Rational Numbers
Theory
A rational number is any number that can be written as a fraction of two integers: fractions, terminating and recurring decimals, percentages and integers are all rational. This Year 9 maths page revises the Year 8 work: converting between fractions, decimals and percentages, writing recurring decimals, and calculating with negative fractions and decimals.
A rational number can be written as \(\dfrac{a}{b}\), where \(a\) and \(b\) are integers and \(b \ne 0\). For example \(-2.5 = -\dfrac{5}{2}\), \(0.35 = \dfrac{7}{20}\), \(75\% = \dfrac{3}{4}\) and \(4 = \dfrac{4}{1}\).
The same rational number can be written as a fraction, a decimal or a percentage. Per cent means "out of \(100\)", so \(37\% = \dfrac{37}{100} = 0.37\).
Dividing the numerator by the denominator turns a fraction into a decimal. A terminating decimal stops, like \(\dfrac{3}{8} = 0.375\). A recurring decimal repeats forever, like \(\dfrac{5}{6} = 0.8333\ldots\). A dot over a digit shows it repeats: \(0.8\dot{3}\). Dots over the first and last digit of a block show the whole block repeats: \(0.\dot{1}\dot{8} = 0.181818\ldots\)
Rational numbers can be negative. The rules for integers still work: adding a negative is subtracting, and for \(\times\) and \(\div\) the same signs give a positive answer and different signs give a negative answer.
Converting between the three forms:
| From | To | How | Example |
|---|---|---|---|
| fraction | decimal | divide top by bottom | \(\dfrac{3}{8} = 3 \div 8 = 0.375\) |
| decimal | percentage | \(\times 100\) | \(0.375 = 37.5\%\) |
| percentage | decimal | \(\div 100\) | \(4\% = 0.04\) |
| decimal | fraction | over \(10\), \(100\), \(1000\), then simplify | \(0.35 = \dfrac{35}{100} = \dfrac{7}{20}\) |
| fraction | percentage | \(\times 100\%\) | \(\dfrac{3}{4} \times 100\% = 75\%\) |
Signs with fractions work the same way as with integers:
How to convert a rational number
- Fraction to decimal: divide the numerator by the denominator. Keep dividing until the digits stop or start repeating.
- Decimal to percentage: multiply by \(100\), which moves the point two places right.
- Percentage to decimal: divide by \(100\), which moves the point two places left.
- Decimal to fraction: write it over \(10\), \(100\) or \(1000\) to match the decimal places, then simplify.
How to calculate with negative fractions and decimals
- Rewrite two signs as one: \(+(-x) = -x\) and \(-(-x) = +x\).
- Add or subtract fractions over a common denominator, using a number line to decide the sign.
- Multiply or divide as usual, then use the sign rule.
- Simplify and check the size and sign.
(i) \(0.36\) as a percentage and as a fraction in simplest form
(ii) \(\dfrac{3}{8}\) as a decimal and as a percentage
(iii) \(1\dfrac{3}{4}\) as a percentage
(iv) \(0.4\%\) as a decimal
(i) Multiply by \(100\) for the percentage. For the fraction, use two decimal places, so write it over \(100\) and divide top and bottom by \(4\):
| \(0.36 \times 100\) | \(=\) | \(\mathbf{36\%}\) |
| \(0.36\) | \(=\) | \(\dfrac{36}{100} = \mathbf{\dfrac{9}{25}}\) |
(ii) Divide the top by the bottom, then multiply by \(100\):
| \(3 \div 8\) | \(=\) | \(\mathbf{0.375}\) |
| \(0.375 \times 100\) | \(=\) | \(\mathbf{37.5\%}\) |
(iii) The whole \(1\) is \(100\%\) and \(\dfrac{3}{4}\) is \(75\%\):
| \(1\dfrac{3}{4}\) | \(=\) | \(1.75 = \mathbf{175\%}\) |
(iv) Divide by \(100\), moving the point two places left:
