Suppose a shop named Tesco is offering discounts on really expensive boxes of biscuit. The number of boxes (n) is directly proportional to the discount (d).
We write;

The following table shows a number of biscuit boxes and their discounts.

Number of boxes |
2 |
4 |
6 |
40 |

Discount (£) |
4 |
8 |
12 |
80 |

If you observe the table carefully you might be able to spot a pattern, when working with proportion in Maths you always have to identify the connection or the pattern in the values in question.

We know that the number of biscuits is directly proportional to the discount which means the more boxes a customer buys the more discount he/she will get. Suppose we wanted to work out the discount for 100 boxes of biscuits, we would need to find the formula for

**d ∝ n** first, then use to find any other value we want to find.
The pattern from our table above is that when we divide the number of boxes with the discount we always get a constant/fixed value. In proportions this value is often referenced as

**k**.

Number of boxes |
2 |
4 |
6 |
40 |

Discount (£) |
4 |
8 |
12 |
80 |

d/n |
2 |
2 |
2 |
2 |

If in our table d/n =2 then;

The formula for

**d ∝ n**
So to work out the discount you will get on a certain number of boxes, simply multiply the number of boxes with 2. For example;
If you buy 100 boxes of biscuits you will get a discount of;

The discount for 100 boxes of biscuits is £200.

## With square roots

This also applies in situations where a figure is proportional to a square root.
The flow rate of a water in a square pipe is proportional to the square of it’s width;
we say;

## Example

width (w^{2}) |
25 |
144 |

flow rate (r) |
125 |
720 |

r / w^{2} |
5 |
5 |

if r/w

^{2} = 5 then the formula is;

So when w = 15 what is r?

## Summary

if

d ∝ v we have found out that;

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Hmm, this is a really detailed and clear explanation on ‘direct proportion’. Keep up the good work and continue posting more of such examples for other topics where applicable, it will definitely help many students who are experiencing difficulties in Mathematics to understand these concepts better!

I will refer my students who are attending my Mathematics tuition lessons in Singapore to this website.

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Thanks for the comment. I will definitely be publishing more maths topics very soon.

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This helped! Alot!

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Thank you

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