If Sue buys A, she will have $\$1.50$ left; if she buys B, she will have $\$2$ left. Given that 2A=3B, how much money does she have?

Question

Problem statement

I can see the answer is B, but only using elimination. Surely I should be able to convert the question to an equation, e.g. 2F = 3L... to make it simpler.

Answer 1

Let $M$ be the amount of money that Sue has. Then we obtain the following system: $$ \begin{cases} 2F = 3L \\ M - F = 1.5 \\ M - L = 2 \end{cases} \implies \begin{cases} 2F = 3L \\ 2M - 2F = 3 \\ 3M - 3L = 6 \end{cases} \implies \begin{cases} 2M - 3L = 3 \\ 3M - 3L = 6 \end{cases} $$ Subtracting the first equation from the second yields $M = 3$, as desired.

Answer 2

Let $x$ be the amount of money that Sue have, $F$ is the price of a fairy floss, $L$ is the price of a lollipop.

$$2F=3L \Rightarrow F=\frac{3L}{2}$$

$$x- F=1.5$$

$$x-L=2 (*)$$

$$x- F=1.5 \Rightarrow x-\frac{3L}{2}=1.5 \Rightarrow \frac{2x}{3}-L=1 \Rightarrow -\frac{2x}{3}+L=-1 (**)$$

$$(*)+(**) \Rightarrow \frac{x}{3}=1 \Rightarrow x=3$$