Fractions, decimals and percentages.
One quantity, several useful forms. Convert confidently without changing the value.
Recognise fraction types · simplify and convert · read recurring notation · convert recurring decimals
One quantity, different forms
Imagine a whole pizza cut into equal slices. A fraction describes how many of those equal parts you have. The denominator names the size of the parts; the numerator counts them. Equal-sized parts are essential to the meaning.
Decimals and percentages describe the same quantities in different languages. A decimal uses place value based on ten; a percentage counts hundredths. We will first recognise these forms, then change between them without changing the amount represented.
Fractions, decimals and percentages describe parts of a whole. Always identify what counts as the whole.
Proper fraction
The numerator is smaller than the denominator: its value is less than 1 for positive fractions.
Improper fraction
The numerator is greater than or equal to the denominator. 74 is greater than 1.
Mixed number
A whole number plus a proper fraction: 134 = 1 + 34 = 74.
Decimal
Place value shows tenths, hundredths, thousandths, … Here 0.75 = 75100.
Percentage
“Per hundred”: 75% = 75100. Percentages can exceed 100%; 125% = 1.25.
The same amount, written three ways. Each little square is 1100 of the whole, or 1%.
Pause and explainWhy must the pieces be equal when we call one piece 14?
The denominator 4 means four equal shares of the same whole. An unequal piece is not automatically one quarter.
Convert and simplify
Cutting each half of a cake into two pieces produces quarters, but the amount of cake has not changed. This is why 12 and 24 are equivalent: they describe the same share using different-sized parts.
Multiplying or dividing both parts of a fraction by the same non-zero number preserves its value. To convert to a decimal, divide numerator by denominator. To express a decimal as a percentage, count how many hundredths it represents by multiplying by 100.
Fraction ↔ decimal
Divide numerator by denominator: 3 ÷ 8 = 0.375.
Use place value in reverse: 0.375 = 3751000 = 38, dividing top and bottom by 125.
Decimal ↔ percentage
Decimal to percentage: multiply by 100.
0.375 = 37.5%.
Percentage to decimal: divide by 100.
12% = 0.12.
Write 225 as an improper fraction, decimal and percentage.
Multiply the whole number by the denominator, then add the numerator.
To reverse 125: divide 12 by 5. The quotient is 2 and the remainder is 2, giving 225.
| Fraction | Decimal | Percentage |
|---|---|---|
| 12 | 0.5 | 50% |
| 14 | 0.25 | 25% |
| 34 | 0.75 | 75% |
| 15 | 0.2 | 20% |
| 18 | 0.125 | 12.5% |
| 13 | 0.333… | 3313% |
| 23 | 0.666… | 6623% |
Pause and explainWhy is 0.4 equal to 40%, rather than 4%?
0.4 is forty hundredths. Four percent would be four hundredths, or 0.04.
Read recurring decimal dots
Some divisions finish, such as 1 ÷ 4 = 0.25. Others keep producing the same digits. A recurring decimal is an exact infinite pattern, not a decimal that we have merely rounded after a few places.
To convert a recurring decimal into a fraction, create two versions with the same repeating tail. Subtracting then removes that tail. The important decision is how far to shift the decimal point so the repeating digits line up; the equations follow from that alignment.
A dot over one digit means that digit repeats forever. Dots over the first and last digits of a block mean the whole block repeats.
Convert 0.17 to a fraction.
- Let x = 0.17777…
- 10x = 1.7777… and 100x = 17.7777…
- Subtract the aligned repeating tails: 100x − 10x = 17.7777… − 1.7777…
- 90x = 16, so x = 1690 = 845.
Convert 0.123 to a fraction, then reverse it.
Let x = 0.123123… Then 1000x = 123.123123…
Subtract x: 999x = 123. Hence x = 123999 = 41333.
Long division of 41 by 333 gives digits 1, 2, 3 with remainders 77, 104, 41. The starting remainder 41 returns, so the block 123 repeats.
Pause and explainWhy does subtracting 0.272727… from 27.272727… help?
The identical infinite repeating parts cancel, leaving exactly 27. This turns the problem into an equation without a recurring tail.
Your turn. Keep the value unchanged.
Before attempting the questions, practise describing one amount in three ways, such as 34, 0.75 and 75%. Explain which operation takes you from one form to the next.
Simplify fractions only by changing numerator and denominator together. For recurring decimals, write the equations with aligned decimal places. Finally, convert your answer back to a decimal to check that the original pattern returns.
Simplify 84126 and write it as a recurring decimal.
Reveal worked answer
Convert 338 to an improper fraction and a decimal.
Reveal worked answer
Write 0.045 as a fraction in simplest form and as a percentage.
Reveal worked answer
Convert 0.272727… and 0.16666… to fractions.