Question

How do you simplify \[5\dfrac{2}{3}-1\dfrac{4}{5}\]?

Answer 1

Hint: To solve the given question, we should know about the improper fractions and proper fraction and how to convert them from one form to another fraction. The expressions of the form \[a\dfrac{b}{c}\] are called improper fractions, we can convert these to the proper fractions form as \[\dfrac{ca+b}{c}\]. Also, we should know how to do the subtraction of the fractions. The subtraction \[\dfrac{a}{b}-\dfrac{c}{d}\] is evaluated as \[\dfrac{a}{b}-\dfrac{c}{d}=\dfrac{ad-bc}{bd}\].

Complete step by step solution:
We are asked to simplify the expression \[5\dfrac{2}{3}-1\dfrac{4}{5}\]. As we can see that it has two improper fractions, so first we have to convert them to proper fraction form. We know that the proper form of an improper fraction \[a\dfrac{b}{c}\] is \[\dfrac{ca+b}{c}\]. Using this, we can find the proper fraction form of the given improper fractions. For \[5\dfrac{2}{3}\], its proper form is \[\dfrac{5\times 3+2}{3}=\dfrac{17}{3}\]. Similarly, for \[1\dfrac{4}{5}\] the proper form is \[\dfrac{5\times 1+4}{5}=\dfrac{9}{5}\].
Thus, the given expression becomes, \[\dfrac{17}{3}-\dfrac{9}{5}\]. We know the subtraction of fractions like \[\dfrac{a}{b}-\dfrac{c}{d}\] is evaluated as \[\dfrac{a}{b}-\dfrac{c}{d}=\dfrac{ad-bc}{bd}\]. Using this, we can evaluate the above subtraction as,
\[\begin{align}
  & \Rightarrow \dfrac{17}{3}-\dfrac{9}{5} \\
 & \Rightarrow \dfrac{17(5)-3(9)}{3\times 5} \\
\end{align}\]
Simplifying the above expression, we get
\[\begin{align}
  & \Rightarrow \dfrac{85-27}{15} \\
 & \Rightarrow \dfrac{58}{15} \\
\end{align}\]
Thus, on simplifying the given expression we get \[\dfrac{58}{15}\].

Note:
We should also know how to convert improper fractions of the form \[-a\dfrac{b}{c}\] to proper fractions as it’s little different from the one we did before. Here, a is a positive integer. The proper form of fraction \[-a\dfrac{b}{c}\] is \[\dfrac{-ac-b}{c}\]. To apply any mathematical operations on improper fraction, we first need to convert it to its proper fraction form.