Percentages

Lesson

We know that whole numbers like $514$514 mean a number with $5$5 hundreds, $1$1 ten and $4$4 ones. Can you see that the units get smaller and and smaller as we go towards the right? But what does a decimal such as $0.45$0.45 mean? Well guess what, we can think of decimals in the same way so $0.45$0.45 can be thought of a number with $4$4 tenths and $5$5 hundredths. Check the diagram below if you're confused!

Well what does this mean when we want to switch between decimals and percentages? Let's see how It actually makes things a lot easier!

Let's take $0.4$0.4 for example. This means we have $0$0 ones and $4$4 tenths. In maths terms that means $4\times\frac{1}{10}$4×110 which equals $\frac{4}{10}$410, and we know this fraction can be converted to $40%$40%!

What about something like $0.75$0.75? Well, we can use the same steps. This number means $7\times\frac{1}{10}+5\times\frac{1}{100}$7×110+5×1100 which becomes $\frac{75}{100}$75100 and that is $75%$75%.

So what can we see so far from these kind of problems? Well what we've done is actually moved the decimal point two places to the right and transformed it into a percentage in both cases. This is because we have actually **multiplied the decimal by $100%$100%**. Try and multiple $0.4$0.4 and $0.75$0.75 by $100%$100% in your calculator and see what you get!

**Express **$0.6$0.6 as a percentage

**Think **of where the decimal place will be

**Do: **$0.6\times100%=60%$0.6×100%=60% (we moved the decimal point two places to the right)

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.

Remember!

Decimal -> Percentage: multiply by $100%$100%

Let's take a quick look at an example, say $0.47$0.47.

To convert this into a percentage let's multiply it by $100%$100%, so

$0.47\times100%$0.47×100% | $=$= | $0.47\times100%$0.47×100% |

$=$= | $47%$47% |

Can you see that multiplying a decimal by $100$100 is the same as moving the decimal point two places to the right?

Let's also summarise this interesting phenomenon:

Remember!

Decimal → Percentage: move the decimal place **right **$2$2 places

These simplified rules are extremely useful when we have more complicated problems, like the examples below:

**Convert **$0.0507$0.0507 into a percentage

**Think **about whether your final answer will be a whole number

**Do: **Let's use the technique of moving the decimal point

So looking from the diagram we know that: $0.0507\times100%=5.07%$0.0507×100%=5.07%

**What percentage** of the whole week does Lana work if she works $0.8$0.8 of the working week?

**Think **about working completely in percentages

**Do: **

$0.8$0.8 | $=$= | $80%$80% |

$\frac{80%\times5}{7}$80%×57 | $=$= | $\frac{400}{7}$4007 $%$% |

$=$= | $57\frac{1}{7}$5717 $%$% |

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.

**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% |

Convert $0.4$0.4 to a percentage.

Convert $0.51$0.51 to a percentage.