Percentages

Lesson

Do you remember how to convert decimals into percentages and vice versa in previous chapters? It is quite simple, and we can summarise it as

Remember!

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

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

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 increasing the place value of the digits by $2$2 places?

Wow, I wonder if there's a similar thing happening when we divide!

Say we wanted to convert $6%$6% into a decimal, so

$6%$6% | $=$= | $\frac{6}{100}$6100 |

$=$= | $0.06$0.06 |

So in THIS case converting to a decimal ends up looking like the place value of each digit decreased by $2$2 places!

Let's also summarise this interesting phenomenon:

Remember!

Decimal → Percentage: **increase** the place value of each digit by $2$2 places

Percentage → Decimal: **decrease** the place value of each digit by $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 changing the place values of digits.

Units | . | Tenths | Hundredths | Thousandths | Ten-Thousandths |
---|---|---|---|---|---|

$0$0 | . | $0$0 | $5$5 | $0$0 | $7$7 |

$5$5 | . | $0$0 | $7$7 |

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

Notice that we dropped the zeros that would have moved up to the Tens and Hundreds columns because they won't have any value. Remember to do this for any zeros at the start of the number to the left of the decimal point or at the end of the number to the right of the decimal point.

For example: $00$00$12.34=12.34$12.34=12.34 and $43.2100=43.21$43.2100=43.21

Similarly, if we have some empty places we fill them with zeros. We can see this in the next example.

**Express **$4.2%$4.2% as a decimal

**Think **about filling in empty places with zeros after changing the place values.

**Do**

Units | . | Tenths | Hundredths | Thousandths |
---|---|---|---|---|

$4$4 | . | $2$2 | ||

$0$0 | . | $0$0 | $4$4 | $2$2 |

$4.2%$4.2% | $=$= | $4.2\div100$4.2÷100 |

$=$= | $0.042$0.042 |

What do you think?

If dividing and multiplying decimals by $100$100 changes the place value of digits by $2$2 places, what do you think will happen if it was $10$10 or $1000$1000?

We know that % just means 'over a hundred'. So multiplying by $100%$100% is the same as multiplying by $\frac{100}{100}$100100 which 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.

Let's say we know that Sam the postman works $60%$60% of the working week (Monday to Friday), but what if we wanted to see how much he works compared to the *whole *week, knowing only the $60%$60% part?

Well we could convert the $60%$60% back into a decimal, which would become $0.6$0.6. $0.6\times5=3$0.6×5=3, so this means that out of the five weekdays, he works $3$3. This means he works $\frac{3}{7}$37 of the whole week, which is $\frac{3}{7}\times100%=42\frac{6}{7}$37×100%=4267%.

Is this a smaller or bigger percentage than of the working week? Why do you think this is so?

So in this example we proceeded like this:

1. convert to a decimal

2. multiply by $5$5

3. divide by $7$7

4. convert back to percentage

Because we converted the percentage to something else and back to a percentage in the end, we can discard steps 1 and 4. So in the end, all we have to do is multiply by $5$5 and divide by $7$7! This is the same as **multiplying by $\frac{5}{7}$57**.

**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 $%$% |

Convert the percentage $749%$749% into a decimal.

Interest rates are often listed as percentages, but decimal values are needed when using them in calculations. What is the decimal representation of each of the following interest rates?

$12%$12%

$8.2%$8.2%

$7.14%$7.14%