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Home > Maths I > Relationships > Direct variation

Maths I

Direct variation

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Finding the variation formula

Variation is another word for proportion. If two quantities A and B are in proportion to one another, we say that A varies directly as B and write:

A \propto B

Example

Let A stand for the amount in pence you pay for bananas. Let B be the number of bananas you buy.

Study the table.

Number of bananas (B)123456
Amount or cost (A)20406080100120

Notice that if you double the number of bananas you buy, you double the amount you have to pay. Or if you multiply the number of bananas by 3, the amount you pay is also multiplied by 3. And so on.

Bananas.

So A varies directly as B -

A \propto B

We can now write this as A = kB, where k is the constant of variation.

Choose any pair of values from the table above, say B = 3 and A = 60.

Put these values into A = kB and we get -

60 = k \times 3 or 3k = 60

Therefore, k = 20 i.e. the constant of variation is 20.

We can now write A = 20B and this is called the variation formula.

The graph of A against B is shown below.

A graph showing how the greater the number of bananas purchased (axis called b), the more pence has to be spent (axis called a). Number of bananas goes from 0 to 6, and each one costs 20 pence, ending up at 6 bananas costing 120 pence.

Notice that the graph is a straight line through the origin.

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