Problem 79 - Elementary Algebra

Question

In the following exercises, translate and solve. \(8.2 \%\) of what number is \(\$ 2.87 ?\)

Step-by-Step Solution

Answer

35

Step 1: Identify the equation

To find the number for which 8.2% is $2.87, start by setting up the equation for the given problem. Let the unknown number be represented by 'x'. We know 8.2% of 'x' is 2.87. The equation is given by: \[ 8.2\text{%} \times x = 2.87 \]Note that 8.2% can be written as a decimal (0.082), so the equation becomes: \[ 0.082 \times x = 2.87 \]

Step 2: Solve for 'x'

To isolate 'x', divide both sides of the equation by 0.082: \[ x = \frac{2.87}{0.082} \]

Step 3: Calculate

Carry out the division to find the value of 'x': \[ x = \frac{2.87}{0.082} \approx 35 \]

Key Concepts

percentage calculations

Percentage calculations are a fundamental part of mathematics. Understanding how to work with percentages is essential for many real-world applications.

In the given exercise, we need to find out what number, when 8.2% is taken, equals $2.87. To find an unknown percentage of a number, follow these steps:

• Convert the percentage to a decimal by dividing by 100.
• Set up an equation where the percentage (as a decimal) times the unknown number equals the given value.
• Solve the equation by isolating the unknown number.

For instance, here we started with 8.2% and converted it to a decimal (0.082). Then, we set up the equation: \[0.082 \times x = 2.87\]. Finally, we solved for 'x' by dividing both sides of the equation by 0.082.

solving equations

Solving equations involves finding the value of an unknown variable that makes the equation true. This is a core skill in algebra and essential for solving real-world problems.

In our task, the equation given was: \[0.082 \times x = 2.87\]. The goal is to isolate the variable 'x' (the unknown number). To do this:

• We first recognize that 0.082 (the coefficient) is multiplied by 'x'.
• To isolate 'x', we need to perform the opposite operation, which is division.

Dividing both sides by 0.082 results in: \[ x = \frac{2.87}{0.082} \]. This simplifies the equation and allows us to solve for 'x'. The final step is performing the division to get approximately 35.

decimal conversion

Decimal conversion is the process of converting a percentage to a decimal. This step is crucial because most mathematical operations require values to be in the same format.

To convert a percentage to a decimal:

• Divide the percentage by 100.
• For example, 8.2% becomes 0.082 (8.2 ÷ 100).

In the exercise, converting 8.2% to 0.082 allows us to write the equation in a format suitable for algebraic manipulation: \[0.082 \times x = 2.87\]. Without this conversion, solving for 'x' would be complicated.

basic algebra

Basic algebra is the cornerstone of solving mathematical problems involving variables and constants. It includes understanding and manipulating algebraic expressions and equations.

In solving our exercise, we applied basic algebra principles:

• Setting up an equation.
• Converting percentages to decimals.
• Isolating the unknown variable through division.

These steps showcase how foundational algebra skills streamline solving more complex problems. By mastering basic algebra, you gain a versatile toolset for tackling numerous mathematical challenges.