| \(0.4 \div 100\) | \(=\) | \(\mathbf{0.004}\) |
(i) \(\dfrac{5}{16}\)
(ii) \(\dfrac{5}{6}\)
(iii) \(\dfrac{2}{11}\)
(i) \(16 = 2^4\) has only the prime factor \(2\), so the decimal stops:
| \(5 \div 16\) | \(=\) | \(\mathbf{0.3125}\) |
(ii) \(6 = 2 \times 3\). The factor \(3\) makes it recur. After the \(8\), the remainder is always \(2\), so the \(3\) repeats:
| \(5 \div 6\) | \(=\) | \(0.8333\ldots\) |
| \(\) | \(=\) | \(\mathbf{0.8\dot{3}}\) |
(iii) Here a block of two digits repeats, so dot the first and last digit of the block:
| \(2 \div 11\) | \(=\) | \(0.181818\ldots\) |
| \(\) | \(=\) | \(\mathbf{0.\dot{1}\dot{8}}\) |
(i) \(-2.6 + 4.15\)
(ii) \(-\dfrac{2}{3} - \dfrac{1}{4}\)
(iii) the missing number in \(\dfrac{1}{2} + {?} = -\dfrac{1}{3}\)
(i) Start at \(-2.6\) and move \(4.15\) right. The answer is the difference, with the sign of the bigger size:
| \(-2.6 + 4.15\) | \(=\) | \(4.15 - 2.6\) |
| \(\) | \(=\) | \(\mathbf{1.55}\) |
(ii) Use twelfths. Starting below zero and subtracting moves further left:
| \(-\dfrac{2}{3} - \dfrac{1}{4}\) | \(=\) | \(-\dfrac{8}{12} - \dfrac{3}{12}\) |
| \(\) | \(=\) | \(\mathbf{-\dfrac{11}{12}}\) |
(iii) Undo the adding by subtracting \(\dfrac{1}{2}\):
| \({?}\) | \(=\) | \(-\dfrac{1}{3} - \dfrac{1}{2}\) |
| \(\) | \(=\) | \(-\dfrac{2}{6} - \dfrac{3}{6}\) |
| \(\) | \(=\) | \(\mathbf{-\dfrac{5}{6}}\) |
Check: \(\dfrac{3}{6} + \left(-\dfrac{5}{6}\right) = -\dfrac{2}{6} = -\dfrac{1}{3}\).
(ii) How many \(\dfrac{3}{8}\) kg bags can be filled from \(6\) kg of rice?
(iii) Evaluate \(-\dfrac{3}{4} \times \dfrac{2}{9}\).
(i) Change to minutes first: \(2\) hours \(= 120\) minutes. "Of" means multiply:
| \(\dfrac{3}{5} \times 120\) | \(=\) | \(\dfrac{360}{5}\) |
| \(\) | \(=\) | \(\mathbf{72}\ \textbf{minutes}\) |
(ii) "How many \(\dfrac{3}{8}\)s fit into \(6\)?" is a division. Flip the second fraction and multiply:
| \(6 \div \dfrac{3}{8}\) | \(=\) | \(\dfrac{6}{1} \times \dfrac{8}{3}\) |
| \(\) | \(=\) | \(\mathbf{16}\ \textbf{bags}\) |
(iii) Cancel first, then use the sign rule. Different signs give a negative:
| \(-\dfrac{3}{4} \times \dfrac{2}{9}\) | \(=\) | \(-\dfrac{1}{2} \times \dfrac{1}{3}\) |
| \(\) | \(=\) | \(\mathbf{-\dfrac{1}{6}}\) |
Common pitfalls
Frequently asked questions
What is a rational number?
A number that can be written as a fraction \(\dfrac{a}{b}\) of two integers, with \(b \ne 0\). Integers, fractions, terminating decimals, recurring decimals and percentages are all rational.
How do you change a fraction into a percentage?
Multiply the fraction by \(100\%\). For example, \(\dfrac{3}{8} \times 100\% = 37.5\%\). You can also divide to get a decimal first, \(0.375\), then multiply by \(100\).
How do you know if a fraction gives a recurring decimal?
Simplify it, then look at the prime factors of the denominator. Only \(2\)s and \(5\)s means the decimal terminates. Any other prime, such as \(3\), \(7\) or \(11\), means it recurs.
What does a dot over a decimal digit mean?
The digit repeats forever: \(0.8\dot{3} = 0.8333\ldots\) When a block repeats, dots go over its first and last digits: \(0.\dot{1}\dot{8} = 0.181818\ldots\)
How do you subtract a fraction from a negative fraction?
Use a common denominator and think of a number line. Starting below zero and subtracting moves further left, so \(-\dfrac{2}{3} - \dfrac{1}{4} = -\dfrac{8}{12} - \dfrac{3}{12} = -\dfrac{11}{12}\).