need help writing an equation.

[sk8er]

Sally is given 20.00 cash,john is given 50.00cash, and pete is given 75.00cash. If they all get to share in a 200.00 bonus, what would their percentage of the bonus be plus the cash they were given so that each person receives an equal amount?

 

[soroban]

Hello, sk8er!

Sally is given $20 cash, John is given $50 cash, and Pete is given $75 cash.
If they all get to share in a $200 bonus, what would their percentage of the bonus be
plus the cash they were given so that each person receives an equal amount?

\(\displaystyle \text{Let: }\:\begin{Bmatrix}S &=& \text{Sally's share} \\ J &=& \text{John's share} \\ P &=& \text{Pete's share} \end{Bmatrix}\)

\(\displaystyle \text{We have: }\:\begin{Bmatrix}S + J + P &=& 200 & [1] \\ 20 + S &=& 50 + J & [2] \\ 50 + J &=& 75 + P & [3] \\ 20+S &=& 75 + P & [4] \end{Bmatrix}\)

From \(\displaystyle [4]\!:\;S \:=\ + 55\)

From \(\displaystyle [3]\!:\;J \:=\+25\)

Substitute into \(\displaystyle [1]\!:\;(P+55) + (P+25) + P \:=\:200 \quad\Rightarrow\quad 3P \:=\:120 \)

Hence: .\(\displaystyle P \,=\,40 \quad\Rightarrow\quad J \,=\,65 \quad\Rightarrow\quad S \,=\,95\)


Their percentages of the bonus were: .\(\displaystyle \begin{Bmatrix}\text{Sally:} & \dfrac{95}{200} &=& 47.5\% \\ \\ \text{John:} & \dfrac{65}{200} &=& 32.5\% \\ \\ \text{Pete:} & \dfrac{40}{200} &=& 20.0\% \end{Bmatrix}\)