Question

Translate to an equation and solve. After two months of fund-raising, an organization raised \(72 \%\) of its goal. If the current amount raised is \(\$ 61,200,\) what is the goal amount?

Short answer

Answer: $85,000

Step-by-step solution

Set up an equation

Let's represent the goal amount as 'x'. We know that 72% of the goal amount is equal to $61,200. So, we can write the equation as: 0.72x = 61,200

Solve the equation

To find the value of x, which represents the goal amount, we need to get rid of the multiplier (0.72) by dividing both sides of the equation by it: x = 61,200 / 0.72

Calculate the goal amount

Carry out the division to find the value of x: x = 85000 So, the goal amount is \(\$ 85,000\).

Key concepts

Translating Word Problems

When facing a word problem, the first task is to convert the scenario described in words into an algebraic equation. This involves identifying the variables and the relationships between them.

• Reading carefully: Focus on the key information given. In our fund-raising example, we focus on the percentage of the goal reached and the actual amount raised.
• Identifying variables: Determine what you are solving for, which is the organization's goal. We represent this unknown value with a variable, commonly 'x'.
• Understanding relationships: Relate the information provided to the variable. The '72%' mentioned is crucial, as it describes the portion of the total goal reached.
• Setting up an equation: Combine all these elements to form an equation. If 72% of the goal (0.72x) equals $61,200, we get the equation 0.72x = 61,200.

By following these steps, translating word problems into algebraic expressions becomes more manageable, paving the way for successful problem-solving.

Percentages in Algebra

Understanding percentages is vital when solving algebra problems involving proportions. Here's how to work with percentages:

• Converting percentages to decimals: To use percentages in calculations, convert them into decimal form by dividing by 100. In our problem, 72% becomes 0.72.
• Interpreting percentages: Recognize that 'percentage of' means multiplying the percentage (in decimal form) by the whole. So, 72% of the goal amount is 0.72 times the goal (x).

Remember, when a problem involves finding a percentage of a certain value, the percentage must first be converted to a decimal to be multiplied with the given value correctly. This fundamental concept aids in setting up accurate algebraic equations for such problems.

Equation Solving Steps

The process of solving equations in algebra is systematic and can be applied to a variety of problems. To solve basic algebra equations, follow these simple steps:

• Setting up the equation: As discussed previously, translate the problem into an algebraic equation using the proper mathematical operations.
• Isolate the variable: In our case, we began with 0.72x = 61,200. The goal is to isolate 'x' by performing the same operation on both sides of the equation.
• Operating the numbers: We divide both sides of the equation by 0.72 to cancel the multiplier, leaving x by itself on one side. After doing this, we get x = 61,200 / 0.72.
• Calculating the solution: Now, perform the actual division to find the value of 'x'. The division yields x = 85,000, which represents the desired goal amount.

The equation-solving pathway leads to the solution systematically. Regardless of the complexity of the algebra problem, understanding and following these steps will immensely help in reaching the answer.