Sum and subtraction of fractions

For example, we want to sum $$\dfrac{1}{5}$$ and $$\dfrac{3}{5}$$. We draw both fractions as partitions of rectangles. The fraction $$\dfrac{1}{5}$$ is:

And the fraction $$\dfrac{3}{5}$$ is:

We want to do the sum, ie, to have at the same time the painted rectangles about the first fraction and then about the second one.

The result is $$4$$ painted rectangles:

Thus, the sum of $$\dfrac{1}{5}$$ and $$\dfrac{3}{5}$$ is $$\dfrac{4}{5}$$. $$$\dfrac{1}{5}+\dfrac{3}{5}=\dfrac{4}{5}$$$

To subtract $$\dfrac{5}{7}$$ to the fraction $$\dfrac{9}{7}$$, we start drawing the rectangles. The fraction $$\dfrac{9}{7}$$ is:

And the fraction $$\dfrac{5}{7}$$ is:

So if we subtract the five painted rectangles of the second fraction to the first one, the result is:

(We have painted the red ones with lilac and we have removed the blue ones)

So: $$$\dfrac{9}{7}-\dfrac{5}{7}=\dfrac{4}{7}$$$

To sum $$\dfrac{3}{12}$$ and $$\dfrac{1}{6}$$ we do: $$$\dfrac{3}{12}=\dfrac{3\cdot6}{12\cdot6}=\dfrac{18}{72}$$$ and

$$$\dfrac{1}{6}=\dfrac{1\cdot12}{6\cdot12}=\dfrac{12}{72}$$$

And now we can sum: $$$\dfrac{3}{12}+\dfrac{1}{6}=\dfrac{18}{72}+\dfrac{12}{72}=\dfrac{18+12}{72}=\dfrac{30}{72}$$$ Finally, we must simplify the fraction: $$$\left.\begin{array}{l} 30=2\cdot3\cdot5 \\ 72=2^3\cdot 3^2 \end{array} \right\} \Rightarrow g.c.d(30,72)=2\cdot3=6$$$

So $$$\dfrac{30}{72}=\dfrac{30:6}{72:6}=\dfrac{5}{12}$$$

The result is: $$$\dfrac{3}{12}+\dfrac{1}{6}=\dfrac{5}{12}$$$

Using the same example, $$$\dfrac{3}{12}+\dfrac{1}{6}=\dfrac{(3\times 6)+(12\times 1)}{12\times 6}=\dfrac{18\times 12}{72}=\dfrac{30}{72}$$$

And the last step is simplify the fraction.

Using our example, we calculate the $$l.c.m.$$ of $$12$$ and $$6$$: $$$\left.\begin{array}{l} 6=2\cdot3 \\ 12=2^2\cdot 3 \end{array} \right\} \Rightarrow l.c.m(6,12)=2^2\cdot3=12$$$ so we need equivalent fractions with denominators $$12$$. For this one $$\dfrac{3}{12}$$, the result is ovbiuosly the very same fraction, but let's see a method to find it in general.

For the fraction $$\dfrac{1}{6}$$, the number $$m$$ is: $$$m=\dfrac{lcm(6,12)}{6}=\dfrac{12}{6}=2$$$ and for the other fraction $$\dfrac{3}{12}$$: $$$m=\dfrac{lcm(6,12)}{12}=\dfrac{12}{12}=1$$$

So now, the last step is multiply the numerator and denominator of each fraction by its number $$m$$ and then sum the numerators. Finally, we must simplify the fraction, if it's possible.

Using the last example: $$$\dfrac{1}{6}=\dfrac{1\cdot 2}{6\cdot 2}=\dfrac{2}{12}$$$ and $$$\dfrac{3}{12}=\dfrac{3\cdot 1}{12\cdot 1}=\dfrac{3}{12}$$$ The sum is: $$$\dfrac{1}{6}+\dfrac{3}{12}=\dfrac{2}{12}+\dfrac{3}{12}=\dfrac{2+3}{12}=\dfrac{5}{12}$$$ And this result is already simplified.