4.01 Converting between percentages, fractions, and decimals

We know from the definition that 1\% represents 1 out of 100. In other words, 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: \dfrac{1}{100} and 0.01.

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

Examples

Example 1

Consider the grid below.

a

How many squares are shaded?

Worked Solution

Create a strategy

Count the number of squares shaded.

Apply the idea

There are 5 rows of 10 squares shaded so there are 50 squares shaded.

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b

What percentage of the grid is shaded?

Worked Solution

Create a strategy

Find the fraction shaded then convert to a percentage.

Apply the idea

We have 50 out of 100\ squares in the grid are shaded. We can express this as fraction \dfrac{50}{100}.

\displaystyle \text{Percent}	\displaystyle =	\displaystyle \frac{50}{100} \times 100 \%	Multiply by 100\%
	\displaystyle =	\displaystyle 50\%	Evaluate

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c

What fraction of the grid does this percentage represent?

A

One quarter

B

One tenth

C

One half

D

One fifth

Worked Solution

Create a strategy

Simplify the fraction from part (a).

Apply the idea

\displaystyle \frac{50}{100}

\displaystyle =

\displaystyle \frac{1}{2}

Simplify the fraction

The correct option is C.

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

For example, 33\% represents "33 out of 100" which can be written as the fraction \dfrac{33}{100}.

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

Examples

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

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

Examples

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

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

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

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

Examples

To convert from a decimal into a percentage, we can just reverse the above steps. We can convert any decimal into a percentage by multiplying the decimal by 100, which is equivalent to increasing the place value of each digit by two places, and attaching a \% symbol.

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

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.

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.

Remember that 100\% is equal to one whole. This means that we can convert from percentages by dividing by 100\% and convert into percentages by multiplying by 100\%.

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

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

Examples

Now that we are able to convert between these three ways to represent values, we can compare them.

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

Examples