Direct & Inverse Proportion (Edexcel GCSE Maths)

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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.

$y, x^{2}$

We are told this is DIRECT proportion.

$y = k x^{2}$

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

$\begin{array}{rcl} 18 & = & k {(3)}^{2} \\ k & = & \frac{18}{9} \\ k & = & 2 \end{array}$

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

$y = 2 x^{2}$

Use this formula to find y when x = 4.

$\begin{array}{rcl} y & = & 2 \times 4^{2} \\ y & = & 2 \times 16 \\ y & = & 32 \end{array}$

$y = 32$

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
- $x : \frac{1}{y}$ will always be the same
- there will be some value of k such that $x = \frac{k}{y}$
- the graph will be related to that graph of $y = \frac{k}{x}$

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 $A = \frac{k}{B}$
  - If A is inversely proportional to the square of B use the formula $A = \frac{k}{B^{2}}$
- 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

$t, n$

We are told this is INVERSE proportion

$t = \frac{k}{n}$

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

$\begin{array}{rcl} 70 & = & \frac{k}{4} \\ k & = & 70 \times 4 \\ k & = & 280 \end{array}$

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

$t = \frac{280}{n}$

(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 $t = 18$

$\begin{array}{rcl} 18 & = & \frac{280}{n} \\ n & = & \frac{280}{18} \\ n & = & 15.55 . . . \end{array}$

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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