Question

How do you solve \[\dfrac{2}{3}r+\dfrac{3}{4}=\dfrac{7}{12}\]?

Answer 1

Hint: In order to find the solution of the given question that is to solve \[\dfrac{2}{3}r+\dfrac{3}{4}=\dfrac{7}{12}\] multiply both sides by the common denominator then factorize to find common factors in numerator and denominator. After that you will be able to find the value of \[r\].

Complete step by step solution:
According to the question, given equation in the question is as follows:
\[\dfrac{2}{3}r+\dfrac{3}{4}=\dfrac{7}{12}\]
To simplify the above equation further and find the value of the variable \[r\], multiply both the sides with a common denominator which is the \[LCM\left( 3,4,12 \right)\].
So, first we will calculate the \[LCM\left( 3,4,12 \right)\].
\[\begin{align}
  & 2\left| \!{\underline {\,
  3\text{ }4\text{ }12 \,}} \right. \\
 & 2\left| \!{\underline {\,
  3\text{ }4\text{ 6} \,}} \right. \\
 & 2\left| \!{\underline {\,
  3\text{ 1 3} \,}} \right. \\
 & \text{ }\left| \!{\underline {\,
  \text{1 1 1} \,}} \right. \\
\end{align}\]
Therefore, \[LCM\left( 3,4,12 \right)=2\times 2\times 3=12\] which is nothing but a common denominator.
Now multiply the term \[12\], to the given equation, we will have:
\[\Rightarrow 12\left( \dfrac{2}{3}r+\dfrac{3}{4} \right)=12\left( \dfrac{7}{12} \right)\]
After this simplify the above equation by opening the bracket with the help of multiplication, we will have:
\[\Rightarrow \dfrac{12\times 2}{3}r+\dfrac{12\times 3}{4}=\dfrac{12\times 7}{12}\]
Now solve it further and factorize the above equation to find common factors in numerator and denominator, we will have:
\[\Rightarrow \left( 4\times 2 \right)r+3\times 3=7\]
After this simplify the above equation by opening the bracket with the help of multiplication, we will have:
\[\Rightarrow 8r+9=7\]
Now subtract the term \[9\]to both the sides of the above equation, we will have:
\[\Rightarrow 8r+9-9=7-9\]
After this simplify the above equation with the help of subtraction, we will have:
\[\Rightarrow 8r=-2\]
Now divide the term \[8\]to both the sides of the above equation, we will have:
\[\Rightarrow \dfrac{8r}{8}=\dfrac{-2}{8}\]
After this simplify the above equation with the help of division, we will get:
\[\Rightarrow r=-\dfrac{1}{4}\]

Therefore, after solving the given equation \[\dfrac{2}{3}r+\dfrac{3}{4}=\dfrac{7}{12}\] the value of the variable \[r\] is \[-\dfrac{1}{4}\].

Note: Students make mistakes like miscalculation of terms while calculating the common denominator. They add the denominators which is completely wrong and leads to the wrong answer. It’s important to remember that the common denominator is nothing but equal to the least common multiple (LCM) of denominators.