Question

The Extreme Rock Climbing Club planned a climbing expedition. The total cost was $1400, which was tobe divided equally among the members going. Prior to the trip, 5 members decided not to go. If thecost per person increased by $14, how many people went on the expedition?

Answer 1

originally, the cost was to be divided among all members

$\frac{1400}{m}=x$

Where m is the number of members and x was the cost per person originally

When 5 people decided not to go, x was increased $14 and m decreased 5...

$\frac{1400}{m-5}=x+14$

Now, we have two equations that we need to solve...

$\frac{1400}{m}=x,\: \frac{1400}{m-5}=x+14$$\begin{gathered} \frac{1400}{m}=x \\ - \\ \underline{\frac{1400}{m-5}=x+14} \\ \frac{1400}{m}-\frac{1400}{m-5}=x-\mleft(x+14\mright) \end{gathered}$

if we subtract the two expressions, we get the above

Now, we simplify...

$\begin{gathered} \frac{1400}{m}-\frac{1400}{m-5}=x-\mleft(x+14\mright) \\ -\frac{7000}{m\left(m-5\right)}=-14 \end{gathered}$

Now, we solve for m

$\begin{gathered} -\frac{7000}{m\left(m-5\right)}=-14 \\ -\frac{7000}{m\left(m-5\right)}m\mleft(m-5\mright)=-14m\mleft(m-5\mright) \\ -7000=-14m\mleft(m-5\mright) \\ -7000=-14m^2+70m \\ -14m^2+70m=-7000 \\ -14m^2+70m+7000=-7000+7000 \\ -14m^2+70m+7000=0 \\ m_{1,\: 2}=\frac{-70\pm\sqrt{70^2-4\left(-14\right)\cdot\:7000}}{2\left(-14\right)} \\ m_{1,\: 2}=\frac{-70\pm\:630}{2\left(-14\right)} \\ m_1=\frac{-70+630}{2\left(-14\right)},\: m_2=\frac{-70-630}{2\left(-14\right)} \\ m1=-20,\: m2=25 \end{gathered}$

we obtain that m=25, meaning there are 25 members, so 25-5=20, this is, 20 people went to the expedition