Question

Solve the following
\[\dfrac{-9}{10}+\dfrac{32}{15}\]

Answer 1

Hint: We are asked to add two fractions. We will first learn how to add two fractions. After that we will make the denominator in both the fractions equal by first finding the lowest common multiple (LCM) of the value in the denominator that is 10 and 15 and then when the denominator is equal, we will simply add the numerator with the same denominator and will get our required answer.

Complete step by step answer:
We are asked to add two fractions. One fraction is \[\dfrac{-9}{10}\] and the other is \[\dfrac{32}{15}.\] Before we start solving, we should learn about fractions. We can only add two fractions if their denominators are equal. If their denominators are equal then we just add the numerator keeping the denominator as same. For example, \[\dfrac{1}{2}+\dfrac{3}{2}.\] As we can see the denominators are the same so we add as follows.
\[\dfrac{1}{2}+\dfrac{3}{2}=\dfrac{1+3}{2}=\dfrac{4}{2}\]
On simplifying this, we get,
\[\Rightarrow \dfrac{1}{2}+\dfrac{3}{2}=2\left[ \text{As }\dfrac{4}{2}=2 \right]\]
Now in our fraction, we have \[\dfrac{-9}{10}\] and \[\dfrac{32}{15}.\] The denominator is not the same, so we have to make it equal. To do so we have to find a number that is common to both 10 and 15. So, we will find the LCM of 10 and 15, we get,
\[10=2\times 5\]
\[15=3\times 5\]
\[LCM\left( 10,15 \right)=2\times 3\times 5=30\]
So, we will make the denominator as 30 in both the fractions. In \[\dfrac{-9}{10},\] we will multiply and divide by 3. So, we get,
\[\dfrac{-9}{10}\times \dfrac{3}{3}=\dfrac{-27}{30}\]
\[\Rightarrow \dfrac{-9}{10}=\dfrac{-27}{30}......\left( i \right)\]
In \[\dfrac{32}{15},\] we will multiply and divide by 2, so we get,
\[\dfrac{32}{15}=\dfrac{32}{15}\times \dfrac{2}{2}\]
\[\Rightarrow \dfrac{32}{15}=\dfrac{64}{30}......\left( ii \right)\]
Now,
\[\dfrac{-9}{10}+\dfrac{32}{15}\]
Using (i) and (ii), we get,
\[\dfrac{-9}{10}+\dfrac{32}{15}=\dfrac{-27}{30}+\dfrac{64}{30}\]
As denominator is the same, so,
\[\Rightarrow \dfrac{-9}{10}+\dfrac{32}{15}=\dfrac{-27+64}{30}\]
On simplifying, we get,
\[\Rightarrow \dfrac{-9}{10}+\dfrac{32}{15}=\dfrac{37}{30}\]
Hence, we get \[\dfrac{-9}{10}+\dfrac{32}{15}=\dfrac{37}{30}.\]

Note:
Students need to remember that when we subtract two terms, the term after the subtraction sign of the biggest term is considered. For example, 2 – 4 = – 2 because 4 has a negative sign and 4 is greater. Similarly, – 27 + 64 = 37. So, do not confuse with – 37. Here 64 is the biggest and 64 is positive. So, we will get 37 only. Also, remember that multiplying and dividing by the same number at the same time will not affect the original term, that is why \[\dfrac{-9}{10}=\dfrac{-9}{10}\times \dfrac{3}{3}=\dfrac{-27}{30}.\]