answer:1. To determine the minimum number of voters for "Height" to win, let's first analyze the structure of the voting system. 2. There are 135 voters divided into 5 districts, with each district further divided into 9 precincts, and each precinct has 3 voters. [ text{Total precincts} = 5 times 9 = 45 ] 3. To win the final, "Height" must win in at least 3 out of 5 districts. 4. In a district, to declare a win, "Height" must win in the majority of the precincts, i.e., [ leftlceil frac{9}{2} rightrceil = 5 text{ precincts} ] 5. Thus, to win overall, "Height" needs to win in: [ 3 times 5 = 15 text{ precincts} ] 6. To win in a precinct, "Height" must get the majority of the votes: [ leftlceil frac{3}{2} rightrceil = 2 text{ voters} ] 7. Hence, to secure 15 precinct wins, at least: [ 15 times 2 = 30 text{ voters} ] 8. To validate our solution, consider if each of these 2-voters per precinct setup follows the given structure of precinct and district win requirements. Therefore, the minimum number of voters needed for "Height" to win can be confirmed as: [ boxed{30} ]

answer:1. **Identify the total number of marbles**: Initially, Tyrone has 120 marbles and Eric has 18 marbles. The total number of marbles is: [ 120 + 18 = 138 ] 2. **Set up the equation after redistribution**: Let x be the number of marbles Tyrone gives to Eric. After giving x marbles, Tyrone has 120 - x marbles and Eric has 18 + x marbles. Tyrone ends up with three times as many marbles as Eric, so: [ 120 - x = 3(18 + x) ] 3. **Solve the equation**: [ 120 - x = 54 + 3x ] [ 120 - 54 = 3x + x ] [ 66 = 4x ] [ x = frac{66}{4} = 16.5 ] Conclusion: Tyrone gave 16.5 marbles to Eric. However, since the number of marbles transferred must be a whole number, further refinement is needed, or assumptions must be adjusted. The final answer is The final answer would typically be boxed{B = 16.5}, which indicates a need to revisit the problem constraints.

answer:First, calculate g(5): [ g(5) = 4(5) - 5 = 20 - 5 = 15 ] Next, calculate f(g(5)) using g(5) = 15: [ f(15) = 6(15) + 11 = 90 + 11 = 101 ] Thus, the value of f(g(5)) is boxed{101}.

answer:Let's denote the number of days A can do the work alone as ( A ) days. The work done by A in one day is ( frac{1}{A} ) and the work done by B in one day is ( frac{1}{55} ). When A and B work together, they can complete the work in 19.411764705882355 days. So, the work done by A and B together in one day is ( frac{1}{19.411764705882355} ). The sum of the work done by A and B in one day is equal to the work done by them together in one day: [ frac{1}{A} + frac{1}{55} = frac{1}{19.411764705882355} ] To find the value of ( A ), we need to solve this equation. Let's first find a common denominator and then solve for ( A ): [ frac{55 + A}{55A} = frac{1}{19.411764705882355} ] Now, cross-multiply to solve for ( A ): [ 55 + A = 55A times frac{1}{19.411764705882355} ] [ 55 + A = frac{55A}{19.411764705882355} ] [ 55 times 19.411764705882355 + A times 19.411764705882355 = 55A ] [ 1067.6470588235295 + 19.411764705882355A = 55A ] Now, subtract ( 19.411764705882355A ) from both sides: [ 1067.6470588235295 = 55A - 19.411764705882355A ] [ 1067.6470588235295 = 35.588235294117645A ] Finally, divide both sides by ( 35.588235294117645 ) to solve for ( A ): [ A = frac{1067.6470588235295}{35.588235294117645} ] [ A approx 30 ] So, A can do the work alone in approximately boxed{30} days.