Question

How do you solve $x+\dfrac{2}{3}=\dfrac{3}{5}$ by clearing the fractions?

Answer 1

Hint: We will look at the meaning of the phrase "clearing the fractions". This will give us a rough idea of the steps we need to do to clear the fractions. We will find the least common multiple of the numbers in the denominators. Then we will multiply the entire equation by this least common multiple. After that, we will solve the obtained equation to find the value of the variable.

Complete step-by-step solution:
The phrase "clearing the fractions" means that we have to eliminate the denominators from the equation. This will help us in the simplification of the given equation. To eliminate the denominators, let us find the least common multiple of the numbers in the denominators. The denominators are 3 and 5. The first few multiples of 3 are 6, 9, 12, 15, 18 and so on. The first few multiples of 5 are 10, 15, 20, 25 and so on. We can see that 15 is a common multiple of 3 and 5. Also, no other multiple before 15 is common to both 3 and 5. Hence, 15 is the least common multiple of 3 and 5. Multiplying the given equation by 15, we get the following,
$\begin{align}
  & 15x+15\times \dfrac{2}{3}=15\times \dfrac{3}{5} \\
 & \Rightarrow 15x+5\times 2=3\times 3 \\
 & \therefore 15x+10=9 \\
\end{align}$
Now, we will shift the term 10 to the other side of the equation in the following manner,
$\begin{align}
  & 15x=9-10 \\
 & \Rightarrow 15x=-1 \\
 & \therefore x=-\dfrac{1}{15} \\
\end{align}$
Thus, we have obtained the solution of the given equation by clearing the fractions.

Note: The concept of the least common multiple and the highest common factor is very important. The least common multiple of two numbers is the smallest number that can be divided by the two numbers. The highest common factor of two numbers is the highest number that divides the given two numbers. If the highest common factor of two numbers is 1, then the least common multiple of these two numbers is their product.