Problem 22 - Algebra 1

Question

Solve the percent problem. \(\$ 4\) is \(2.5 \%\) of what amount?

Step-by-Step Solution

Answer

$4 is 2.5% of $160.

Step 1: Translate the Problem to an Equation

The problem says: $4 is 2.5% of an amount X. We can express this as an equation: \(4 = 0.025 * X\), where 2.5% has been converted into decimal form (0.025).

Step 2: Solve for X

In order to solve for X, we would need to divide both sides of the equation by 0.025. This gives us: \(X = \frac{4}{0.025}\).

Step 3: Perform the Calculation

Finally, we calculate the right side of the equation to find the value of X. This gives: \(X = 160\).

Key Concepts

Translating Word Problems to Equations

Understanding how to interpret words into equations is a foundational skill in algebra, particularly when dealing with percent problems. When you read a word problem, look for keywords and phrases that indicate mathematical operations. For example, if you encounter the phrase 'is 2.5% of,' it suggests a part-whole relationship where you are dealing with a percentage of a certain quantity. You will want to identify the 'part' (in this case, \(4), the 'percent' (2.5%), and the 'whole' which is the unknown quantity we are looking to find.

To create the equation, start by representing the unknown quantity with a variable, such as 'X'. The word 'is' translates into the equals sign '=', and 'percent of' refers to multiplication after converting the percent to a decimal. Therefore, the statement '\)4 is 2.5% of what amount?' becomes the equation 4 = 0.025 * X.

Solving Equations

Once we have translated our word problem into an equation, the next step is to solve for the unknown variable. Solving equations generally involves isolating the variable on one side of the equation. In our example, to find out what amount corresponds to \(4 being 2.5% of it, we set up the equation as 4 = 0.025 * X.

To isolate the variable X, we need to get rid of the multiplication by 0.025. To do this, we do the opposite operation—division—on both sides of the equation. So, we divide the \)4 by 0.025, which is the same as multiplying by the reciprocal of 0.025. This allows us to solve for X as follows: \(X = \frac{4}{0.025}\). Once X is isolated, we can perform the calculation to find the value of X.

Percent to Decimal Conversion

Percent to decimal conversion is a vital skill for solving percent problems algebraically. A 'percent' is derived from the Latin 'per centum', meaning 'by the hundred'. Essentially, a percentage is a fraction with a denominator of 100. To convert a percent to a decimal, simply divide by 100. This process removes the percent sign and moves the decimal point two places to the left.

For instance, to convert 2.5% to a decimal, divide 2.5 by 100, which equals 0.025. This conversion is crucial because we typically do not perform mathematical operations with percentages directly in equations. Instead, we use their decimal equivalents which allow us to use standard arithmetic operations such as addition, subtraction, multiplication, and division, as required in our equation.