Question

A person spends \[\dfrac{1}{3}\] part of his income on food, \[\dfrac{1}{4}\] part on house rent and remaining Rs.630 on other items. The house rent is: -
(a) Rs.504
(b) Rs.1512
(c) Rs.378
(d) None of these

Answer 1

Hint: Assume x as the total income of the person. Find the amount that the person spends on food and house rent by calculating \[{{\left( \dfrac{1}{3} \right)}^{rd}}\] part of x and \[{{\left( \dfrac{1}{4} \right)}^{th}}\] part of x respectively. Now, take the sum of these parts and add it with 630 and equate it with x. Solve the obtained linear equation in x to find the total income of the person. Now, substitute this value of x in the expression for house rent, i.e. \[\dfrac{x}{4}\], to get the answer.

Complete step by step answer:
Here, we have been given that a person spends \[\dfrac{1}{3}\] part of his income on food, \[\dfrac{1}{4}\] part on house rent and remaining Rs.630 on other items. We have been asked to calculate the amount he spends on house rent.
Now, let us assume that the total income of the person is Rs.x. So, according to the given question, we have,
\[\Rightarrow \] Amount he spend on food = \[\dfrac{1}{3}x=\dfrac{x}{3}\]
\[\Rightarrow \] Amount he spend on house rent = \[\dfrac{1}{4}x=\dfrac{x}{4}\]
\[\Rightarrow \] Amount he spend on other items = 630
Since, it is not given that he saves any money therefore we will assume that all his money gets spent. Therefore, the sum of all the money spent on different things listed will be equal to his total income. So, we have,
\[\begin{align}
  & \Rightarrow \dfrac{x}{3}+\dfrac{x}{4}+630=x \\
 & \Rightarrow \dfrac{7x}{12}+630=x \\
\end{align}\]
\[\begin{align}
  & \Rightarrow x-\dfrac{7x}{12}=630 \\
 & \Rightarrow \dfrac{5x}{12}=630 \\
 & \Rightarrow x=\dfrac{12\times 630}{5} \\
\end{align}\]
\[\Rightarrow x=\] Rs.1512
Therefore, the total income of the person is Rs.1512. Now, we have to find the money spent on house rent. So, we have,
\[\Rightarrow \] Money spend on house rent = \[\dfrac{x}{4}=\dfrac{1512}{4}\]= Rs.378

So, the correct answer is “Option c”.

Note: One may note that we do not have to assume the money spent on food and house rent as different variables as it may confuse us. We just have to assume one variable and carry our calculation using that assumption. Remember that the value of x is not our answer, we have to substitute it in the expression \[\dfrac{x}{4}\] to get the answer. Sometimes in a hurry students just write the value of x as an answer. So, the question must be read carefully.