$38$ bottles of soda was consumed by $18$ women. Some took $2$ and others took $3$ . (A) How many women took $2$ sodas? (B) How many women took $3$ sodas?
I thought I might use simultaneous equations though I dont know how to come up with it . Or maybe I should use quadratic? I dont really know which I should use and how I should use it. Please help and show the working step by step.
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Let the number of women who took two bottles be labelled $w_{2}$, similarly for the women who took three bottles, call them $w_{3}$.
Then it is clear that $w_{2}+w_{3}=18$.
The other relation is that the total bottles of soda (being $38$) must equal $2w_{2}+3w_{3}$. Hence your simultaneous system looks like \begin{eqnarray} w_{2}+w_{3} &=& 18 \\ 2w_{2}+3w_{3} &=& 38 \end{eqnarray} Multiplying the top relation by $2$ yeilds \begin{equation} 2w_{2}+2w_{3}=36 \end{equation} Subtrating this from the lower relation gives \begin{equation} 2w_{2}+3w_{3}-2w_{2}-2w_{3} = 38-36 \end{equation} Or $$w_{3}=2$$ Which means that there were two women who took three bottles, meaning sixteen women took two bottles. We can verify this by inserting the above value for $w_{3}$ into the first relation at the outset, \begin{equation} w_{2}+2=18 \end{equation}
Hint for alternate solution: If all of the women took two bottles, how many bottles would've been taken? How many women need to take one more bottle if $38$ bottles have been taken all in all?
Simultaneous equations: Let $x$ be the number of women who took two bottles, and $y$ the number of women who took $3$. Then we have $$ \cases{x + y = 18\\ 2x + 3y = 38} $$ where the first line says that there are $18$ women in total, and the second line says that $38$ bottles have been taken.