A group of three friends ate dinner at a restaurant. When they settled the check and tip, Peter paid \( \frac{4}{5} \) as much as John paid, and John paid \( \frac{1}{3} \) as much as Ralph paid. What fraction of the check and tip did John pay?

To solve this, let's denote the amount John paid as \( J \). According to the problem, we have the following relationships: 1. Peter paid \( P = \frac{4}{5} J \). 2. Ralph paid \( R = 3J \). Now, we can express the total amount paid by the three friends: \[ P + J + R = \frac{4}{5} J + J + 3J. \] Combining these amounts: \[ P + J + R = \frac{4}{5} J + \frac{5}{5} J + \frac{15}{5} J = \frac{4+5+15}{5} J = \frac{24}{5} J. \] To find John’s fraction of the total check and tip, we need to find the fraction of the total amount that John contributed: \[ \text{Fraction paid by John} = \frac{J}{P + J + R} = \frac{J}{\frac{24}{5} J} = \frac{5}{24}. \] Thus, the fraction of the check and tip that John paid is \( \frac{5}{24} \).