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There is a family who loves to play video games. The family plays an app called GoBlocks, which allows users to play a wide variety of games and talk to each other online. The family can build together, play together, and purchase in-game accessories to personalize their avatars. The application is free, but there are in-app purchases made with GoBucks. With these GoBucks, users can purchase accessories for their avatars or buy special abilities.


  1. $\frac{2}{3}$ of the budget for GoBucks goes to the adults in the family, and the rest goes to the children in the family. The total budget can be calculated using the following modeled equation:

$\frac{2}{3}x = 120$


AWhat does the constant 120 mean in the equation?
BWhat does the variable $x$ represent in the equation? Solve for $x$.
CHow much money will go to the children in the family?

Verified Answer :

  • The constant \(120\) represents the amount of GoBucks allocated to the adults in the family.
  • The variable \(x\) represents the total GoBucks budget, which is calculated to be \(\boxed{180}\).
  • The amount allocated to the children is \(\boxed{60}\).

Explanation

  1. Problem Analysis

We are given information about a family's budget for "GoBucks," used for in-game purchases in the GoBlocks application. The budget is divided between adults and children. Specifically, \(2/3\) of the total budget is allocated to the adults, and the remaining \(1/3\) is allocated to the children. We are provided with two equations: \(\frac{2}{3}x = 120\) and \(\frac{1}{3}x = 60\). Our goal is to interpret the values in these equations and find the total budget and the children's portion.

  1. Interpretation of the Constant 120

The constant \(120\) in the equation \(\frac{2}{3}x = 120\) represents the specific portion of the total GoBucks budget that is allocated to the adults in the family. Since the problem states that \(2/3\) of the budget is for the adults, this constant directly corresponds to that share.

  1. Representing and Solving for x

The variable \(x\) represents the total budget available for GoBucks in the family. To solve for \(x\), we can use the equation \(\frac{2}{3}x = 120\). By multiplying both sides of the equation by \(\frac{3}{2}\), we get:
\[x = 120 \times \frac{3}{2} = 60 \times 3 = 180.\]
Thus, the total budget for GoBucks is \(180\). We can verify this with the second equation: \(\frac{1}{3} \times 180 = 60\), which is consistent with the problem statement.

  1. Children's Budget Allocation

The children's portion of the budget is given as \(1/3\) of the total budget. Using the calculated total budget \(x = 180\), the amount allocated to the children is \(\frac{1}{3} \times 180 = 60\). This matches the value provided in the second equation of the problem.

Examples

Understanding how to break down a total budget into proportions is a foundational skill in personal finance and project management. For example, if you are organizing a small event with a fixed budget for food, decorations, and entertainment, you can model these costs as fractions of your total budget. By solving for the total budget and its individual components, you ensure that every part of your plan is fully funded and that your financial resources are distributed as intended.

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