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Do you need a common denominator for dividing and multiplying fractions?
No you do not.
Do you need a common denominator to divide fractions?
No. Dividing fractions is achieved by inverting the divisor and multiplying the resulting fractions. To multiply fractions the numerators are multiplied together to form the new numerator and the denominators are multiplied together to form the new denominator.
Adding and subtracting unlike denominators?
Fractions can only be added or subtracted if the denominators are the same. If the denominators are different, then the fractions need to be made into equivalent fractions with the same denominator. The new denominator can be found simply by multiplying the denominators together, but this can lead to some large fractions with which to work. A better new denominator is the lowest common multiple of (all the) denominators. (Once the new denominator is found, the fractions' new numerators are found by multiplying their current numerator by the new denominator divided by their current denominator to make their equivalent fractions with the new denominator.) Once all the fractions are converted into equivalent fractions with the new denominator then the fractions can be added or subtracted, with the result being simplified (if possible).
Do you need a common denominator for all fractions before doing the order of operations?
You DO need a common denominator to add, subtract, or compare fractions. You DO NOT need a common denominator to multiply or divide fractions.
Why do fractions need to have a common denominator?
They need a common denominator when you add them. For example, if you want to add 1/2 and 1/3, you need to change the fractions to equivalents that share the same denominator, so you can combine them. 1/2 + 1/3 = 3/6 + 2/6 = 5/6
