Q: A beginning math student asked, "Why do you need a common denominator when adding fraction but not when multiplying?"
A: Because when you add things, they must be of the same kind. For example, apples can be added to apples to find a total number of apples.
When you multiply things they can be of different kinds, although different rules apply. For example, the number of apples that fit in a crate can be multiplied by the number of crates to get the total number of apples.
A "common denominator" follows similar rules. Some number of twelfths is of the same kind as some other number of twelfths, so they can be added. For example, 4/12 can be added to 3/12 to give 7/12. Do you see? Just as you can add apples to apples to get a total number of apples, you can add twelfths to twelfths to get a total number of twelfths.
LCM and GCD - Least Common Multiple and Greatest Common Divisor... A common denominator is a common multiple of the original denominators, so the LCM is a good common denominator.
Word Problems - Translating the English problem into mathematical notation, substitution, percents, arithmetic using fractions, Distance = Rate * Time, compound interest, "work" problems, "related rates" problems, linear Programming, and more.
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