Fractions, decimals and percentages

A percentage is a number out of 100, so 35% means 35/100. Fractions, decimals and percentages are three ways of writing the same amount.

To convert:

1. Decimal to percentage: × 100.
2. Percentage to decimal: ÷ 100.
3. Fraction to decimal: divide the top by the bottom.
4. Decimal to fraction: write it over 10, 100 or 1000, then simplify.

Use the steps from the table section, then simplify any fraction.

Example: write 0.35 as a fraction and as a percentage.

1. 0.35 has 2 decimal places, so 0.35 = 35/100
2. Divide top and bottom by 5: 7/20
3. 0.35 × 100 = 35%

Example: write 3/8 as a decimal and as a percentage.

1. 3 ÷ 8 = 0.375
2. 0.375 × 100 = 37.5%

Example: write 45% as a fraction in its simplest form.

1. 45% = 45/100
2. Divide top and bottom by 5: 9/20

To compare amounts written in different ways, change them all to decimals (or all to percentages) first.

A terminating decimal stops, like 0.375. A recurring decimal repeats forever, like 0.333…

Dots show which digits repeat:

• 0.3̇ means 0.333… (3 recurring)
• 0.1̇2̇ means 0.121212… (the dots mark the first and last digits of the repeating block)
• 0.416̇ means 0.41666… (only the 6 repeats)

A fraction in its simplest form gives a terminating decimal if its denominator has no prime factors other than 2 and 5.

Example: do 7/40 and 5/12 terminate?

1. 40 = 2³ × 5, so 7/40 terminates: 7 ÷ 40 = 0.175
2. 12 = 2² × 3, and 3 isn't 2 or 5, so 5/12 recurs: 5 ÷ 12 = 0.41666… = 0.416̇

Use algebra to make the repeating part cancel out.

1. Call the decimal x.
2. Multiply by 10 if one digit repeats, by 100 if two digits repeat, and so on.
3. Subtract x, so the recurring parts cancel.
4. Solve for x and simplify.

Example: write 0.2̇7̇ as a fraction.

1. x = 0.272727…
2. 100x = 27.272727…
3. 100x − x: 99x = 27
4. x = 27/99 = 3/11

Harder example: write 0.13̇ as a fraction.

1. x = 0.1333…
2. 10x = 1.333…
3. 10x − x: 9x = 1.2
4. x = 1.2/9 = 12/90 = 2/15