In this post, let's look at example with 3 fractions.
What is \(\dfrac{1}{2} + \dfrac{3}{3} + \dfrac{9}{4}\)?
Strategy
First, let's see what the least common multiple (LCM) is of the denominators.
\(2, 4, 6, 8, 10, 12, 14, 16, 20, 22, 24..\)
\(3, 6, 9, 12, 15, 18, 21, 24, 27, 30..\)
\(4, 8, 12, 16, 20, 24, 28, 32, 36, 40...\)
We can see that the LCM is \(12 \) because it is the lowest number common to the three lists.
Next, let's make new fractions with denominators that equal the LCM by multiplying the numerators and denominators by certain numbers. These numbers are often found in the denominator of other fractions alone or as a product of the original numbers.
\(\dfrac{1}{2} * \dfrac{6}{6} = \dfrac{6}{12}\) ## (\(6\) is a product of \(2\) and \(3\) which are in the denominators of the other fractions in the problem.
\(\dfrac{3}{3} * \dfrac{4}{4} = \dfrac{12}{12}\) ## \(4\) is the denominator of another fraction in the problem.
\(\dfrac{9}{4} * \dfrac{3}{3} = \dfrac{27}{12}\) ## \(3\) is the denominator of another fraction in the problem.
Finally, let's add up new fractions (adding numerators and leaving denominators the same) and reduce the final answer via the greatest common factor (GCF) of the numerator and denominator, if it is divisible.
\(\dfrac{6}{12} + \dfrac{12}{12} + \dfrac{27}{12} = \dfrac{45}{12} = \dfrac{15}{4} \) ## original answer is divisible
Finding the GCF of \(45\) and \(12\)
\(1, 2, 3, 5, 9, 15, 45\)
\(1, 2, 3, 4, 6, 12\)
The GCF or highest number of the two lists is \(3\). In that case, we had divided the answer's numerator and denominator by \(3\).