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Fractions are unavoidable and sooner or later we all have to learn to work with fractions. The mathematical usage of the word fraction has a very clear everyday connotation as a part of a bigger object. It would be unthinkable nowadays to just introduce fractions as a pair of numbers and postulate their basic properties. Still, to express fractions one needs a pair of numbers with a meaning and intuition attached to them.

When multiplying fractions, the numerators (top numbers) are multiplied together and the denominators (bottom numbers) are multiplied together. To divide fractions, rewrite the problem as multiplying by the reciprocal (multiplicative inverse) of the divisor. To add fractions that have the same, or a common, denominator, simply add the numerators, and use the common denominator. However, fractions cannot be added until they are written with a common denominator. The figure below shows why adding fractions with different denominators is incorrect.

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