L2 · The Constant of Proportionality k

Learning goal

“I can find the constant of proportionality k = a/b for a proportional relationship, use it to generate equivalent ratios and fill in missing quantities, and explain why k must be identical in every row - with k allowed to be a decimal, a fraction, or less than 1.”

Constant of proportionality (k) — the fixed multiple relating two directly proportional quantities: if a = k·b in every row, that multiplier k is the constant of proportionality, and k = a/b.

Directly proportional — two quantities are directly proportional when one is always the same multiple of the other: as one grows, the other grows by the same factor.

Equivalent ratios — ratios with the same value and the same k: 8 : 2, 12 : 3, and 20 : 5 are equivalent because each gives k = 4.

Worked example

Juice is mixed from syrup: 2 cups of syrup make 8 cups of juice; 3 cups of syrup make 12 cups of juice. Juice j is directly proportional to syrup s. Step 1: Name the direction - juice in terms of syrup: j = k·s. Find k from the first row: k = j/s = 8/2 = 4. Step 2: Audit k on the second row: 12/3 = 4. Same value. That sameness is not a coincidence - it IS the proportionality. k is called CONSTANT because every equivalent ratio returns it. Step 3: Use k both ways the curriculum asks. Generate an equivalent ratio: s = 5 → j = 4 × 5 = 20. Find a missing quantity: j = 28 → s = 28 ÷ 4 = 7. Say it plainly: k = a/b is a division you can run on ANY row - and a relationship only earns the name "proportional" when every row returns the same k.

Check yourself — explain it

Why must k come out identical for the rows (2, 8) and (3, 12)?

Paired problem — example still visible

y is directly proportional to x, with (x, y) = (5, 15) and (9, 27). Find k, then find y when x = 12. -> k = 15/5 = 3 (check: 27/9 = 3 ✓); y = 3 × 12 = 36.

Faded completion — last step is yours

"w = k·t, and when t = 4, w = 10. So k = 10 ÷ ___ = ___." -> 4; 2.5

Ready to practise?

10 scored questions with a 3-rung hint ladder. Mastery needs 8/10 plus both gate questions.

Start practice →Take the unit quiz