Question

At a supermarket, John spent \[\dfrac{1}{2}\] of his money on fresh fruits and vegetables, \[\dfrac{1}{3}\] on meat products and \[\dfrac{1}{{10}}\] on bakery products. If he spent the remaining $\$ 6$ on candy, how much did John spend at the supermarket?
A) $\$ 60$
B) $\$ 80$
C) $\$ 90$
D) $\$ 120$
E) $\$ 180$

Answer 1

Hint: We will first assume that he has some money as x. Then, we will add all the money he spends on fruits, vegetables, meat products and the bakery products and substitute it equal to x and thus get the answer.

Complete step-by-step solution:
Let us assume that he has $\$ x$ with him initially.
Now, since we are given that John spent \[\dfrac{1}{2}\] of his money on fresh fruits and vegetables.
$ \Rightarrow $ He spent \[\dfrac{1}{2} \times \$ x = \$ \dfrac{x}{2}\] on fresh fruits and vegetables. ………….(1)
Now, since we are also given that John spent \[\dfrac{1}{3}\] on meat products.
$ \Rightarrow $ He spent \[\dfrac{1}{3} \times \$ x = \$ \dfrac{x}{3}\] on meat products. ………….(2)
Now, since we are also given that John spent \[\dfrac{1}{{10}}\] on bakery products.
$ \Rightarrow $ He spent \[\dfrac{1}{{10}} \times \$ x = \$ \dfrac{x}{{10}}\] on meat products. ………….(3)
Now, John spent the left 6 dollars on candy. ………….(4)
Adding all the equations (1), (2), (3) and (4), we will get back to the total amount we were given initially.
$ \Rightarrow \$ \left( {\dfrac{x}{2} + \dfrac{x}{3} + \dfrac{x}{{10}} + 6} \right) = \$ x$
Since we have dollar signs on both sides, we can remove it from both the sides and thus we will get:-
$ \Rightarrow \dfrac{x}{2} + \dfrac{x}{3} + \dfrac{x}{{10}} + 6 = x$
Taking x common from the terms which have x in LHS, we will then obtain:-
$ \Rightarrow x\left( {\dfrac{1}{2} + \dfrac{1}{3} + \dfrac{1}{{10}}} \right) + 6 = x$
Solving the parenthesis on LHS by taking the LCM of the terms, we will then get:-
$ \Rightarrow x\left( {\dfrac{{15 + 10 + 3}}{{30}}} \right) + 6 = x$
Soling the terms inside the parenthesis, we will then get:-
$ \Rightarrow \dfrac{{28}}{{30}}x + 6 = x$
Taking simplified terms in LHS to obtain:-
$ \Rightarrow \dfrac{{14}}{{15}}x + 6 = x$
Rearranging the terms to get the following expression:-
$ \Rightarrow x - \dfrac{{14}}{{15}}x = 6$
Simplifying the LHS to get the following expression:-
$ \Rightarrow \dfrac{{15 - 14}}{{15}}x = 6$
$ \Rightarrow \dfrac{x}{{15}} = 6$
$ \Rightarrow x = 6 \times 15 = 90$
This means that he had $\$ 90$ in the start.

$\therefore $ The correct option is (C).

Note: The students must note that sometimes we use the assumption value as $\$ 100$. But we cannot do that here, because we are given that he spent $\$ 6$ among them for the candy which will create the problem, if we are just given the proportion of everything he spent on, we are bound to find the correct answer.
The students must not finally forget to put the units after the amount because it is after all money. Just writing 90 does not indicate that we are talking about money.