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How do you simplify \[-\dfrac{2}{7}+\left( -\dfrac{2}{5} \right)\]?

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Last updated date: 29th Feb 2024
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IVSAT 2024
Answer
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Hint: To simplify the expression we will use the Least Common Multiple (LCM). We will first make the different denominators of both the fraction same by taking the LCM of each other. The numerators of the respective fractions are multiplied by the factor by which the denominators are made similar to the LCM of the denominator.

Complete step by step answer:
According to the given question, we have to solve the given expression.
LCM of any two numbers would mean the product of the least common factors of the two numbers involved.
For example- LCM of 4 and 8 will be
\[\begin{align}
  & 4=2\times 2 \\
 & 8=2\times 2\times 2
\end{align}\]
\[LCM(4,8)=8\]
So we will start solving the expression by taking the LCM of 7 and 5 which is:
\[7=7\times 1\]
\[5=5\times 1\]
\[LCM(7,5)=35\]
Hence we get the LCM as 35.
For the fraction \[-\dfrac{2}{7}\], the denominator will be multiplied by 5 to get equal with the LCM and so the numerator will also be multiplied by 5.
Similarly, the fraction \[-\dfrac{2}{5}\], denominator will be multiplied by 7 to get equal with the LCM and so the numerator will also be multiplied by 7.
Therefore, based on the above statements we have the expression as:
\[\Rightarrow -\dfrac{2\times 5}{7\times 5}+\left( -\dfrac{2\times 7}{5\times 7} \right)\]
Multiplying each of the component with the associated factor we have,
\[\Rightarrow -\dfrac{10}{35}+\left( -\dfrac{14}{35} \right)\]
It can also be grouped as:
\[\Rightarrow \dfrac{-10-14}{35}\]
Doing the required subtraction, we have,
\[\Rightarrow \dfrac{-24}{35}\]

Therefore, we have the required simplified form \[ \dfrac{-24}{35}\]

Note: The given expression has a negative sign within a parenthesis, it should not be misinterpreted and should be solved appropriately. LCM should not be confused with HCF which is the Highest Common Factor. Taking the above example of LCM to explain the HCF.
For example- HCF of 4 and 8
\[\begin{align}
  & 4=2\times 2 \\
 & 8=2\times 2\times 2
\end{align}\]
\[HCF(4,8)=4\]
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