A basic rule is that you can only add "like" quantities. You can apples to apples and get apples. You can add oranges to oranges and get oranges. Etc... If you add oranges to apples you get so many fruit. You have to change the apples to fruit and change the oranges to fruit, then you add fruit to fruit and get fruit.
Now suppose you want to add 2 fifths to 3 sevenths? You have to change the fifths and sevenths to something that is common to both. We can change fifths and sevenths to thirty-fifths or seventieths or several other things. Since thirty-fifths is the smallest, and hence, the easiest to work with we will convert to thirty-fifths so that both 2/5 and 3/7 are "like quantities". They will both be 35ths. Here is the procedure...
\(\displaystyle \frac{2}{5} = \frac{howmany}{35}\) ? If we multiply both numerator and denominator by 7 we get \(\displaystyle \frac{2\cdot 7}{5\cdot 7}=\frac{14}{35}\).
Use the same procedure for the other fraction.
\(\displaystyle \frac{3}{7} = \frac{howmany}{35}\) ? If we multiply both numerator and denominator by 5 we get \(\displaystyle \frac{3\cdot 5}{7\cdot 5}=\frac{15}{35}\).
Now our fractions are expressed as like quantities.\(\displaystyle \frac{2}{5} + \frac{3}{7} = \frac{14}{35} + \frac{15}{35} = \frac{14+15}{35}=\frac{29}{35}\).
Hope that helps.
