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

`(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...

`(1400)/(m-5)=x+14`

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

`(1400)/(m)=x,\: (1400)/(m-5)=x+14`
`\begin{gathered} (1400)/(m)=x \\ - \\ \underline{(1400)/(m-5)=x+14} \\ (1400)/(m)-(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} (1400)/(m)-(1400)/(m-5)=x-\mleft(x+14\mright) \\ -(7000)/(m\left(m-5\right))=-14 \end{gathered}`

Now, we solve for m

`\begin{gathered} -(7000)/(m\left(m-5\right))=-14 \\ -(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)=(-70\pm√(70^2-4\left(-14\right)\cdot\:7000))/(2\left(-14\right)) \\ m_(1,\: 2)=(-70\pm\:630)/(2\left(-14\right)) \\ m_1=(-70+630)/(2\left(-14\right)),\: m_2=(-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