**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*

- Other problems may involve a variable
- For example
*y*is directly proportional to the square of*x*means that*y*=*kx*^{2}*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*=*kx*^{3}- Each of these would have a different type of graph, depending on the function

- 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 =***k*B - If
**A**is**directly**proportional to the square of**B**use the formula**A =***k*B^{2}

- For example if
- STEP 2
**Find**by substituting the values given in the question into your formula and solving**k** - 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

- 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

*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.

We are told this is DIRECT proportion.

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

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

Use this formula to find *y* when *x* = 4.

**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 two quantities are
- If
*x*is inversely proportional to*y,*then- will always be the same
- there will be some value of
*k*such that - the graph will be related to that graph of

- 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 - If
**A**is**inversely****B**use the formula

- For example if
- STEP 2
**Find**by substituting the values given in the question into your formula and solving**k** - 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

- STEP 1

The time, hours, it takes to complete a project varies inversely proportional to the number of people working on it, .

If 4 people work on the project it takes 70 hours to complete.

(a)

Write an equation connecting and .

Identify the two variables

We are told this is INVERSE proportion

We can now find k using and (given in words in the question)

We can now write the full formula/equation in and

(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 when

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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