Parent Volunteer As the treasurer of her daughter's Girl...

Parent Volunteer As the treasurer of her daughter's Girl Scout troop, Laney collected money for some girls and adults to go to a 3-day camp. Each girl paid $\$ 75$ and each adult paid $\$ 30$ . The total amount of money collected for camp was $\$ 765 .$ If the number of girls is three times the number of adults, how many girls and how many adults paid for camp?

Key Concepts

Unit Pricing and Total Revenue

This concept involves understanding how different items, each with its own unit price, contribute to a total sum. In scenarios like fundraising or ticket sales, the total revenue comes from the sum of individual contributions, each calculated by multiplying the quantity by the unit price. This approach is essential for forming the equations that ultimately lead to solving the problem.

Ratios and Proportional Reasoning

Many problems involve relationships where one quantity is a multiple or a fraction of another, which is described by a ratio. Understanding ratios helps in setting up correct expressions and equations that reflect the proportional relations between different groups or values within the problem.

Substitution Method

This is a common algebraic technique for solving systems of equations. It involves solving one equation for one variable and substituting that expression into another equation. This method effectively reduces a system of equations to a single equation with one unknown, making it easier to find the solution.

Systems of Linear Equations

This concept involves setting up two or more equations that work together to model a situation where multiple related quantities are involved. In problems like this, each equation represents a different condition or relationship, and solving the system yields the values of the unknowns that satisfy all conditions simultaneously.

Translating Word Problems into Equations

This key concept is about reading a real-life scenario and converting the described relationships, ratios, and totals into mathematical expressions or equations. This translation process is critical in ensuring that all elements of the problem are accurately represented and can be manipulated mathematically.

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