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Definition of fraction
Fraction is made up of numerator and denominator whereby the top number is the numerator and the bottom number is the denominator.
| \(numerator \over denominator\) |
The denominator represents the whole while the numerator represents part of the whole.
Imagine cutting a pizza into eight equal parts and you take one piece of the eight parts. That is \(1 \over 8\).
2. Proper fraction & improper fraction
When the numerator is smaller than the denominator, the fraction is called a proper fraction. For instance, if you take two parts of the cut pizza that is a proper fraction \(2 \over 8\) . If you take three parts of the cut pizza, you still have a proper fraction \(3 \over 8\).
On the other hand, when the numerator is equal to or more than the denominator, the fraction is called an improper fraction. If you have two similar pizzas and you take all the parts from the first pizza and two parts from the second pizza, that gives you an improper fraction \(10 \over 8\).
Remember that \(8 \over 8\) is also an improper fraction because the numerator equals the denominator.
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=18
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Proper fraction
78
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Improper fraction
118
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3. Basic operations
Equivalent fractions can have different numerators and different denominators but they represent the same value. For example, \(2 \over 3\) , \(4 \over 6\) and \(6 \over 9 \) are equivalent fractions because they represent the fraction \(2 \over 3\).
| \({2 \over 3} = {4 \over 6} = {6 \over 9}\) |
Equivalent fractions are obtained by multiplying both the numerator and the denominator of a fraction by the same whole number.
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\({4 \over 6}={2 \times {2\over3}}={2×2 \over 2×3}\)
\({6 \over 9} = {3 \times {2 \over 3}}={3×2 \over 3×3}\)
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You can add and subtract fractions by following these steps:
- check if the denominators are the same
- if the denominators are the same, add or subtract the numerators
- write the answer over the denominator
- if the denominators are not the same, convert to equivalent fractions to have same denominator and repeat the above step
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\({1 \over 4}+{2 \over 4}={1+2 \over 4}={3 \over 4}\)
\({1 \over 6}+{2 \over 3}={1 \over 6}+{4 \over 6}={1+4 \over 6}={5 \over 6}\)
\({2 \over 3}-{1 \over 2}={4 \over 6}-{3 \over 6}={4-3 \over 6}={1 \over 6}\)
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To multiply fractions, multiply the numerators together and multiply the denominators together. Simplify your answer.
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\({1 \over 4} \times {2 \over 4}={{1×2} \over {4×4}}={2 \over 16}={1 \over 8} \)
\({1 \over 6} \times {2 \over 3}={{1×2} \over {6×3}}={2 \over 18}={1 \over 9} \)
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To divide fractions, take the reciprocal of the divisor and then multiply as above.
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\({1 \over 4} \div {2 \over 4}={1 \over 4} \times {4 \over 2}={{1×4} \over 4×2}={4 \over 8}={1 \over 2}\)
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You can perform a combination of operations in the same way.
\(({1 \over 6}+{1 \over 3}) \times {1 \over 4}\)
\(=({1 \over 6}+2 \times {1 \over 3}) \times {1 \over 4}\) Multiply by 2 to get same denominator
\(=({1 \over 6} + {2×1 \over 2×3}) \times {1 \over 4}\)
\(=({1 \over 6}+{2 \over 6}) \times {1 \over 4}\)
\(={(1+2) \over 6} \times {1 \over 4}\) Add the numerators
\(={3 \over 6} \times {1 \over 4}\)
\(={1 \over 2} \times {1 \over 4}\) Simplify
\(={1 \over 8}\) Multiply the numerators and the denominators
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