Inverse Proportion

1Summary 重點摘要

Learn what inverse proportion is, know how two numbers change opposite ways.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Step 1: What inverse proportion isInverse proportion describes the relation where one quantity increases while the other decreases: as x grows, y shrinks, and as x shrinks, y grows, while their product never changes. Everyday examples abound — over a fixed journey, the faster you walk, the shorter the time, so speed and time are inversely proportional; likewise, the more workers on a job, the fewer days it takes. The key is not the phrase 'one up, one down' but verifying that the product stays constant, as explained in the final step.
2. Step 2: The formula xy = kInverse proportion can be written as xy = k, where k is the constant of proportionality, a fixed number. The equation says that no matter how x and y change, their product always equals k. It can also be written as y = k/x, which displays the inverse relation clearly. Compared with direct proportion y = kx, the crucial difference is that direct proportion keeps the quotient fixed (y ÷ x = k) while inverse proportion keeps the product fixed, so never mix the two formulas and decide which type the question involves before answering.
3. Step 3: How to find kGiven one corresponding pair of x and y, k is found by multiplying the two numbers: if x = 2 when y = 10, then k = 2 × 10 = 20, meaning xy = 20. This is the standard opening move of every inverse proportion question — take one pair (x, y) from the given information, multiply to get k, then write the complete equation. After finding k, state the concrete equation xy = 20 immediately; this makes the next substitution easy and your working clear to the reader.
4. Step 4: How to find an unknownWith xy = k established, the y for any x is simply y = k ÷ x. For example, given xy = 20, when x = 5 we get y = 20 ÷ 5 = 4. The worked example follows the same flow: from x = 4 and y = 6 we get k = 4 × 6 = 24; substituting x = 8 gives y = 24 ÷ 8 = 3. Notice that doubling x halves y (6 becomes 3) — this 'opposite multiples' check quickly confirms whether an answer is reasonable and is worth a few seconds in an examination.
5. Step 5: The graph of inverse proportionPlotted on a coordinate plane, inverse proportion gives a curved hyperbola, not a straight line. The curve has two notable features: it never crosses either axis, because x = 0 or y = 0 would make the product zero rather than a non-zero k; and the larger k is, the farther the curve sits from the origin. This contrasts sharply with the straight line through the origin in direct proportion, and graph identification is a common multiple-choice question — remember that for k > 0 the curve lies only in the first and third quadrants.
6. Step 6: How to judge, with common examplesBefore solving an application question, confirm the relation really is inverse. The test: multiply two corresponding pairs; the relation is inverse only if the products agree. For example, x = 2 with y = 12 and x = 3 with y = 8 both give 24, so they are inversely proportional; but if x = 3 came with y = 10, the product would be 30 and the relation would not be inverse. Other common examples: with a fixed amount of work, more people means less time; sharing a fixed cake among more people gives smaller pieces; with a fixed force, a larger contact area gives smaller pressure. One rising while the other falls is never enough — the product must stay constant.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Inverse proportion: xy = k
• k = x × y (constant)
• x bigger y smaller
• Graph is curve
• Many examples in life