Direct and inverse proportional relationships · Higher
도입됨| English | 의미 |
|---|---|
| constant of proportionality/ˈkɒnstənt ɒv prəˌpɔːʃəˈnælɪti/ | 비례 상수 |
Buying twice as much fabric at a fixed price doubles the bill. Sharing one fixed job among twice as many equally productive workers halves its duration.
- Buying twice as much fabric at a fixed price doubles the bill. Sharing one fixed job among twice as many equally productive workers halves its duration.
- This lesson studies constant of proportionality 比例常数: The fixed multiplier in a stated proportional relationship.
Choose the mathematical structure
- Direct proportion y=kx preserves y/x for nonzero x and gives a straight line through the origin. Inverse proportion y=k/x preserves xy and gives a reciprocal curve, with x=0 excluded. State the fixed assumptions: a start-up charge breaks direct proportion, while changing productivity can break inverse proportion. Higher constructs the equation from a known pair; Foundation interprets and uses a given equation.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
A 5-yuan start-up fee makes C=5+3L. Why is cost no longer directly proportional to length?
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Direct proportion needs y=kx, a straight line THROUGH THE ORIGIN with constant y/x. The fixed fee lifts the line off the origin, so the ratio C/L changes with L.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
For a given fabric cost C=3L, lengths 2,4,6 m cost 6,12,18 yuan: C/L=3. A 5-yuan fixed fee instead gives C=5+3L, which is linear but not directly proportional. For a given fixed-job model t=24/w, workers 2,4,6 need 12,6,4 hours, and wt=24 worker-hours. Doubling workers halves time. Higher construction: if y is directly proportional to x and y=18 at x=6, k=18/6=3, hence y=3x. If t is inversely proportional to w and t=6 at w=4, k=tw=24, hence t=24/w. These models assume a constant rate and exclude impossible negative worker counts.
Direct and inverse proportional relationships
Direct proportion y=kx preserves y/x for nonzero x and gives a straight line through the origin
Classify the worked-case statements, then explain the units or invariant that justifies each decision.
Use C=3L to find C for L=6.
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Substitute L=6: C=18.
Test a tempting shortcut
- An increasing graph need not be direct proportion. An inverse relationship is proportional to 1/x, not to -x. Keep a product constant for inverse proportion and a quotient constant for direct proportion.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Every straight line represents direct proportion. This claim is false. Explain which definition or assumption it violates.
Use t=24/w to find t for w=4.
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Substitute w=4: t=24/4=6 hours.
Every straight line represents direct proportion.
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An increasing graph need not be direct proportion. An inverse relationship is proportional to 1/x, not to -x. Keep a product constant for inverse proportion and a quotient constant for direct proportion.
Interpret a new situation
- AQA R10/R13/R14 covers numerical, algebraic and graphical direct/inverse proportion. Interpret the units of k and the gradient. In Foundation use the supplied equations; Higher also constructs and interprets them from the context.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find the constant product wt for this inverse model.
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Multiply: w×(24/w)=24 for w≠0.
Match each model to its identifying property.
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Check the invariant: quotients for direct, products for inverse. A straight line alone proves nothing without the origin.
Use this in your course
- 8300 · Higher · 3.3. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
The fixed multiplier in a stated proportional relationship. Choose the relationship, show the method, check its assumptions and interpret the result.