2.13 Convert fractions, decimals and percents

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 help 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 shaded squares in the row and the number of rows.

Apply the idea

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

5\times 10 = 50

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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 shaded. We can express this as fraction \dfrac{50}{100}.

Recall that percent means parts per hundred.

This means that \dfrac{50}{100}= 50\%

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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

Observe the size of the shaded part of the grid model.

We can also find an equivalent fraction in (b) with a numerator of 1 (one).

Apply the idea

There are 50 shaded squares out of 100 squares. This can be written as \dfrac{50}{100} as a fraction. Looking at the model we can see that the shaded part is half the grid.

The numerator and denominator of \dfrac{50}{100} can also be divided by 50 to find an equivalent fraction with one in the numerator.

\displaystyle \frac{50 \div 50}{100\div 50}

\displaystyle =

\displaystyle \frac{1}{2}

Divide by 50 to find equivalent 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, 47\% represents "47 out of 100" which can be written as the fraction \dfrac{47}{100}.

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

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 in decimal form as 0.01. 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.

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.

For example, 0.08 is 8 hundredths or 8\%.

A percentage is limited to representing hundredths, so smaller units like thousandths cannot be represented by whole number percentages such as 0.0035 which is 0.35\%.

Remember to attach the \% symbol to decimal at the same time as increasing the place values.

Examples

We now know that decimals, fractions and percentages are just different ways of showing the same value:

1\% can be written as \dfrac{1}{100} or 1 hundredth or 0.01

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

Examples