Question

How do you simplify \[\dfrac{5}{12}+\dfrac{2}{3}?\]

Answer 1

Hint: We are given \[\dfrac{5}{12}+\dfrac{2}{3}\] and we have to simplify and solve this. To do so we will learn how fractions are added and then we will learn that denominators must be equal before adding. We will make this denominator equal by multiplying by the appropriate term. Once it is the same we add the term of the numerator and then simplify and find our answer.

Complete step by step answer:
We are given an expression \[\dfrac{5}{12}+\dfrac{2}{3}\] and we have to simplify this expression and solve our problem. We are simply given the addition of two fractions. We will first learn how to add or subtract the fraction and then we will solve our given problem. To add any fraction we will first make their denominator equal and then we will simply add the new fraction obtained. For example, we have \[\dfrac{1}{2}+\dfrac{1}{3}.\] So, the denominator is 2 and 3. Their lowest common multiple is 6. So, we make the denominator as 6 in each of the given fractions. So in \[\dfrac{1}{2},\] we multiply numerator and denominator by 3. So, we will get,
\[\dfrac{1}{2}=\dfrac{1}{2}\times \dfrac{3}{3}\]
\[\Rightarrow \dfrac{1}{2}=\dfrac{3}{6}.....\left( i \right)\]
Now in \[\dfrac{1}{3}\] we will multiply the numerator and denominator by 2, so we get,
\[\dfrac{1}{3}=\dfrac{1}{3}\times \dfrac{2}{2}\]
\[\Rightarrow \dfrac{1}{3}=\dfrac{2}{6}......\left( ii \right)\]
Now we will solve our question \[\dfrac{1}{2}+\dfrac{1}{3}\] using (i) and (ii), we get,
\[\dfrac{1}{2}+\dfrac{1}{3}=\dfrac{1}{6}+\dfrac{2}{6}\]
Now we keep the denominator the same and simply adding the numerator, we will get,
\[\dfrac{1}{2}+\dfrac{1}{3}=\dfrac{3+2}{6}\]
\[\Rightarrow \dfrac{1}{2}+\dfrac{1}{3}=\dfrac{5}{6}\]
So, \[\dfrac{1}{2}+\dfrac{1}{3}=\dfrac{5}{6}.\]
Now, we will use these steps on our given problem. In \[\dfrac{5}{12}+\dfrac{2}{3},\] the denominator of fraction are 12 and 3 and the lowest common multiple is 12. So, we made the denominator as 12. In \[\dfrac{5}{12},\] we multiply and divide by 1, so we get,
\[\dfrac{5}{12}=\dfrac{5}{12}\times \dfrac{1}{1}=\dfrac{5}{12}.....\left( iii \right)\]
In \[\dfrac{2}{3},\] we multiply numerator and denominator by 4, so,
\[\dfrac{2}{3}=\dfrac{2}{3}\times \dfrac{4}{4}\]
\[\Rightarrow \dfrac{2}{3}=\dfrac{8}{12}\]
So,
\[\Rightarrow \dfrac{2}{3}=\dfrac{8}{12}....\left( iv \right)\]
Now,
\[\dfrac{5}{12}+\dfrac{2}{3}\]
\[\Rightarrow \dfrac{5}{12}+\dfrac{8}{12}......\left[ \text{using }\left( iii \right)\text{ and }\left( iv \right) \right]\]
Added numerator term while keeping denominator as 12.
\[\Rightarrow \dfrac{5}{12}+\dfrac{2}{3}=\dfrac{13}{12}\]

Note:
Remember when we add or subtract fractions, we need to make the denominator equal, if we do like \[\dfrac{5}{12}+\dfrac{2}{3}=\dfrac{5+2}{12+3},\] that is added numerator to numerator and denominator to denominator then it will be an incorrect step, so we need to be careful while solving fractions.