Question

The sum of two numbers is $17$. One number is $3$ less than $\dfrac{2}{3}$ of the other number. What is the lesser number?

Answer 1

Hint: Here, in the given question, we are given that the sum of two numbers is $17$. One number is $3$ less than $\dfrac{2}{3}$ of the other number and we need to find the lesser number. We will first make the equation according to the information provided in the equation. Let us assume the two numbers be $x$ and $y$. We will find one number in terms of other, after this we will substitute it in the equation. From here we will find one number and using this we will find the other number.

Complete step by step answer:
We are given that ‘sum of two numbers is $17$.’Let the two unknown numbers be $x$ and $y$.
$ \Rightarrow x + y = 17...\left( i \right)$
As it is given that ‘one number is $3$ less than $\dfrac{2}{3}$ of the other number.’ Let’s say that $y$ is the ‘one number’. If $y$ is $3$ less than $\dfrac{2}{3}$ of $x$, this is written as:
$ \Rightarrow y = \dfrac{2}{3}x - 3$

Now, we will substitute the above-written value of $y$ in equation $\left( i \right)$.
$ \Rightarrow x + \dfrac{2}{3}x - 3 = 17$
Take LCM of the like terms
$ \Rightarrow \dfrac{{3x + 2x}}{3} - 3 = 17$
$ \Rightarrow \dfrac{{5x}}{3} - 3 = 17$
Add $3$ to both sides of the equation.
$ \Rightarrow \dfrac{{5x}}{3} - 3 + 3 = 17 + 3$
$ \Rightarrow \dfrac{{5x}}{3} = 20$
On cross multiplication, we get
$ \Rightarrow 5x = 20 \times 3$
On multiplication of terms, we get
$ \Rightarrow 5x = 60$

Divide both sides by $5$
$ \Rightarrow \dfrac{{5x}}{5} = \dfrac{{60}}{5}$
$ \Rightarrow x = \dfrac{{60}}{5}$
On division, we get
$ \Rightarrow x = 12$
Now, we will substitute the value of $x$ in the equation $\left( i \right)$.
$ \Rightarrow 12 + y = 17$
$ \Rightarrow y = 17 - 12$
On subtraction of terms, we get
$ \therefore y = 5$

Therefore, the lesser number is $5$.

Note: All we did is convert the given word problem into an algebraic equation. The key to solve this type of questions is to understand the statement given in the question and use it to form an algebraic equation, it all depends on the language of the problem and how we grasp it. We should take care of the calculations so as to be sure of our final answer.