2. Rational numbers

Lesson

We use percentages everywhere in our daily lives, from taxes to discounts to nutritional information. We know that a $50%$50% discount will save us money, and $95%$95% sugar-free drink is healthier than a $20%$20% sugar-free drink, but what do these numbers really mean?

Percentage

A percentage is an amount out of $100$100, denoted by the symbol $%$%.

We know from the definition that $1%$1% represents $1$1 out of $100$100. In other words, $1%$1% is equal to one hundredth.

We have encountered hundredths before when looking at place values and there were two different ways that we represented them:

$\frac{1}{100}$1100 and $0.01$0.01

Both of these ways to write one hundredth are also ways to write $1%$1% and we will be using these ways to helps us convert between percentages, fractions and decimals.

Consider the grid below.

How many squares are shaded?

What percentage of the grid is shaded?

What fraction of the grid does this percentage represent?

one quarter

Aone tenth

Bone half

Cone fifth

Done quarter

Aone tenth

Bone half

Cone fifth

D

We can convert quite easily between fractions and percentages by remembering what percentages represent. Percentages represent a value out of $100$100, and any value out of $100$100 can be written as a fraction with the denominator $100$100.

For example, $47%$47% represents "$47$47 out of $100$100" which can be written as the fraction $\frac{47}{100}$47100.

Similarly, $\frac{33}{100}$33100 represents "$33$33 out of $100$100" which can be written as the percentage $33%$33%.

As we can see, using a denominator of $100$100 can help us convert between these two types of values.

Percentages into fractions

We can convert any percentage into a fraction by writing the percentage value as the numerator and $100$100 as the denominator. After doing this, we can simplify the fraction to get it into its simplest form.

Write $24%$24% as a fraction in its simplest form.

**Think**: We can convert our percentage into a fraction by writing the value in the percentage as the numerator and writing $100$100 as the denominator.

**Do**: Since the value in the percentage in $24$24, our fraction will be:

$\frac{24}{100}$24100

We can then simplify the fraction by dividing both the numerator and denominator by their greatest common factor $4$4 to get:

$\frac{6}{25}$625

Since $6$6 and $25$25 share no common factors except for $1$1, this fraction is in its simplest form.

Write $250%$250% as a mixed number in its simplest form.

**Think**: Again, we can convert our percentage into a fraction by writing the value in the percentage as the numerator and $100$100 as the denominator.

**Do**: Since the value in the percentage is $250$250, our fraction will be:

$\frac{250}{100}$250100

We can then simplify the fraction by dividing both the numerator and denominator by their greatest common factor $50$50 to get:

$\frac{5}{2}$52

Since $5$5 and $2$2 share no common factors except for $1$1, this fraction cannot be simplified further. However, the question asks us to write it as a mixed number rather than an improper fraction.

Since $5\div2$5÷2 is equal to $2$2 remainder $1$1, we can write $2$2 as the whole number part and $1$1 as the numerator of the mixed number. This will give us:

$2\frac{1}{2}$212

**Reflect**: Notice that in both cases we followed the same steps to convert from a percentage into a fraction. Regardless of what the value in the percentage is, we can always represent it as a fraction with denominator $100$100.

To convert a fraction into a percentage, we can just reverse these steps:

Fractions into percentages

We can convert any fraction into a percentage by finding its equivalent fraction that has a denominator of $100$100. After this, we can write the value in the numerator followed by the $%$% symbol to represent the percentage.

Write $\frac{3}{4}$34 as a percentage.

**Think**: We want to find the fraction that is equivalent to $\frac{3}{4}$34 and has a denominator of $100$100. We can do this by multiplying both the numerator and denominator by the same number such that the denominator becomes $100$100.

**Do**: Since $4\times25$4×25 is equal to $100$100, we know that we need to multiply the numerator and denominator of our fraction by $25$25. This will give us:

$\frac{75}{100}$75100

Now that the fraction has a denominator of $100$100, we can convert it to a percentage by attaching a $%$% symbol to the numerator, giving us:

$75%$75%

Write $1\frac{42}{200}$142200 as a percentage.

**Think**: To convert this mixed number into a percentage, we need to make it an improper fraction with a denominator of $100$100.

**Do**: Remembering the $1\frac{42}{200}$142200 is equal to $1+\frac{42}{200}$1+42200. We can then use the fact that $1$1 is equivalent to $\frac{200}{200}$200200. This means that $1\frac{42}{200}$142200 is equal to $\frac{200}{200}+\frac{42}{200}$200200+42200, which we can evaluate to be the improper fraction:

$\frac{242}{200}$242200

We can then simplify this fraction to have a denominator of $100$100 by dividing both the numerator and denominator by $2$2. This will give us:

$\frac{121}{100}$121100

Now that the fraction has a denominator of $100$100, we can convert it to a percentage by attaching a $%$% symbol to the numerator, giving us:

$121%$121%

**Reflect**: In both examples we aimed to find the equivalent fraction that had a denominator of $100$100. This is because putting the fraction into this form allows us to convert it directly to a percentage.

It should be noted that, in the second example, while we converted from a mixed number to an improper fraction, we do not have to do this.

Remembering that one whole is equivalent to $100%$100%, we could have ignored the whole number part of the mixed number and added on $100%$100% after converting the fraction part into a percentage.

We can convert between decimals and percentages by taking advantage of the hundredths place value. We know that $1%$1% represents $1$1 hundredth which we can write as $0.01$0.01 as a decimal. Using the same logic, we can convert larger percentages.

Percentages into decimals

We can convert any percentage into a decimal by dividing the percentage value by $100$100, which is equivalent to decreasing the place value of each digit by two places, and removing the $%$% symbol.

