Question

Rob spends $\dfrac{1}{2}$​ of his monthly paycheck, after taxes, on rent. He spends $\dfrac{1}{3}$​ on food and $\dfrac{1}{8}$​ on entertainment. If he donates the entire remainder, \[\$500\], to charity, what is Rob's monthly income, after taxes?
A) \[\$12,000\]
B) \[\$13,200\]
C) \[\$14,000\]
D) \[\$15,000\]

Answer 1

Hint: First assume the monthly income after the taxes. Then, find the money spends on rent, food, and entertainment in terms of the assumption. Then subtract all the spending and equate it to the donation.

Complete step by step answer:
Given: - Rob spends $\dfrac{1}{2}$​ of his monthly paycheck, after taxes, on rent. He spends $\dfrac{1}{3}$​ on food and $\dfrac{1}{8}$​ on entertainment. If he donates the entire remainder, \[\$500\], to charity. After that take L.C.M. and solve it to get the monthly income of Bob.
Let Rob’s monthly income after taxes be \[\$x\].
The money spends on rent is $\dfrac{1}{2}x$.
The money spends on food is $\dfrac{1}{3}x$.
The money spends on entertainment is $\dfrac{1}{8}x$.
So, the amount left after all the spending be,
$ \Rightarrow x - \dfrac{1}{2}x - \dfrac{1}{3}x - \dfrac{1}{8}x$
So, the money donated to the charity is,
$ \Rightarrow x - \dfrac{1}{2}x - \dfrac{1}{3}x - \dfrac{1}{8}x = 500$
Take L.C.M. on the left side,
$ \Rightarrow \dfrac{{24x - 12x - 8x - 3x}}{{24}} = 500$
Subtract the values in the numerator,
$ \Rightarrow \dfrac{1}{{24}}x = 500$
Multiply both sides by 24,
$ \Rightarrow x = 24 \times 500$
Multiply the terms on the right side,
$\therefore x = \$ 12000$
Thus, Bob's monthly income, after taxes is \[\$ 12,000\].

Hence, option (A) is the correct answer.

Note:
A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable, where the highest power of the variable is one.
If the linear equation has only a single variable then it is called a linear equation in one variable.
For solving an equation having only one variable, the following steps are followed
(i) Using LCM, clear the fractions if any.
(ii) Simplify both sides of the equation.
(iii) Isolate the variable.
(iv) Verify your answer.