Proportion
Section: Algebra and graphs 2 | Syllabus: Cambridge IGCSE Mathematics (0580)
Direct Proportion
Extended Only
If y is directly proportional to x, written y ∝ x, then y=kx for some constant k called the constant of proportionality. As x increases, y increases in the same ratio - if x doubles, y also doubles. This idea extends beyond simple proportion: y can be directly proportional to x², x³ or √x.
- y ∝ x means y=kx
- y ∝ x^2 means y=kx^2; y ∝ x^3 means y=kx^3; y ∝ √(x) means y=k√(x)
- Find k first by substituting a known pair of values, then use the completed equation to find any other unknown
Worked Example: Direct Proportion
- Question: y is directly proportional to x. y = 20 when x = 4. Find y when x = 7.
- Step 1: Write the equation and substitute the known pair: y=kx 20=k4
- Step 2: Solve for k: k=5, so y=5x
- Step 3: Substitute x = 7: y=57=35
- Answer: y = 35
Worked Example: Direct Proportion with a Power
- Question: y is directly proportional to x². y = 45 when x = 3. Find y when x = 5.
- Step 1: Write the equation and substitute the known pair: y=kx^2 45=k9
- Step 2: Solve for k: k=5, so y=5x^2
- Step 3: Substitute x = 5: y=525=125
- Answer: y = 125
Common Mistakes
MistakeWriting y = kx even when the statement says y is proportional to x² or x³
Fixmatch the power in the equation exactly to the power stated in the proportionality sentence
MistakeSubstituting the new x-value before finding k
Fixalways find k using the first given pair of values, and write out the full equation, before using it to find anything else
Inverse Proportion
Extended Only
If y is inversely proportional to x, written y ∝ (1/x), then y=(k/x). As x increases, y decreases - if x doubles, y is halved. This also extends to y proportional to (1/x^2), (1/x^3), and (1/√(x)).
- y ∝ (1/x) means y=(k/x)
- y ∝ (1/x^2) means y=(k/x^2), and similarly for (1/x^3) and (1/√(x))
- The method is identical to direct proportion: find k from a known pair, then substitute to find the unknown
Worked Example: Inverse Proportion
- Question: y is inversely proportional to x. y = 8 when x = 3. Find y when x = 6.
- Step 1: Write the equation and substitute the known pair: y=(k/x) 8=(k/3)
- Step 2: Solve for k: k=24, so y=(24/x)
- Step 3: Substitute x = 6: y=(24/6)=4
- Answer: y = 4
Worked Example: Inverse Proportion with a Power
- Question: y is inversely proportional to x². y = 2 when x = 5. Find x when y = 50.
- Step 1: Write the equation and substitute the known pair: y=(k/x^2) 2=(k/25)
- Step 2: Solve for k: k=50, so y=(50/x^2)
- Step 3: Substitute y = 50: 50=(50/x^2) x^2=1
- Answer: x = 1
Common Mistakes
MistakeWriting y = k/x when the statement says "inversely proportional to x²", giving the wrong power in the denominator
Fixmatch the power in the denominator exactly to the power stated in the proportionality sentence
MistakeAssuming inverse proportion means y decreases by the same amount each time, like a linear sequence
Fixinverse proportion means y is divided by the same factor that x is multiplied by, not reduced by a constant subtraction
Proportion in Context
Extended Only
Proportion problems are often set in a real-world context, where the type of proportion must first be identified from the wording before it can be written as an equation.
- Look for phrases such as "directly proportional to", "inversely proportional to", or "varies as" to identify the type of proportion
- Identify which variable depends on which - e.g. "the volume is proportional to..." means volume is the subject of the equation
- Once the equation is set up, the method is the same as before: find k, then solve for the unknown
Worked Example: Proportion in a Real-World Context
- Question: The volume, V cm³, of a candle is directly proportional to the cube of its height, h cm. A candle of height 4 cm has a volume of 32 cm³. Find the volume of a candle of height 6 cm.
- Step 1: Write the equation and substitute the known pair: V=kh^3 32=k64
- Step 2: Solve for k: k=0.5, so V=0.5h^3
- Step 3: Substitute h = 6: V=0.5216=108
- Answer: V = 108 cm³
Common Mistakes
MistakeSetting up the equation with the variables in the wrong roles, e.g. writing h = kV³ instead of V = kh³
Fixidentify the subject of the sentence first - "V is proportional to..." always makes V the subject of the equation
MistakeForgetting to apply the stated power to the x-value when substituting to find k
Fixapply the power written in the proportionality statement to the known value before substituting, not after
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