Problem 4 - Prealgebra

Question

Solve each of the following problems by first restating it as one of the three basic percent problems of Section 7.2 . In each case, be sure to show the equation. A family spends 450 dollar every month on food. If the family's income each month is 1,800 dollar what percent of the family's income is spent on food?

Step-by-Step Solution

Answer

25% of the family's income is spent on food.

Step 1: Understanding the Problem

We need to find out what percent of the family's income is spent on food. The monthly expenditure on food is $450, and the total monthly income is $1,800.

Step 2: Identifying the Basic Percent Problem

This is a 'what percent of' problem, where we need to find the percentage referenced to another value. We are looking for the percentage of $1,800 that $450 represents.

Step 3: Translating to an Equation

Use the basic percentage formula: \(\text{Part} = \text{Percent} \times \text{Whole}\). Here, \(\text{Part} = 450\), \(\text{Whole} = 1800\), and we are solving for \(\text{Percent}\). This gives us the equation: \(450 = x \times 1800\).

Step 4: Solving for the Percent

Rearrange the equation to solve for \(x\): divide both sides by 1800: \(x = \frac{450}{1800}\).

Step 5: Calculating the Percent

Compute the value of \(x\) as a decimal, and then convert it to a percentage: \(x = \frac{450}{1800} = 0.25\), which corresponds to 25%.

Key Concepts

Percent Calculation

Understanding percent calculations is essential for solving many everyday problems, like determining how much of your income is spent on food, for example. In our exercise, we're interested in finding out the percentage of the family's income that goes towards groceries. This involves comparing a part (money spent on food) to the whole (total income). The goal is to express the part as a portion of the whole in percentage terms.

When calculating percentages, here’s a streamlined approach to follow:

• Identify the figures involved: the part (in this exercise, $450) and the whole (here, $1,800).
• Convert the part-to-whole relationship into a percentage equation.
• Solve the equation to find out what percentage the part is of the whole.

We use this method because percentages provide a simple way to compare different quantities by expressing them in terms of 100.

Basic Percentage Formula

The basic percentage formula is your go-to equation when tackling percent-related problems. It consists of setting up an equation that relates the part (what we're interested in) to the whole amount (total we're comparing against). The formula is expressed as:

• \[ \text{Part} = \text{Percent} \times \text{Whole} \]

This formula is handy because it can be rearranged depending on what you need to find. In the exercise, we need to discover the percentage of the family's income that goes to food expenses. For such problems, the formula can be manipulated to solve for the percent:

• \[ \text{Percent} = \frac{\text{Part}}{\text{Whole}} \times 100 \]

Here, the part is \(450 (money spent on food), and the whole is \)1,800 (total income). Rearranging and solving this equation will give us the percentage we want.

Prealgebra

Prealgebra forms the foundational skills needed to tackle percentage calculations. It involves understanding basic arithmetic operations like multiplication, division, and the relationships between numbers. In problems related to percents, prealgebra helps with setting up and interpreting equations.

For instance, in our problem, prealgebra guides us to rework the equation \[ 450 = x \times 1800 \]to solve for \( x \), representing the percent. This involves dividing both sides by 1800 to isolate \( x \). Such steps require understanding the logic of equations:

• If two sides of an equation are equal, you can perform the same operation on both sides without changing the equation's equality.
• Rearranging equations to isolate variables often involves basic division and multiplication.

Mastering these prealgebra techniques is crucial as they apply to numerous math problems, including percent calculations.