Middle Years

# 4.02 Conversions

Lesson

### Percentages and fractions

Every percentage can be thought of as a fraction with a denominator of $100$100.  The word percent actually comes from per centum, which in Latin means for every one hundred.

For example, $3%$3% would mean $3$3 for every $100$100, which is another way of saying $3$3 out of $100$100

To convert any percentage to a fraction, take the number in front of the percent sign and put it as the numerator of a fraction with a denominator of $100$100, or in other words, divide by $100$100.

### Special cases

$33\frac{1}{3}$3313% and $66\frac{2}{3}$6623% are special percentages. Try and put $\frac{1}{3}$13 and $\frac{2}{3}$23 into your calculator and seeing what decimal it becomes! Now try putting those percentages into the calculator. All four values turn into one of two recurring decimals $0.3333$0.3333... and $0.6666$0.6666... So it's important to remember that $33\frac{1}{3}$3313% = $\frac{1}{3}$13 and $66\frac{2}{3}$6623% = $\frac{2}{3}$23.

### Mixed numbers and percentages

Not all fractions turn into nice percentage values, for example, $\frac{4}{7}$47. The usual steps to convert this to a percentage would be to multiply it by $100%$100% .  This gives: $\frac{4}{7}\times100%=\frac{400}{7}%$47×100%=4007%.

Because this is an improper fraction percentage, it's hard to understand it when looking at it straight away, that's why it'll be easier to change it into a mixed number, which is $57\frac{1}{7}%$5717%. Now we can look at it straight away and understand this is around $57%$57%

#### Worked examples

##### Example 1

Convert $\frac{3}{5}$35 into a percentage.

Do:

 $\frac{3}{5}$35​ $=$= $\frac{3}{5}\times100%$35​×100% To convert a decimal or fraction to a percentage multiply by $100%$100% $=$= $\frac{300%}{5}$300%5​ Multiply numerators $=$= $60%$60% Simplify if possible
##### Example 2

Convert $\frac{16}{3}$163 into a percentage

Remember that you can have percentages that are more than $100%$100%

Do:

  $=$= $\frac{16}{3}\times100%$163​×100% Multiply by $100%$100% $=$= $\frac{1600%}{3}$1600%3​ Multiply numerators $=$= $533\frac{1}{3}%$53313​% Simplify
##### Example 3

Express $\frac{4}{13}$413 as a percentage, rounded to $2$2 decimal places

Do:

 $\frac{4}{13}$413​ $=$= $\frac{4}{13}\times100%$413​×100% Multiply by $100%$100% $=$= $\frac{400%}{13}$400%13​ Multiply numerators $=$= $30.7692\dots%$30.7692…% Evaluate and consider if you need to round up or down $=$= $30.77%$30.77% 2 d.p. Round to $2$2 decimal places

##### Example 4

Convert $65%$65% into a fraction.

Do:

 $65%$65% $=$= $\frac{65}{100}$65100​ To convert a percentage to a fraction or decimal divide by $100$100 $=$= $\frac{13}{20}$1320​ Simplify by dividing both the numerator and denominator by a common factor of $5$5

Fractions to percentages

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

Percentage → Fraction:     divide by $100%$100%

#### Practice questions

##### Question 1

Convert $\frac{3}{4}$34 into a percentage.

##### Question 2

Xanthe and Jimmy are spellchecking an article before it is printed. Xanthe checks $\frac{3}{5}$35 of the article and Jimmy checks $34%$34% of the article.

1. What percentage of the article have they checked altogether?

2. What percentage still needs to be checked?

#### Worked examples

##### example 5 - tricky decimal points

Express $5.8%$5.8% as a simplified fraction

Think about how to get rid of that decimal point in the without changing the answer.

Do:

 $5.8%$5.8% $=$= $\frac{5.8}{100}$5.8100​ To convert to a fraction divide by $100$100 $=$= $\frac{58}{1000}$581000​ Multiply the numerator and denominator by $10$10 $=$= $\frac{29}{500}$29500​ Simplify by dividing the numerator and denominator by a common factor of $2$2
##### example 6 - mixed numbers

QuestionWhat is $12\frac{4}{5}$1245 $%$% as a fraction?

Think: Let's change the mixed number into a single fraction first.

