Problem 52 - Elementary Algebra

Question

Answer the question by setting up and solving an appropriate equation. \(55 \%\) of what number is \(38.5\) ?

Step-by-Step Solution

Answer

The number is 70.

Step 1: Understand the problem

The problem is asking us to find a number such that 55% of this number is equal to 38.5. To solve this, we need to represent the unknown number with a variable.

Step 2: Represent the unknown

Let the unknown number be represented by the variable \(x\). We need to find \(x\) such that 55% of \(x\) equals 38.5.

Step 3: Set up the equation

Express 55% as a decimal, which is 0.55. The equation representing "55% of \(x\) is 38.5" is given by: \[ 0.55x = 38.5 \]

Step 4: Solve for x

To find \(x\), divide both sides of the equation by 0.55: \[ x = \frac{38.5}{0.55} \] Calculating this gives: \[ x \approx 70 \]

Step 5: Verify the solution

Check the calculation by multiplying 70 by 0.55. If the result is 38.5, then the solution is correct. \[ 0.55 \times 70 = 38.5 \] The calculation is correct.

Key Concepts

Solving Equations

Solving equations is all about finding the value of unknowns that make the equation true. In our example, the goal is to find a number which, when 55% of it is taken, results in 38.5. This involves establishing a balance. We set up an equation based on the given relationship. The equation states that a certain percentage of a number equals another given number. This is written as a mathematical statement describing the problem.

To solve the equation, follow these steps:

• Set up the equation based on the problem statement. Here, the relationship is represented by the equation: \(0.55x = 38.5\).
• The goal is to isolate the unknown variable, \(x\), to find its value. This involves performing mathematical operations that maintain the balanced nature of the equation.
• In this example, you'll divide both sides of the equation by 0.55 to solve for \(x\). The steps you take should maintain the equation's equality.

The solution of equations requires patience and careful manipulation of the numbers involved. Practice helps you get more comfortable with recognizing how to handle different types of problems.

Unknown Variables

In algebra, unknown variables often symbolize values we need to find. They are usually denoted by letters like \(x\), \(y\), or \(z\). In the example, we used \(x\) as the placeholder for our unknown number.

• Think of variables as little mystery boxes. Once you solve the equation, you figure out what number is inside them.
• Assigning a variable helps create a mathematical statement that mirrors the real-world question you're trying to answer. Here, \(x\) was used to represent the unknown number that 55% equaled 38.5.
• Once the solution is obtained, replacing the variable with the found number gives you the answer.

Handling unknown variables gets easier with practice, as you start seeing patterns in how they relate to other quantities in equations.

Conversion of Percentages to Decimals

Understanding how to convert percentages to decimals is crucial in solving percentage problems. Since mathematical equations handle decimals more naturally than percentages, converting them early in the process simplifies the solution.

• A percentage represents a fraction out of 100, so converting it to a decimal involves dividing by 100. For instance, 55% is converted by computing 55 ÷ 100 = 0.55.
• This decimal form can then be used in equations, making calculations straightforward and eliminating the extra steps that percentages introduce.
• Converting correctly ensures the solution remains precise and reflects the problem accurately.

Getting comfortable with these conversions allows for more efficient problem-solving as you encounter more complex algebraic equations involving percentages.