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Direct & Inverse Proportion

Direct Proportion

What is direct proportion?

  • Proportion is a way of talking about how two variables are related to each other
  • Direct proportion means that as one variable goes up the other goes up by the same factor
    • This means that the ratio between the two amounts will always stay the same
  • If x and y are directly proportional then
    • x : y will always be the same
    • there will be some value of k such that y = kx
    • their graph is a linear (straight line) graph, with gradient k

cie-igcse-1-10-direct-proportion-graph

  • Other problems may involve a variable being directly proportional to a function of another variable
  • For example
    • y is directly proportional to the square of x means that y = kx2
    • y is directly proportional to the square root of x means that y = k√x
    • y is directly proportional to the cube of x means that y = kx3
      • Each of these would have a different type of graph, depending on the function

How do we deal with direct proportion questions?

  • Direct proportion questions can be dealt with in the same way
  • STEP 1
    Identify the two variables and set up the formula in terms of those variables
    • For example if A is directly proportional to B, then the two variables are A and B
    • If A is directly proportional to B use the formula A = kB
    • If A is directly proportional to the square of B use the formula A = kB2
  • STEP 2
    Find k by substituting the values given in the question into your formula and solving
  • STEP 3
    Write the formula for A in terms of B by substituting in your value of k
  • STEP 4
    Use the formula to find the required quantity

Exam Tip

  • Even if the question doesn’t ask for a formula it is always worth working one out and using it in all but the simplest cases

Worked example

y is directly proportional to the square of x.
When x = 3, y = 18.

Find the value of y when x = 4.

Identify the two variables.

y comma space x to the power of 2 space end exponent

We are told this is DIRECT proportion.

y equals k x squared

We can now find k using y = 18 when x = 3.

table row 18 equals cell k open parentheses 3 close parentheses squared end cell row k equals cell 18 over 9 end cell row k equals 2 end table

We can now write the full formula/equation in x and y .

y equals 2 x squared

Use this formula to find y when x = 4.

table row y equals cell 2 cross times 4 squared end cell row y equals cell 2 cross times 16 end cell row y equals 32 end table

bold italic y bold equals bold 32

Inverse Proportion

What is inverse proportion?

  • Inverse proportion means as one variable goes up the other goes down by the same factor
    • If two quantities are inversely proportional, then we can say that one is directly proportional to the reciprocal of the other 
  • If x is inversely proportional to y, then
    • x space colon space 1 over y will always be the same
    • there will be some value of k such that x space equals space k over y
    • the graph will be related to that graph of y space equals space k over x

cie-igcse1-10-inverse-proportion-graph

How do we deal with inverse proportion questions?

  • Inverse proportion questions can be dealt with in a similar way to direct proportion questions
    • STEP 1
      Identify the two variables and set up a formula in terms of those variables
      • For example if A is inversely proportional to B, then the two variables are A and B
      • If A is inversely proportional to B use the formula bold italic A space equals space k over bold italic B
      • If A is inversely proportional to the square of B use the formula bold italic A space equals space k over bold italic B squared
    • STEP 2
      Find k by substituting the values given in the question into your formula and solving
    • STEP 3
      Write the formula for A in terms of B by substituting in your value of k
    • STEP 4
      Use the formula to find the required quantity

Worked example

The time, t hours, it takes to complete a project varies inversely proportional to the number of people working on it, n.
If 4 people work on the project it takes 70 hours to complete.

(a)

Write an equation connecting t and n.


 Identify the two variables

t comma space n

We are told this is INVERSE proportion

t equals k over n

We can now find k using n equals 4 and t equals 70 (given in words in the question)

table row 70 equals cell k over 4 end cell row k equals cell 70 cross times 4 end cell row k equals 280 end table

We can now write the full formula/equation in n and t

bold italic t bold equals bold 280 over bold italic n

(b)

Given that the project needs to be completed within 18 hours, find the minimum number of people needed to work on it.


Use the formula to find n when t equals 18

table row 18 equals cell 280 over n end cell row n equals cell 280 over 18 end cell row n equals cell 15.55 space... end cell end table

A sensible answer would be a whole number (as it is a number of people)

16 people is the minimum number of workers required to finish the project in 18 hours

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