For example, $83%$83% represents $83$83 hundredths. This is $0.83$0.83 when written as a decimal.

Similarly, $0.65$0.65 is equal to $65$65 hundredths. We can write this as the percentage $65%$65%.

As we can see, We can convert from a percentage into a decimal by thinking of the percentage value as a number of hundredths.

Write $179%$179% as a decimal.

**Think**: Since $179%$179% represents $179$179 hundredths, the decimal we are converting to should also represent $179$179 hundredths.

**Do**: We can write the decimal representing $179$179 hundredths by filling in the place values with the digits of $179$179, starting from the hundredths place. This will give us:

$1.79$1.79

Write $23.5%$23.5% as a decimal.

**Think**: Since $23.5%$23.5% represents $23.5$23.5 hundredths, the decimal we are converting to should also represent $23.5$23.5 hundredths. How can we represent half a hundredth?

**Do**: We can represent half a hundredths with $5$5 thousandths. This means that the decimal we convert to should represent $23$23 hundredths and $5$5thousandths. This will give us:

$0.235$0.235

**Reflect**: In both examples we wrote the percentage as a decimal representing the number of hundredths indicated by the percentage.

Notice that converting the percentages into decimals had the same effect as decreasing the place value of the digits in the percentage by two places, then removing the $%$% symbol. This is equivalent to dividing by $100$100 and removing the $%$% symbol.

To convert from a decimal into a percentage, we can just reverse the steps.

Decimals into percentages

We can convert any decimal into a percentage by multiplying the decimal by $100$100, which is equivalent to increasing the place value of each digit by two places, and attaching a $%$% symbol.

Write $0.314$0.314 as a percentage.

**Think**: We can convert this decimal into a percentage by increasing the place value of each digit by two places and attaching a $%$% symbol to the result.

**Do**: Increasing the place value of each digit by two places will give us:

$31.4$31.4

Then we can attach the $%$% symbol to give us:

$31.4%$31.4%

Caution

A percentage is limited to representing hundredths, so smaller units like thousandths cannot be represented by whole number percentages.

While this conversion was shown in two steps, both steps should be done at the same time. This is because while $0.314=31.4%$0.314=31.4%, the middle step is not actually true: $0.314\ne31.4$0.314≠31.4.

Remember to attach the $%$% symbol to decimal at the same time as increasing the place values. This also applies for when we are reversing these steps to convert percentages into decimals.

Write $3.9$3.9 as a percentage.

**Think**: We can convert this decimal into a percentage by increasing the place value of each digit by two places and attaching a $%$% symbol to the result.

**Do**: Increasing the place value of each digit by two places will give us $390$390 (remembering to fill any empty places with a zero).

As we have been warned, we do not write this as a step. Instead, we attach the $%$% symbol before writing it as a percentage, giving us:

$390%$390%

**Reflect**: In both examples, we increased the place value of each digit in the decimal by two places and then attached a $%$% symbol, filling in empty places with zeros where necessary.

As with any skill in mathematics, there is always another way. While the methods above use the meaning of percentages, fractions and decimals to make the conversions, another way to convert between them is to treat the$%$% symbol like a unit.

What does this mean?

Remember that $100%$100% is equal to one whole.

This means that we can convert from percentages by dividing by $100%$100% and convert into percentages by multiplying by $100%$100%.

Write $3%$3% as a decimal.

**Think**: We can convert from a percentage by dividing by $100%$100%.

**Do**: When dividing $3%$3% by $100%$100%, the $%$% symbols will cancel out to give us:

$3\div100$3÷100

Since dividing by $100$100 has the same effect as decreasing the place value of each digit by two places, $3\div100$3÷100 will be equal to:

$0.03$0.03

Write $\frac{3}{8}$38 as a percentage.

**Think**: We can convert to a percentage by multiplying by $100%$100%.

**Do**: Multiplying $\frac{3}{8}$38 by $100%$100% will give us:

$\frac{3}{8}\times100%$38×100%

Now that we have a percentage value, we just need to evaluate the multiplication. Remembering that $100$100 is equal to $\frac{100}{1}$1001, we can multiply the numerators together and the denominators together. This will give us:

$\frac{300}{8}%$3008%

We can simplify this fraction by dividing both the numerator and denominator by their greatest common factor $4$4 to give us:

$\frac{75}{2}%$752%

Since $75\div2$75÷2 is equal to $37$37 remainder $1$1, we can write $37$37 as the whole number part and $1$1 as the numerator of the mixed number. This will give us the mixed number percentage:

$37\frac{1}{2}%$3712%

Or we could write it as the decimal percentage:

$37.5%$37.5%

**Reflect**: Since $100%$100% is equal to $1$1, multiplying or dividing by it does not change the value of the numbers we are converting, it only changes whether they are percentage values or not.

Caution

Although we are treating the $%$% symbol like a unit, it is **not** a unit. This is because it represents "out of $100$100" which is not a unit of measurement.

These are some common conversions that we can remember to help us convert between percentages, fractions and decimals.

Convert between percentages, fractions and decimals to complete the table below.

Write the answers as mixed number percentages and simplified mixed numbers where necessary.

Fraction Decimal Percentage $\frac{11}{100}$11100 $\editable{}$ $\editable{}$ $\editable{}$ $1.83$1.83 $\editable{}$ $\frac{5}{8}$58 $\editable{}$ $\editable{}$

Represent and determine equivalencies among fractions, mixed numbers, decimals, and percents (on the state assessment, items measuring this objective are assessed without the use of a calculator)