1. **State the problem:** We have a choir with 384 people total, consisting of sopranos (S), altos (A), tenors (T), and bases (B). We need to find the number of each. 2. **Write down the given information as equations:** - Total people: $$S + A + T + B = 384$$ - Altos are twice sopranos: $$A = 2S$$ - Tenors are 12 more than bases: $$T = B + 12$$ - Sopranos plus altos is three times tenors plus bases: $$S + A = 3(T + B)$$ 3. **Substitute known relations into the equations:** From $$A = 2S$$ and $$T = B + 12$$, substitute into total: $$S + 2S + (B + 12) + B = 384$$ Simplify: $$3S + 2B + 12 = 384$$ $$3S + 2B = 372$$ From $$S + A = 3(T + B)$$ and $$A = 2S$$, $$T = B + 12$$: $$S + 2S = 3((B + 12) + B)$$ $$3S = 3(2B + 12)$$ $$3S = 6B + 36$$ Divide both sides by 3: $$\cancel{3}S = \cancel{3}(2B + 12)$$ $$S = 2B + 12$$ 4. **Substitute $$S = 2B + 12$$ into $$3S + 2B = 372$$:** $$3(2B + 12) + 2B = 372$$ $$6B + 36 + 2B = 372$$ $$8B + 36 = 372$$ Subtract 36: $$8B = 336$$ Divide both sides by 8: $$\cancel{8}B = \cancel{8}42$$ $$B = 42$$ 5. **Find other values:** $$S = 2B + 12 = 2(42) + 12 = 84 + 12 = 96$$ $$A = 2S = 2(96) = 192$$ $$T = B + 12 = 42 + 12 = 54$$ 6. **Check total:** $$S + A + T + B = 96 + 192 + 54 + 42 = 384$$ correct. **Final answer:** $$S = 96$$ $$A = 192$$ $$T = 54$$ $$B = 42$$