Question

How do you simplify \[\dfrac{5}{7}-\dfrac{2}{3}\]?

Answer 1

Hint: A rational number is a number that can be expressed as a fraction, for instance \[\dfrac{1}{2}\] or \[\dfrac{11}{39}\]. To subtract two rational numbers with the same denominator, it is very easy where we can just subtract the numerators with the denominator being the same. If the denominators are different, we can’t directly subtract them. We have to make the denominators the same by taking LCM and then subtracting.

Complete step by step answer:
As per the given question, we have to simplify the given expression which is the subtraction of two rational numbers. And, the given expression is \[\dfrac{5}{7}-\dfrac{2}{3}\].

Here, we need to understand two important facts. The first fact is that we can’t directly add or subtract counts in fractions unless their size indicators are the same where count is numerator and size indicator is denominator. And the second fact is that multiplying by 1 doesn’t change the value of something but does change the way it looks.

In the given question, the denominators of the rational numbers are different. So, we take their LCM. That is, LCM of 7 and 3 is 21. Thus, we make the denominators as 21 by the following process.

Firstly, we have to multiply each rational number by 1. Then, we get
\[\Rightarrow \dfrac{5}{7}-\dfrac{2}{3}=\left( \dfrac{5}{7}\times 1 \right)-\left( \dfrac{2}{3}\times 1 \right)\]
Now, we multiply \[\dfrac{5}{7}\] by \[\dfrac{3}{3}\] and \[\dfrac{2}{3}\] by \[\dfrac{7}{7}\] to get 21 in the denominator. That is, we get
\[\Rightarrow \left( \dfrac{5}{7}\times \dfrac{3}{3} \right)-\left( \dfrac{2}{3}\times \dfrac{7}{7} \right)\to \dfrac{15}{21}-\dfrac{14}{21}\]
As the denominator is the same, we can directly subtract 14 from 15. Then, we get
\[\Rightarrow \dfrac{15}{21}-\dfrac{14}{21}=\left( \dfrac{15-14}{21} \right)=\dfrac{1}{21}\]

\[\therefore \dfrac{1}{21}\] is the simplified form of \[\dfrac{5}{7}-\dfrac{2}{3}\].

Note: A common mistake made while subtracting two rational numbers is by subtracting directly without checking the denominators. We have to take care while making the denominators the same in case of different denominators. If we write LCM of 7 and 3 as 20 instead of 21, then we would have the wrong solution. We should avoid calculation mistakes to get the correct answer.