1. **State the problem:** There are 55 musicians playing pieces on oboe, double bass, or trombone. Each piece is either in a major or minor key. 2. **Given data:** - Total musicians: 55 - Pieces in minor key: $\frac{3}{5} \times 55 = 33$ - Pieces in major key: $55 - 33 = 22$ - Double bass players: 7 - Oboe major pieces: 1 - Double bass minor pieces: 2 - Trombone total players: 29 (given) 3. **Find missing values:** - Total minor pieces = 33 - Minor pieces on oboe and trombone: $33 - 2 = 31$ - Trombone minor pieces: Let it be $x$ - Oboe minor pieces: $31 - x$ 4. **Total players per instrument:** - Oboe total = major + minor = $1 + (31 - x) = 32 - x$ - Double bass total = major + minor = $m + 2 = 7$ so major on double bass $m = 5$ - Trombone total = major + minor = $t + x = 29$ 5. **Total major pieces = 22:** - Major on oboe = 1 - Major on double bass = 5 - Major on trombone = $t$ So, $1 + 5 + t = 22 \Rightarrow t = 16$ 6. **Trombone minor pieces:** - $t + x = 29 \Rightarrow 16 + x = 29 \Rightarrow x = 13$ 7. **Oboe minor pieces:** - $31 - x = 31 - 13 = 18$ 8. **Summary:** | Instrument | Major | Minor | Total | |-------------|-------|-------|-------| | Oboe | 1 | 18 | 19 | | Double bass | 5 | 2 | 7 | | Trombone | 16 | 13 | 29 | 9. **Find largest proportion of major pieces:** - Oboe: $\frac{1}{19}$ - Double bass: $\frac{5}{7}$ - Trombone: $\frac{16}{29}$ 10. **Compare fractions:** - $\frac{5}{7} \approx 0.714$ - $\frac{16}{29} \approx 0.552$ - $\frac{1}{19} \approx 0.053$ Largest proportion is for double bass. 11. **Simplify fraction for double bass:** $\frac{5}{7}$ is already in simplest form. **Final answer:** The double bass has the largest proportion of pieces played in a major key, which is $\frac{5}{7}$.