Do:

 $12\frac{4}{5}%$1245​% $=$= $\frac{64}{5}%$645​% Convert to a single improper fraction (think:$5\times12+4=64$5×12+4=64fifths) $=$= $\frac{64}{5}\div100$645​÷100 Convert to a fraction by dividing by $100$100 $=$= $\frac{64}{5}\times\frac{1}{100}$645​×1100​ Change to multiplication by the reciprocal $=$= $\frac{64}{500}$64500​ Evaluate $=$= $\frac{16}{125}$16125​ Simplify by dividing the numerator and denominator by a common factor of $4$4

#### Practice questions

##### Question 3

Express the fraction $\frac{9}{22}$922 as a percentage, writing your answer with two decimal places.

##### Question 4

Convert the mixed number $2\frac{16}{25}$21625 to a percentage.

### Percentages and decimals

You can convert percentages to decimals in just one step, but you might also find it helpful to think about a percentage as a fraction first, and then a decimal.

To convert a percentage to a decimal, the key is to remember that 'per cent' means 'per one hundred'.  So every percentage is a value in the hundredths.

Did you know?

You can convert a percentage to a decimal in one go, or you can convert first to a fraction, and then a decimal.

#### Examples

Percentage to decimal:

$40%=\frac{40}{100}=0.4$40%=40100=0.4

$81%=\frac{81}{100}=0.81$81%=81100=0.81

$132%=\frac{132}{100}=1.32$132%=132100=1.32

$31.5%=\frac{31.5}{100}=0.315$31.5%=31.5100=0.315

$24\frac{1}{4}%=\frac{24.25}{100}=0.2425$2414%=24.25100=0.2425

Decimal to percentage:

$0.01=1%$0.01=1%

$0.1=0.10=10%$0.1=0.10=10%

$0.23=23%$0.23=23%

$0.815=81.5%$0.815=81.5%

$1.63=163%$1.63=163%

Note: a value in the units place means that we end up with a percentage in the hundreds.

Decimals to percentages

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

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

#### Practice questions

##### Question 5

A bushfire moves through an area of land, burning $20%$20% of the land.

1. Express how much of the land is burnt as a decimal to one decimal place.

2. Express how much of the land is not burnt as a decimal to one decimal place.

##### Question 6

Convert $60%$60% to a decimal.

##### Question 7

Convert $0.51$0.51 to a percentage.

### Comparing and ordering fractions, percentages and decimals

When comparing any values, we always need to have them all in the same form. This means that when we are comparing percentages, fractions and decimals, we need to convert them so that they are all of one type.

#### Worked example

##### Example 7

Arrange the following in the ascending order: $34%$34%, $\frac{7}{20}$720, $0.345$0.345

Think: If we want all three values to be of the same type, they must either all be percentages, all be fractions with the same denominator, or all have the same number of decimal places.

Do: For this example, we will convert them all to be fractions with a denominator of $100$100.

$34%=\frac{34}{100}$34%=34100

$\frac{7}{20}=\frac{35}{100}$720=35100

$0.345=\frac{34.5}{100}$0.345=34.5100

Now that the values are all of the same type, we can compare them. The largest is $\frac{7}{20}$720 and the smallest is $34%$34%. So arranging them in ascending order gives us:

$34%$34%, $0.345$0.345, $\frac{7}{20}$720

Reflect: In order to compare the three values we converted them to all be of the same form.

#### Practice question

##### Question 8

In this question we will be working with the numbers $\frac{1}{4}$14, $60%$60% and $0.3$0.3.

1. Convert $\frac{1}{4}$14 into a percentage. Do not round your answer.

2. Convert $0.3$0.3 into a percentage.

3. Which of the following arranges $\frac{1}{4}$14, $60%$60% and $0.3$0.3 from largest to smallest?

$60%$60%, $\frac{1}{4}$14, $0.3$0.3

A

$60%$60%, $0.3$0.3, $\frac{1}{4}$14

B

$\frac{1}{4}$14, $60%$60%, $0.3$0.3

C

$0.3$0.3, $\frac{1}{4}$14, $60%$60%

D

$60%$60%, $\frac{1}{4}$14, $0.3$0.3

A

$60%$60%, $0.3$0.3, $\frac{1}{4}$14

B

$\frac{1}{4}$14, $60%$60%, $0.3$0.3

C

$0.3$0.3, $\frac{1}{4}$14, $60%$60%

D