## What you need to know

Here we will be going through how to convert between fractions, decimals, and percentages (in all directions).

**Decimals and Percentages**

Converting between decimals and percentages is nice and straightforward.

• Convert decimal to percentage – multiply by 100 (shift decimal point right two places).

• Convert percentage to decimal – divide by 100 (shift decimal point left two places).

Example: a) Write 37% as a decimal.

b) Write 0.548 as a percentage.

a) To convert this to a decimal we will divide by 100, so we get

37 \div 100 = 0.37

b) To convert this to a percentage we will times by 100, so we get

0.548 \times 100 = 54.8\%

**Decimals and Fractions**

Converting between decimals and fractions is a little more work.

• Convert fraction to decimal – treat the fraction like a division and divide the number on the top by the number on the bottom. There are ways we can make this process easier, which we’ll see in the examples below.

• Convert decimal to fraction – write the decimal as a fraction with 1 on the bottom. Then, keep multiplying top and bottom by 10 until the decimal becomes a whole number.

Example: a) Write \frac{12}{25} as a decimal.

b) Write \frac{11}{8} as a decimal.

a) Dividing 12 by 25 doesn’t sound too pleasant, but there is a way we can change the fraction (before dividing top by bottom) to make life easier. Notice that 25\times 4=100 and dividing by 100 is straightforward. So, if we times top and bottom by 4 we get

\dfrac{12}{25}=\dfrac{48}{100}=0.48

So, we have successfully converted this fraction to a decimal.

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b) In this case, there is no nice shortcut. We’ll just have to divide 11 by 8 in whichever way feels most comfortable – here, we’ll go for the bus stop method (make sure to put lots of zeroes after your decimal point!) Doing this, we get the picture shown on the right, and we see that the result is

\dfrac{11}{8} = 1.375

Example: Write 4.56 as a fraction in its simplest form.

Any number divided by 1 is equal to itself, so we can write 4.56 as \frac{4.56}{1}. Now, if we multiply top and bottom by 100 (or multiply by 10 twice, if you’re not sure why we choose 100), and we’ll see that we get

\dfrac{4.56}{1}=\dfrac{4.56\times 100}{1\times 100}=\dfrac{456}{100}

Success! We’ve written the decimal as a fraction. All that remains now is to simplify it. Cancelling out factors (until there are no common factors left), we get

\dfrac{456}{100} = \dfrac{228}{50} = \dfrac{114}{25}

**Fractions and Percentages**

We’ll use the tools we’ve already learned to help us convert between fractions and percentages.

• Convert percentages to fractions – a percentage is already out of 100, so we must put the value in a fraction over 100. Then, if necessary, multiply top and bottom by powers of 10 to make the values into whole numbers.

• Convert fractions to percentages – firstly convert the fraction to a decimal (using the method we’ve seen), then convert that decimal to a percentage (also using the method we’ve seen).

Example: Write 48.1% as a fraction.

As mentioned, percentages are already out of 100 so

48.1\%=\dfrac{48.1}{100}

Then, multiplying top and bottom by 10 (to make the numbers whole) we get

48\% = \dfrac{481}{1000}

Example: Write \frac{4}{5} as a percentage.

Firstly, let’s convert it to a decimal. Notice that if we times top and bottom by 2, the fraction becomes

\dfrac{4}{5}=\dfrac{4\times 2}{5\times 2} = \dfrac{8}{10}

Now, dividing by 10 isn’t too tricky: 8\div 10 = 0.8. Then, to convert this decimal to a percentage, we times by 100:

0.8 \times 100 = 80\%

NOTE: Alternatively, if you can write a fraction with 100 on the denominator, then the value on the top immediately gives you what the fraction would be as a percentage. For example,

\dfrac{34}{50}=\dfrac{68}{100}=68\%

### Example Questions

1) Write \frac{17}{40} as a decimal.

We will treat the fraction like a division and divide 17 by 40. Doing the bus stop method, we see

Therefore, this fraction is equal to the decimal: 0.425.

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2) Write 16.4% as a fraction in its simplest form.

So, we start by write this as a fraction over 100: \dfrac{16.4}{100}.

Then, to make the numbers whole, we will times top and bottom by 10:

\dfrac{16.4\times 10}{100 \times 10}=\dfrac{164}{1,000}

Now, we simplify until there are no common factors left. So, we get the answer to be

\dfrac{164}{1,000}=\dfrac{82}{500}=\dfrac{41}{250}

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3) Write the following numbers in order from smallest to largest.

0.46,\,\,\,\,\,44.6\%,\,\,\,\,\,\dfrac{9}{20},\,\,\,\,\,0.466

To compare the sizes of these numbers, we need to have them all in the same form. Specifically, numbers are easiest to compare if they’re all decimals. So, let’s convert the percentage to a decimal first. Dividing by 100 we see

**44.6\% = 44.6\div 100 = 0.446**

Next, convert the fraction to a decimal. Notice that if we multiply top and bottom of the fraction by 5, it becomes

\dfrac{9}{20}=\dfrac{9\times 5}{20\times 5}=\dfrac{45}{100}

Dividing by 100 is fairly easy, so we get

\dfrac{45}{100}=0.45

Now, all the 4 numbers in decimal form are: 0.46, 0.446, 0.45, 0.466. From smallest to largest, this is

0.446,\,\,\,0.45,\,\,\,0.46,\,\,\,0.466

Finally, writing the values in order and in their original forms, we get

44.6\%,\,\,\,\dfrac{9}{20},\,\,\,0.46,\,\,\,0.466

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