Question

Add an expression the sum as a mixed fraction: $\dfrac{-31}{6}$ and $\dfrac{-27}{8}$

Answer 1

Hint: We are supposed to add the two fractions. To add the fractions, we need to make the denominators the same. To make the denominators, we will take the least common multiple of the two denominators and multiply the denominator and numerator of each fraction with appropriate factor. After we successfully add the fraction, we will convert it into a mixed fraction.

Complete step by step answer:
The two fractions with us are $\dfrac{-31}{6}$ and $\dfrac{-27}{8}$. The denominators of the two given fractions are 6 and 8 and thus we need to find the least common multiple of 6 and 8.
To find the LCM, we will first find the prime factors of 6 and 8 separately.
Prime factors of 6: 2 $\times $ 3
Prime factors of 8: 2 $\times $ 2 $\times $ 2
To get the LCM, we need to multiply all the factors occurring in the factorisation, only once if they are common, otherwise all the factors.
Therefore, LCM: 2 $\times $ 3 $\times $ 2 $\times $ 2 = 24
Thus, now we can make the denominator the same and the common denominator will be 24.
We will multiply the denominator and numerator of $\dfrac{-31}{6}$ by 4.
$\Rightarrow \dfrac{-31}{6}\times \dfrac{4}{4}=\dfrac{-124}{24}$
We will multiply the denominator and numerator of $\dfrac{-27}{8}$ by 3.
$\Rightarrow \dfrac{-27}{8}\times \dfrac{3}{3}=\dfrac{-81}{24}$
Now we can add these improper fractions.
$\Rightarrow \dfrac{-124}{24}+\dfrac{-81}{24}=\dfrac{-205}{24}$
$\dfrac{-205}{24}$ cannot be simplified further.
Now, to convert this improper fraction into mixed fraction, we have to split the numerator with one of the parts being a complete multiple of the denominator.
 Thus, we will split 205 into 192 and 13, where 192 is 8 times 24.
\[\begin{align}
  & \Rightarrow \dfrac{-205}{24}=\dfrac{-\left( 192+13 \right)}{24} \\
 & \Rightarrow \dfrac{-205}{24}=-8+\dfrac{-13}{24} \\
\end{align}\]
Therefore, the improper fraction will be \[-8\dfrac{13}{24}\].

Note: Fraction cannot be added if the denominators are not same. Moreover, only improper fractions can be converted into mixed fractions.
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