Direct Proportion

1Summary 重點摘要

Learn what direct proportion is, know how two numbers change together.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Step 1: The meaning of direct proportionDirect proportion describes the relation between two quantities: when x increases, y increases in the same ratio; when x decreases, y decreases in the same ratio; and the quotient y/x stays constant. For apples at 5 each, buying 2 costs 10 and buying 4 costs 20, with 102 and 204 both equal to 5. Why must the ratio be constant? It guarantees the two quantities scale by the same factor. A common mistake is claiming direct proportion from the fact that both quantities rise, without checking the ratio.
2. Step 2: The expression of direct proportionA direct proportion can be written as y = kx, where k is a fixed constant called the constant of proportionality, meaning y is always k times x. For apples at 5 each, the total cost of x apples is y = 5x with k = 5; for a wage of 60 per hour, the pay for x hours is y = 60x with k = 60. Why introduce k? It turns the proportional relation into a number that can be found, making the problem computable. A common mistake is confusing y = kx with y = x + k; the latter adds a fixed amount each time and is not a proportion.
3. Step 3: Finding the constant kWith one pair of corresponding values x and y, the constant follows from k = y ÷ x. For example, if y = 10 when x = 2, then k = 10 ÷ 2 = 5 and the relation is y = 5x; verify with another pair, and if y = 15 when x = 3, the quotient 153 is again 5, confirming the proportion. Why check a second pair? One pair alone cannot confirm a constant ratio; two or more agreeing pairs are safer. A common mistake is writing k = x ÷ y, which gives the reciprocal and ruins the later calculation.
4. Step 4: Finding unknown valuesOnce k is known, the relation is fixed and any y can be found from x, or any x from y. For example, given y = 5x: when x = 10, y = 5 × 10 = 50; conversely, if y = 35, then x = 35 ÷ 5 = 7. Why does this work? Every pair of values satisfies the same equation in a direct proportion, so one value determines the other. A common mistake is omitting the units after finding y, or substituting into an old ratio instead of the relation just found, giving answers that disagree.
5. Step 5: The graph of direct proportionPlotting y = kx on coordinate axes gives a straight line through the origin (0, 0), whose slope is k. For y = 5x the line passes through (0, 0), (1, 5) and (2, 10). Why must it pass through the origin? When x = 0, y = k × 0 = 0, which is unavoidable for a direct proportion; a line that misses the origin, such as y = 5x + 2, is not a proportion. A common mistake is checking only that the graph is straight while ignoring the origin; a straight line not through the origin is not a direct proportion.
6. Step 6: Everyday and applied examplesDirect proportion appears everywhere in life and science: at constant speed, distance is proportional to time, so at 60 km/h a journey covers 120 km in 2 hours and 240 km in 4 hours; at a fixed hourly wage, total pay is proportional to hours worked; at constant density, mass is proportional to volume; the cost of the same goods is proportional to the quantity bought. Familiarity matters because word problems must first be recognised as proportional before an equation can be formed. A common mistake is treating an inverse situation, where more workers mean fewer days, as direct proportion.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Direct proportion: y = kx
• k = y ÷ x (constant)
• x bigger y also bigger
• Graph is straight line through origin
• Many examples in life