⚖️Proportional Relationships

Two quantities are proportional when one is always the same multiple of the other. Every pair fits a single rule:

y = kx

The number k is the constant of proportionality. It never changes inside one relationship, which is exactly what makes the relationship worth naming.

Rearranging y = kx gives the way to find it:

k = y / x

So k is a ratio, and it is the same ratio for every pair. If 3 metres of ice rope cost 12 gold, then k = 12 / 3 = 4 gold per metre, and 7 metres cost 4 x 7 = 28 gold. One division answers the whole table.

Because k is "the amount of y for one unit of x", it is also called the unit rate. Kilometres per hour, gold per metre, pages per minute: every unit rate you have already met is a constant of proportionality wearing its own units.

The same k hides in a different place in each representation.

In a table, divide each y by its own x and check that the answers agree.

x: 2 5 8 y: 5 12.5 20 -> 5/2 = 2.5, 12.5/5 = 2.5, 20/8 = 2.5, so k = 2.5

In an equation, k is simply the number in front of x. In y = 2.5x, k is 2.5. Nothing is added on the end.

In a graph, k is the height of the line above x = 1. A proportional graph is a straight line through the origin (0, 0), and it always passes through the point (1, k). That point is worth looking for by name: it reads the unit rate straight off the picture. In a graph of gold against metres of rope, (1, 4) says "one metre costs 4 gold".

Any two points do the work as well: from (0, 0) to (8, 20) the line runs 8 across and rises 20, so k = 20 / 8 = 2.5. Rise divided by run — remember that phrase, because it comes back with a new name shortly.

Not every tidy table is proportional, and a table will tell you which of three shapes it has if you ask it two questions.

Is the ratio y / x the same every row? If yes, it is proportional: y = kx.

Are the differences in y the same, for equal steps in x? If yes but the ratios disagree, it is linear but not proportional — a straight line that misses the origin, y = kx + b with b not 0.

If neither is constant, it is neither.

x: 1 2 3 4 y: 3 6 9 12 ratios 3, 3, 3, 3 -> proportional, y = 3x y: 5 8 11 14 ratios 5, 4, 3.67... -> linear, not proportional, y = 3x + 2 y: 2 5 10 17 differences 3, 5, 7 -> neither

The middle row is the one to watch. It climbs by 3 every step, so it feels just as regular as the first — but at x = 0 it sits at 5, not 0, and doubling x does not double y. A constant difference is not a constant ratio.

Sleds, prices and pumps rarely arrive described the same way. One is an equation, one is a table, one is a sentence. They cannot be compared side by side until each has been turned into its own k.

Ren's sled: y = 3x, where y is kilometres in x hours. So k = 3 kilometres per hour.

Kofi's sled: covers 12 kilometres in 5 hours. So k = 12 / 5 = 2.4 kilometres per hour.

3 is greater than 2.4, so Ren's sled is faster — even though 12 is the biggest number written down anywhere. Totals compare totals; only rates compare rates.

On a graph the comparison is visible at a glance: both lines start at (0, 0), and the one that has climbed higher by x = 1 is the faster one. The steeper line has the larger k.

That word steeper is the door out of this craft. The constant of proportionality is the first version of slope, and a line that does not pass through the origin needs one extra number for where it starts. Add that number and y = kx becomes y = mx + b: the same rate, plus a starting height. Everything in this lesson survives the move — k just changes its name to m.