Percentages

Lesson

I'm pretty confident that we've come up with a solid technique that helps us convert from percentages to decimals and back again, can you remember what it is?

It can be summarised neatly as follows:

Decimal → Percentage: multiply by $100%$100%

Percentage → Decimal: change to a fraction (number out of 100) and then convert to a decimal.

A neat trick when multiplying or dividing numbers by $100$100 is that you can think of it as 'moving' the decimal point two places left or right (left for division, right for multiplication). Try testing it out yourself! Of course technically the decimal point never moves but we can think of it as a visual aid.

**Convert **the following percentages into decimals: $49%$49%, $308%$308%, $0.17%$0.17%, $5.4%$5.4%

**Think** about using the 'decimal point trick' to divide by $100%$100% quickly and easily

**Do**

$49%$49% | $=$= | $\frac{49}{100}$49100 |

$=$= | $0.49$0.49 | |

$308%$308% | $=$= | $\frac{308}{100}$308100 |

$=$= | $3.08$3.08 | |

$0.17%$0.17% | $=$= | $0.17\div100$0.17÷100 |

$=$= | $0.0017$0.0017 | |

$5.4%$5.4% | $=$= | $\frac{5.4}{100}$5.4100 |

$=$= | $0.054$0.054 |

**Express** the following decimals as percentages: $8.2$8.2, $0.15$0.15, $0.0709$0.0709, $0.4$0.4

**Think** again about using the 'decimal point trick' for multiplication by $100%$100%

**Do**

$8.2$8.2 | $=$= | $8.20$8.20 |

$=$= | $8.20\times100%$8.20×100% | |

$=$= | $820%$820% | |

$0.15$0.15 | $=$= | $0.15\times100%$0.15×100% |

$=$= | $15%$15% | |

$0.0709$0.0709 | $=$= | $0.0709\times100%$0.0709×100% |

$=$= | $7.09%$7.09% | |

$0.4$0.4 | $=$= | $0.40\times100%$0.40×100% |

$40%$40% |

We know that % just means 'over a hundred'. So multiplying by $100%$100% is the same as multiplying by $\frac{100}{100}$100100which is just $1$1. And since multiplying any number by $1$1 gives you the exact same number then we know that we haven't changed the value when multiplying by $100%$100%- all we're doing is converting it to a percentage.

Reason with linear proportions.