Question

A chair and a table cost $1800$ rupees. If by selling the table at a profit of $15\% $ and a chair at a loss $10\% $ total profit of $6\% $ is made. What is the cost price of a chair?

Answer 1

Hint:Here we will convert the given word statement in the form of the mathematical expression using the percentage concepts and get the two equations with two unknowns and will use elimination method to get the price of the chair.

Complete step by step answer:
Let us assume the price of the table be “x” rupees and the price of the chair be “y” rupees.
Given that: A chair and a table cost $1800$rupees
$ \Rightarrow x + y = 1800$ ….. (A)
Also, given that - If by selling the table at a profit of $15\% $ and a chair at a loss $10\% $total profit of $6\% $ is made. Convert the above given statement in the form of the mathematical expression, also remember that percentage is always expressed as the term in numerator upon the hundred.
$ \Rightarrow \left( {x + \dfrac{{15x}}{{100}}} \right) + \left( {y - \dfrac{{10y}}{{100}}} \right) = \left( {1800 + \dfrac{{1800(6)}}{{100}}} \right)$
Simplify the above expression, taking LCM (least common multiple) on both the sides of the equation –
$ \Rightarrow \dfrac{{100x + 15x}}{{100}} + \dfrac{{100y - 10y}}{{100}} = \dfrac{{180000 + 10800}}{{100}}$

Common denominators from both the sides of the equation cancels each other and therefore remove from the denominators of the both the sides of the equations –
$ \Rightarrow 115x + 90y = 190800$
Take common multiple from both the sides of all the terms and remove them
$ \Rightarrow 23x + 18y = 38160$ …..(B)
To use elimination method, multiply the equation (A) with the number on both the sides of the equation –
$ \Rightarrow 23x + 23y = 41400$
Subtract equation (B) from the above equation –
$ \Rightarrow 23x + 23y - 23x - 18y = 41400 - 38160$
Like terms with the same value and the opposite sign cancels each other.
$ \Rightarrow 23y - 18y = 41400 - 38160$

Simplify the above equation finding the difference of the terms –
$ \Rightarrow 5y = 3240$
Term multiplication one side if moved to the opposite side then it goes to the denominator.
$ \Rightarrow y = \dfrac{{3240}}{5}$
Common factors from the numerator and the denominator cancel each other.
$ \Rightarrow y = 648$
Place the above value in the equation A –
$ \Rightarrow x + 648 = 1800$
Make the subject “x” and move constant term on the right hand side of the equation –
$ \Rightarrow x = 1800 - 648$
Simplify the above expression by finding the subtraction –
$ \therefore x = 1152$

Hence, the price of the table cost Rs. $1152$.

Note:The price of the chair and table can also be calculated by the substitution method. Solve wisely and carefully assuming the variable for the unknown price of the chair and the table. Be good in multiples and division and remove common multiples from the numerator and the denominator cancels each other.