To solve this problem, let's break it down into steps:
1. Identifying the variables:
- Let's denote the amount of money Gerald has initially as [tex]\( G \)[/tex].
- Let's denote the amount of money Jerry has initially as [tex]\( J \)[/tex].
2. Setting up the first equation:
- We know that Gerald has 80% more money than Jerry. Therefore, [tex]\( G = 1.8J \)[/tex].
3. Setting up the second equation:
- If Gerald gives Jerry [tex]$72, the new amount of money Gerald has will be \( G - 72 \).
- The new amount Jerry has will be \( J + 72 \).
- According to the problem, after this transaction, Jerry will have 80% more money than Gerald. Therefore, \( J + 72 = 1.8(G - 72) \).
4. Solving the system of equations:
- We now have two equations:
\[
G = 1.8J \quad \text{(1)}
\]
\[
J + 72 = 1.8(G - 72) \quad \text{(2)}
\]
5. Solve equation (1) for \( G \):
\[
G = 1.8J
\]
6. Substitute \( G \) in equation (2):
\[
J + 72 = 1.8(1.8J - 72)
\]
7. Expand and simplify:
\[
J + 72 = 1.8 \cdot 1.8J - 1.8 \cdot 72
\]
\[
J + 72 = 3.24J - 129.6
\]
\[
72 + 129.6 = 3.24J - J
\]
\[
201.6 = 2.24J
\]
\[
J = \frac{201.6}{2.24}
\]
\[
J = 90
\]
8. Now find \( G \):
\[
G = 1.8J
\]
\[
G = 1.8 \times 90
\]
\[
G = 162
\]
9. Calculate the total amount of money the two brothers have altogether:
\[
\text{Total money} = G + J
\]
\[
\text{Total money} = 162 + 90
\]
\[
\text{Total money} = 252
\]
Therefore, Gerald and Jerry together have $[/tex]252 initially.
