2. (This is the same problem as \( \# 11 \) above, but with one small difference.) Matthew had \( 2 / 3 \) as much money as John. Then John gave Matthew \( \$ 5 \) for one of Matthew's sandwiches. Now the ratio of Matthew's money to John's money is \( 7: 8 \). How much money did John have in the beginning?

To solve this problem, let’s use algebra. Let \( J \) represent the initial amount of money that John had. Then Matthew initially had \( \frac{2}{3} J \). After John gives Matthew \$5, Matthew’s new amount of money becomes \( \frac{2}{3} J + 5 \), and John's new amount becomes \( J - 5 \). According to the problem, the new ratio of their money is \( 7:8 \). We can express this as: \[ \frac{\frac{2}{3} J + 5}{J - 5} = \frac{7}{8} \] Now we’ll cross-multiply: \[ 8\left(\frac{2}{3} J + 5\right) = 7(J - 5) \] \[ \frac{16}{3} J + 40 = 7J - 35 \] Next, multiply everything by 3 to eliminate the fraction: \[ 16J + 120 = 21J - 105 \] Now, rearranging brings: \[ 120 + 105 = 21J - 16J \] \[ 225 = 5J \] \[ J = 45 \] So, John initially had \( \$45 \).