Question

If the list price of a toy is reduced by Rs.$2$, a person can buy $2$ toys more for Rs.$360$. Find the original price of the toy.
A. Rs.$10$
B. Rs.$15$
C. Rs.$20$
D. Rs.$25$

Answer 1

Hint: For the given question, we will have to form a linear equation in two variables. It is in the form of $ax + by + c = 0$, where $a,b,c$ are constants and $x.y$ are variables. Also here, we will have to solve the question by assigning variables to the information which is not known to us.

Complete step by step solution:
Let the original price of the toy be Rs.$x$. Since the number of toys bought is also unknown to us, we will let the number of toys be $y$.
From the question, it can be seen that the amount spent for the toys is Rs.$360$. Therefore, we can form the first equation as
$x \times y = 360$
$ \Rightarrow xy = 360$ {equation (1)}
Further in the question, it is given that if the price of the toy is reduced by Rs.$2$, then $2$more toys can be bought for Rs.$360$. On converting this information into an equation, we will get
$(x - 2)(y + 2) = 360$
On multiplying both the terms to each other on the left hand side of the equation, we will get
$ \Rightarrow xy + 2x - 2y - 4 = 360$
 $ \Rightarrow xy + 2x - 2y = 364$
On substituting the value of equation (1) further, we will get
$ \Rightarrow 360 + 2x - 2y = 364$
On subtracting $360$ from both the sides of the equation, we will get
$ \Rightarrow 2x - 2y = 364 - 360$
$ \Rightarrow 2x - 2y = 4$
On factorising out 2 as the common term on the left hand side of the equation, we will get
$ \Rightarrow 2(x - y) = 4$
$ \Rightarrow x - y = 2$ {equation (2)}
Now to find the value of $x$, we have to solve equation (1) and equation (2). For this we have the value of $y$ from equation (1) such that
$y = \dfrac{{360}}{x}$
On putting this value of y in equation (2), we will get
$ \Rightarrow x - \dfrac{{360}}{x} = 2$
Simplifying the expression on the left hand side of the equation, we will get
$ \Rightarrow \dfrac{{{x^2} - 360}}{x} = 2$
$ \Rightarrow {x^2} - 360 = 2x$
On arranging the whole expression at one side, we have
$ \Rightarrow {x^2} - 2x - 360 = 0$
Now on factorising the given expression, we get
$ \Rightarrow {x^2} - 20x + 18x - 360 = 0$
$ \Rightarrow x(x - 20) + 18(x - 20) = 0$
On combining factors altogether, we will get
$ \Rightarrow (x + 18)(x - 20) = 0$
It must be known that when the product of two expressions is zero, either one of them is actually zero. On applying this fact, we will get
$ \Rightarrow x + 18 = 0$ and $ \Rightarrow x - 20 = 0$
$ \Rightarrow x = - 18$ and $ \Rightarrow x = 20$
Since, the price of an object cannot be negative, therefore
$ \Rightarrow x = 20$
Hence, the original price of the toy is Rs.$20$.

So, the correct answer is option C.

Note:
In the solution given above, both the terms ‘expression’ and ‘equation’ are used. A lot of people get confused between algebraic equations and algebraic expression. Algebraic expressions are made up of variables and constants. These are generally joined with the help of algebraic operations (addition, subtraction, division, multiplication, etc.) as per the need of the question. Here is a simple thing to keep in mind so as to avoid any confusion regarding this- “Expressions make Equations”.