Direct Proportion: y = kx

Direct proportion y = kx: find k from one pair of values, use the formula both ways, and spot a proportional table or graph.

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What Direct Proportion Means (Higher)

Two quantities are directly proportional when one is always the same multiple of the other. Double one, and the other doubles too.

x	4	8	12
y	12	24	36

Here y is always 3 times x. That fixed multiple is called k, the constant of proportionality.

Rule

$y \propto x$ means $y = k x$ for some number k. You find k from one pair of values.

Finding the Formula

The 3-mark question: d is directly proportional to t, and $d = 14$ when $t = 4$ . Find a formula for d in terms of t.

Worked example

1
$d \propto t \Rightarrow d = k t$ — “Directly proportional” means d is always the same multiple of t. Call that multiple k.
2
$14 = k \times 4$ — Put in the pair you are given, d = 14 and t = 4. Now k is the only unknown.
3
$k = \frac{14}{4}$ — Divide both sides by 4 to get k on its own.
4
$k = 3.5$ — 14 ÷ 4 = 3.5. Check: 3.5 × 4 = 14.
5
$d = 3.5 t$ — Write the formula with the number in place of k. The question asks for a formula, so t stays a letter.

Common mistake

Working out $\frac{4}{14}$ instead of $\frac{14}{4}$ . k is the thing on the left divided by the thing on the right: $k = \frac{d}{t}$ .

Using the Formula Both Ways

The next part gives one value and asks for the other. Use the formula $d = 3.5 t$ from the last card.

Worked example

1
$d = 3.5 \times 10$ — To find d when t = 10, put t = 10 into the formula.
2
$d = 35$ — Multiply. t went up, so d went up too, which makes sense.
3
$49 = 3.5 t$ — To find t when d = 49, put d = 49 into the formula. Now t is the unknown.
4
$t = \frac{49}{3.5}$ — Divide both sides by 3.5, the k, to get t on its own.
5
$t = 14$ — 49 ÷ 3.5 = 14. Check: 3.5 × 14 = 49.

Key idea

Check your answer makes sense: d got bigger, so t must get bigger by the same factor.

Key idea

Write k = 3.5 on its own line. Most of the marks are for finding k, so it still scores if the last step slips.

Shortcut: Compare Two Pairs

If you only need a missing value, not the formula, you can skip writing y = kx. Because y ÷ x is always the same k, any two pairs give the same answer:

Formula

\frac{y _{1}}{x _{1}} = \frac{y _{2}}{x _{2}}

$x_{1}$ and $y_{1}$ are the pair you are given. $x_{2}$ is the new value, and $y_{2}$ is the one you want.

Worked example

1
$y \propto x, y = 18 when x = 12$ — The question: find y when x = 20.
2
$\frac{18}{12} = \frac{y _{2}}{20}$ — Put the given pair on the left and the new pair on the right. y₂ is the unknown.
3
$y_{2} = \frac{18}{12} \times 20$ — Multiply both sides by 20 to get y₂ on its own.
4
$\frac{18}{12} = 1.5$ — Simplify the fraction first. This is just k.
5
$y_{2} = 1.5 \times 20 = 30$ — Multiply. Check: 30 ÷ 20 = 1.5, the same as 18 ÷ 12.

Key idea

The same idea with no numbers: if x is multiplied by 3, y is multiplied by 3. If x goes up by 20%, y goes up by 20%.

Common mistake

Flipping one side only, like $\frac{18}{12} = \frac{20}{y _{2}}$ . Keep y on top on both sides.

Is It Proportional? Tables and Graphs

The 2-mark question: “Show that y is directly proportional to x” using a table or a graph.

Worked example

1
$x : 2, 6, 10 y : 7, 21, 35$ — The table. To show proportion you must check every pair, not just one.
2
$\frac{7}{2} = 3.5$ — Divide y by x for the first pair.
3
$\frac{21}{6} = 3.5$ — Second pair: the same answer.
4
$\frac{35}{10} = 3.5$ — Third pair: the same again.
5
$y = 3.5 x$ — Every pair gives the same k = 3.5, so y is directly proportional to x. Write this as your conclusion.

Rule

On a graph, direct proportion is a straight line that goes through $(0, 0)$ . The gradient is k.

Common mistake

Thinking every straight line is proportional. $y = 2 x + 3$ is straight but misses $(0, 0)$ . Try it: x = 1, 2, 4 gives y = 5, 7, 11, and $\frac{5}{1} \neq = \frac{7}{2}$ .

Which Method? Read the Question

The words tell you which formula to write.

You see	You do
“y is directly proportional to x”	$y = k x$
“show that y is proportional”	Divide y by x for each pair
Only a missing value wanted	$\frac{y _{1}}{x _{1}} = \frac{y _{2}}{x _{2}}$ , or find k first
“proportional to x², x³ or √x”	See Proportion to Powers of x
“inversely proportional”	See Inverse Proportion: y = k/x
A recipe or a best buy	See Direct Proportion in Real Life

Key idea

Always find k first, then write the full formula, then use it.

Practise direct proportion: y = kx now

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