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Percentages are mathematical expressions that represent a part of a whole, described as a fraction of 100. In this exercise, we need to understand how percentages apply to the voting choices of students. For example, it is given that 40% of girls voted for "A Night Under the Stars." This means out of all the girls at the school, 40% chose this theme.

Percentages can be converted into decimals for calculations. For instance, 40% becomes 0.4 in decimal form. This conversion allows us to calculate the actual number of girls who voted for this theme.

• Given: 40% of girls, represented as 0.4, voted "A Night Under the Stars."
• Equation: "0.4g = 225" where "g" is the number of girls.

Equations are mathematical statements that express equality between two expressions. They are essential tools for problem-solving, allowing us to find unknown values. In the school voting problem, we set up equations to represent the voting scenarios described.

For example, we used the equation \[ 0.4g = 225 \] to represent the votes by girls for "A Night Under the Stars." Similarly, we set up \[ 0.75b = 480 \] for boys voting for "The Time of My Life." These equations help us determine "g" (the number of girls) and "b" (the number of boys) by isolating the variable on one side.Solving these equations involves straightforward steps:

• For the girls: Divide both sides by 0.4 to find "g."
• For the boys: Divide both sides by 0.75 to find "b."

Ultimately, these represent critical building blocks in understanding the problem's solution.

Vote distribution is the way votes are spread across different choices or categories. In the context of this problem, we analyze how the votes were distributed between the themes and the respective voters.

We know:

• 225 votes were for "A Night Under the Stars."
• 480 votes were for "The Time of My Life."
• 40% of the votes for the first theme were from girls.
• 75% of the votes for the second theme were from boys.

Understanding these distributions allows us to interpret how different groups influenced the final vote totals and enables us to identify any inconsistencies or check the realism of the calculated numbers.

Problem solving involves critical thinking and applying mathematical concepts to find solutions. In this practice, several steps are taken to work through word problems effectively.

First, we define what we need to find - the number of boys and girls. Next, we represent the problem using algebra with the help of equations, as discussed earlier. It's crucial to verify each step and result to confirm accuracy and whether the calculated values make sense given the context.

When a calculation gives a non-whole number, such as 562.5 girls, we reconsider our approach using options available. The given exercise provides multiple-choice answers, helping us guide calculations toward finding a logical result, such as the option that best suits the adjusted counts (e.g., 705 boys and 325 girls). The revision process in problem-solving helps refine solutions, ensuring their reliability and correctness.