Direct Proportion

In a direct proportion, the quotient y/x stays constant when x is nonzero. If x doubles, y doubles; if x triples, y also triples. For example, 4 kg cost €10, so the unit price is 10/4=€2.50/kg. Seven kilograms cost €17.50. Check several pairs for the same quotient. Merely increasing together is not enough: one quantity can rise without being proportional. The graph must also pass through the origin. Units help interpret the quotient: euros divided by notebooks gives a price per notebook.

With a unit-rate method, first calculate the cost of one unit and then scale to the requested amount. If 4 notebooks cost €10, one costs 10/4=€2.50. Seven then cost 7·€2.50=€17.50. This assumes the unit price stays fixed and that no base fee or bulk discount applies. Algebraically, you find the same factor k=10/4 and use y=kx. The unit-rate method and the equation are two ways to represent the same proportional relationship, not competing rules. Measured data rarely lie exactly on a line because of measurement error; the model describes the expected relationship.

The equation y=kx gives y=0 whenever x=0, so the origin (0,0) lies on its graph. A fixed base charge shifts every value: for y=2x+5, the cost is still 5 units when x=0. That relation is linear but not directly proportional. This matters in real situations, such as a taxi fare with a fixed starting fee. For measured data, a graph that appears to pass through the origin is not proof by itself; also check the ratios and measurement uncertainty. A unit-rate calculation starts from a known pair and preserves the same quotient for the requested amount.

k is the constant quotient y/x and tells you how much y corresponds to one unit of x. In a price problem it is the unit price; in uniform motion it can be speed. Its unit follows from the quantities, such as euros per notebook or metres per second. If k=€2.50 per notebook, 8 notebooks cost €20. Without context, k has no fixed unit. Also do not confuse k with the slope of every straight line: direct proportion requires a line through the origin. A nonzero intercept represents a fixed starting amount and rules out direct proportion, even if the line rises.

Use direct proportion only when the ratio remains constant. A one-time fee, tiered pricing, or a change in work rate can violate the assumption. Check whether doubling x also doubles y and whether y would be zero at x=0. For a fixed job with a changing number of workers, an inverse relationship may apply instead. With measured data, rounding can make ratios differ slightly; judge the model and measurement precision rather than demanding exact equality for every value. State which conditions keep k constant, such as unit price or speed.