## Direct proportion

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 |

**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 |

**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 |

^{2}= 5 then the formula is; So when w = 15 what is r?

## Summary

if d ∝ v we have found out that;Number of boxes | 2 | 4 | 6 | 40 |

Discount (£) | 4 | 8 | 12 | 80 |

**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 |

**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 |

^{2}= 5 then the formula is; So when w = 15 what is r?

## Summary

if d ∝ v we have found out that;#### Objectives

A list of sections

#### Tips

A list of tips