Question

How do you simplify \[-\dfrac{1}{9}+\left( -\dfrac{5}{6} \right)\]?

Answer 1

Hint: For solving this type of questions, we should know how to find the least common multiple or lowest common multiple that is L.C.M. And also we should know how to add or subtract fractions. First, we will rearrange the equation. Then, we will take the L.C.M. of the denominators, so that we can add the fractions easily. After adding them, we will get the solution.

Complete step-by-step solution:
Let us solve this question.
In this question, we have asked to simplify the term which is given. The term is \[-\dfrac{1}{9}+\left( -\dfrac{5}{6} \right)\].
Let us simplify that given term.
\[\Rightarrow -\dfrac{1}{9}+\left( -\dfrac{5}{6} \right)\]
After taking out the negative from the parenthesis, we get
\[\Rightarrow -\dfrac{1}{9}-\dfrac{5}{6}\]
Now, we will take negative as common from the above equation, we get
\[\Rightarrow -\left( \dfrac{1}{9}+\dfrac{5}{6} \right)\]
Now, we will add the term which is given in the parenthesis or bracket by taking the L.C.M of both the denominators in the above equation.
Hence, we will find the L.C.M. of 9 and 6.
\[\begin{align}
  & 3\left| \!{\underline {\,
  9,6 \,}} \right. \\
 & 3\left| \!{\underline {\,
  3,2 \,}} \right. \\
 & 2\left| \!{\underline {\,
  1,2 \,}} \right. \\
 & 1\left| \!{\underline {\,
  1,1 \,}} \right. \\
\end{align}\]
\[L.C.M.=3\times 3\times 2\times 1=18\]
So, the L.C.M of 9 and 6 will be 18.
In the equation \[-\left( \dfrac{1}{9}+\dfrac{5}{6} \right)\], for further solving, first we will take denominator as L.C.M. of both the denominators and for the numerator, we will divide the L.C.M. by the denominator. The number we get from here is multiplied with the numerator. Then, we will get the number as a result in the numerator.
Therefore, we can write the equation \[-\left( \dfrac{1}{9}+\dfrac{5}{6} \right)\] as
\[\Rightarrow -\left( \dfrac{2\times 1+3\times 5}{18} \right)\]
\[\Rightarrow -\left( \dfrac{2+15}{18} \right)\]
\[\Rightarrow -\left( \dfrac{17}{18} \right)\]
Hence, the simplified value of \[-\dfrac{1}{9}+\left( -\dfrac{5}{6} \right)\] is \[-\left( \dfrac{17}{18} \right)\].

Note: We can solve this question in a different way.
We will have to multiply both the fractions in numerator and denominator with the number which is given in the denominator of different fraction.
For example, if we have given the equation \[-\dfrac{1}{9}-\dfrac{5}{6}\] which we have to simplify
Then, we can write the equation \[-\dfrac{1}{9}-\dfrac{5}{6}\] as
\[-\dfrac{1}{9}-\dfrac{5}{6}=-\dfrac{1\times 6}{9\times 6}-\dfrac{5\times 9}{6\times 9}\]
Hence, the above equation can be written as
\[\Rightarrow -\dfrac{1}{9}-\dfrac{5}{6}=-\dfrac{6}{54}-\dfrac{45}{54}=-\dfrac{51}{54}\]
We know that the multiplication factor of 51 is 3 and 17. And, the multiplication factor of 54 is 2, 3, and 3.
Hence, \[-\dfrac{51}{54}\] can be written as \[-\dfrac{3\times 17}{2\times 3\times 3\times 3}\] which is also can be written as \[-\dfrac{17}{18}\].
So, we have got the same value from this method also.