{x_2}$.

Graph. Plot $y$ against $x$. If $y = kx$, the graph is a straight line through the origin with slope $k$. No straight-line graph through the origin? Then not direct proportion.

Examples that ARE direct proportion.

• Cost and number of identical items. $5$ apples at $\mathbb{?}\,10$ each: cost $= 10 \cdot n$.
• Distance and time at constant speed. Distance $=$ speed $\times$ time.
• Mass and volume of one liquid. Density is constant for a given material.
• Earnings and hours worked at a fixed wage.

Examples that are NOT direct proportion.

• Side and area of a square. Area $= s^2$, not $k \cdot s$. Doubling $s$ quadruples the area.
• Number of workers and time to finish a job. More workers $\Rightarrow$ less time , this is inverse proportion.
• Phone bill = fixed rental + (rate $\times$ minutes). There is a non-zero constant when minutes $= 0$, so the graph doesn't pass through the origin.
• Age and height. They both grow but not in a constant ratio.

A clean test. Tabulate $\dfrac{y}{x}$ for several data points. If the ratio is constant, you have direct proportion. If not, you don't.

How to use it. Once you know $y \propto x$ with constant $k$:

• Given any $x$, compute $y = kx$.
• Given any $y$, compute $x = y/k$.

If you don't know $k$ but know one $(x_1, y_1)$, you can find any other $y$ from $\dfrac{y}{x} = \dfrac{y_1}{x_1}$.

Worked examples

Example 1. A wire of length $7$ m weighs $42$ g. Find the weight of $25$ m of the same wire.

• $k = 42 / 7 = 6$ g per metre.
• For $25$ m: $25 \cdot 6 = 150$ g.

Example 2. A car covers $180$ km in $3$ hours. How long to cover $480$ km at the same speed?

• Speed $= 60$ km/h. Time $= 480 / 60 = 8$ hours.

Example 3. The cost of $4$ books is $\mathbb{?}\,260$. Find the cost of $11$ books.

• Cost per book $= 65$. Total $= 11 \cdot 65 = \mathbb{?}\,715$.

Example 4. Determine whether perimeter and side of a regular hexagon are in direct proportion.

• Perimeter $= 6 \cdot \text{side}$, so $P = 6 s$. Yes, with $k = 6$.

Try it yourself

1. If $6$ litres of paint cover $72$ m$^2$, how many m$^2$ will $10$ litres cover?
2. If $9$ students share $54$ chocolates equally, how many chocolates for $14$ students at the same rate? Is this direct proportion?
3. A car uses $4$ L of fuel to travel $52$ km. How much fuel for $208$ km?
4. Is the relationship between side of a square and its diagonal a direct proportion? If yes, what is $k$?
5. The marks of a student in $5$ subjects average $76$. Total marks $= ?$ Is this a proportion?
6. A printer prints $90$ pages in $6$ minutes. How long for $300$ pages?
7. Determine whether $x = 1, 2, 3, 4$ and $y = 4, 7, 10, 13$ are in direct proportion.
8. The temperature in Fahrenheit and Celsius are NOT in direct proportion. Why? (Hint: $0\,^\circ C = 32\,^\circ F$.)

Activity / Insight

Plot it. Take any of the proportional pairs in the worked examples and plot them on graph paper. You should get a straight line through the origin. Plot side vs area of a square (use $s = 1, 2, 3, 4$). The points trace a curve (a parabola), not a line , a vivid demonstration that this is NOT direct proportion.