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!

Going back from percentages to decimals is quite simple if we work backwards. Remember that $30%$30% is $\frac{30}{100}$30100. So, $30%=\frac{30}{100}$30%=30100 $=$= $0.3$0.3. You may notice that this is the same as decreasing the place value of each digit by two places. To look at this another way, $30%$30% is $\frac{30}{100}=\frac{3}{10}$30100=310 as a fraction, which can be described as $3$3 tenths. This also describes the decimal $0.3$0.3.

Since the percent sign represents the phrase "parts of $100$100", we can always convert percentages to decimals by dividing by $100$100. This can always be done by decreasing the place value of each digit in the percentage by two places.

**Calculate **$85%$85% as a decimal

**Think **about decreasing the place value of each digit, especially after looking at the previous example.

**Do: **$85%=\frac{85}{100}$85%=85100 = $0.85$0.85 (decreasing the place value of each digit by two places)

Remember!

Percentage → Decimal: turn to a fraction (since % means out of 100) and then convert to decimal.

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

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

$=$= | $0.06$0.06 |

We know that dividing by $100$100 is the same as decreasing the place value by two places!

Remember!

Percentage → Decimal: decrease the palce value of each digit by $2$2 places

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

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

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

**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 moves the decimal point two places, what do you think will happen if it was $10$10 or $1000$1000?

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**.

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.

For example: when converting $4.2%$4.2% into a decimal we drop the percent sign and divide by $100$100. We can do this by decreasing the place value of each digit by two places. But notice, this has the same effect as moving the decimal point two places to the left!

Similarly, when multiplying numbers by $100$100 we can move the decimal point two places to the right to match the effect of increasing the place value of each digit by two places.

**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% | $=$= | $\frac{0.17}{100}$0.17100 |

$=$= | $0.0017$0.0017 | |

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

$=$= | $0.054$0.054 |

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%