Problem 97 - Elementary Algebra

Question

A bacon cheeseburger at a popular fast food restaurant contains 2070 milligrams (mg) of sodium, which is \(86 \%\) of the recommended daily amount. What is the total recommended daily amount of sodium?

Step-by-Step Solution

Answer

2406.98 mg

Step 1: Identify the Given Information

A bacon cheeseburger contains 2070 mg of sodium, which is 86% of the recommended daily amount.

Step 2: Set Up an Equation

Let the total recommended daily amount of sodium be denoted as \( x \). According to the problem, 2070 mg represents 86% of \( x \). This can be written as an equation: \( 0.86x = 2070 \)

Step 3: Solve for \( x \)

To find \( x \), divide both sides of the equation by 0.86:\[ x = \frac{2070}{0.86} \]

Step 4: Calculate the Value of \( x \)

Perform the division to find the value:\[ x = 2406.98 \] Rounding to two decimal places, the total recommended daily amount of sodium is approximately 2406.98 mg.

Key Concepts

Percentage Calculations

Understanding percentages is essential in many real-life situations, including this problem about sodium content. A percentage represents a part out of 100. In this exercise, we are given that 2070 mg of sodium is 86% of the recommended daily intake. To find the full amount, we set up an equation where 2070 mg represents 86% of an unknown total amount.

Here's a quick guide on handling percentages:

• Convert the percentage to a decimal by dividing by 100 (i.e., 86% becomes 0.86).
• Multiply the total amount (that you want to find) by this decimal to get the part of the total.

In our example, 2070 mg is obtained by multiplying the unknown value (recommended daily intake) by 0.86. This is written as:

\(2070 = 0.86x \). Once the equation is set, solving for the unknown becomes the next step.

Basic Algebra

Algebra helps us solve for unknown values, just like in this sodium content problem. We start by setting up an equation based on the given information. In algebra, we often use variables (like x) to represent unknown quantities.

Here, we used x as the recommended daily intake and set up the equation as 2070 mg representing 86% of x: \(0.86x = 2070 \). To isolate the variable x, we need to perform operations that 'undo' what has been done to it.

Let's walk through these steps:

• Start with the equation: \[ 0.86x = 2070 \]
• To solve for x, divide both sides by 0.86: \[ x = \frac{2070}{0.86} \]

This isolated x, allowing us to find the value easier.

Solving Equations

Solving equations involves finding the value of the unknown variable that makes the equation true. In the sodium content problem, we needed to find the recommended daily intake (x).

After setting up the equation \(0.86x = 2070 \), we solved for x by dividing both sides by 0.86. This operation is critical because it 'cancels out' the multiplication by 0.86, isolating x on one side of the equation: \[ x = \frac{2070}{0.86} \].

Once we have our isolated variable, we perform the actual division to find x:
\ x = 2406.98 \ Therefore, the recommended daily amount of sodium is approximately 2406.98 mg.

Remember, solving equations might require various steps, such as adding, subtracting, multiplying, or dividing both sides by the same number. Each step helps us systematically isolate the unknown variable.