Question

Evaluate \[\dfrac{2}{3} + \dfrac{{ - 4}}{5} + \dfrac{1}{3} + \dfrac{2}{5}\].

Answer 1

Hint:This is a very basic question simply based on fraction and operations on fraction. We know that fractions with the same denominator can be added directly and those with different denominators need to make the denominator the same. We will use this principle only to evaluate the sum above.

Complete step by step answer:
Given the sum is \[\dfrac{2}{3} + \dfrac{{ - 4}}{5} + \dfrac{1}{3} + \dfrac{2}{5}\]. Now we will solve this step by step. First we will rearrange the term for our convenience. For that, take the terms with the same denominator closer to each other.
\[\dfrac{2}{3} + \dfrac{{ - 4}}{5} + \dfrac{1}{3} + \dfrac{2}{5} = \dfrac{2}{3} + \dfrac{1}{3} + \dfrac{2}{5} + \dfrac{{ - 4}}{5}\]
Now just add the numerators of those with the same denominators.
\[\dfrac{2}{3} + \dfrac{{ - 4}}{5} + \dfrac{1}{3} + \dfrac{2}{5} = \dfrac{{2 + 1}}{3} + \dfrac{{2 - 4}}{5}\]
On adding we get,
\[ \dfrac{2}{3} + \dfrac{{ - 4}}{5} + \dfrac{1}{3} + \dfrac{2}{5}= \dfrac{3}{3} + \dfrac{{ - 2}}{5}\]

Now the denominators are different. So we will take LCM of the fractions by cross multiplying.
\[ \dfrac{2}{3} + \dfrac{{ - 4}}{5} + \dfrac{1}{3} + \dfrac{2}{5}= \dfrac{{3 \times 5 + \left( { - 2} \right) \times 3}}{{3 \times 5}}\]
On taking the product we get,
\[\dfrac{2}{3} + \dfrac{{ - 4}}{5} + \dfrac{1}{3} + \dfrac{2}{5} = \dfrac{{15 - 6}}{{15}}\]
On subtracting,
\[\dfrac{2}{3} + \dfrac{{ - 4}}{5} + \dfrac{1}{3} + \dfrac{2}{5} = \dfrac{9}{{15}}\]
This can be the answer. But for moe simplified form we will divide the numerator and denominator by 3,
\[\dfrac{2}{3} + \dfrac{{ - 4}}{5} + \dfrac{1}{3} + \dfrac{2}{5} = \dfrac{{9 \div 3}}{{15 \div 3}}\]
On dividing we get,
\[\therefore \dfrac{2}{3} + \dfrac{{ - 4}}{5} + \dfrac{1}{3} + \dfrac{2}{5}= \dfrac{3}{5}\]
Now this is the simplified form.

So, \[\dfrac{2}{3} + \dfrac{{ - 4}}{5} + \dfrac{1}{3} + \dfrac{2}{5} = \dfrac{3}{5}\].

Note:Here we have to note two things. Most important is, fraction can be directly added only if denominators are the same and not the numerators. Their comonness will not help in addition. Next is the product of minus and plus signs minus and not plus. Lastly, we can cancel that fraction \[\dfrac{3}{3}\] and write 1 in that place and continue for the solution instead. Also note that, we simplify any fraction by finding the HCF of numerator and denominator.