Problem 69 - Elementary and Intermediate Algebra

Question

Translate each problem to an equation. Do not solve. Recycling. Currently, Americans recover \(33.4 \%\) of all municipal solid waste. This is the same as recovering 85 million tons. What is the total amount of waste generated?

Step-by-Step Solution

Answer

0.334W = 85

Step 1 - Define the Variables

Let the total amount of waste generated be represented by the variable \( W \) (in million tons).

Step 2 - Convert the Percentage to a Decimal

The problem states that Americans recover \(33.4 \%\) of the waste. Convert this percentage to a decimal by dividing by 100: \[ 33.4 \% = \frac{33.4}{100} = 0.334 \]

Step 3 - Set Up the Equation

The number of tons of waste recovered is \(85\) million tons. Set up the equation by recognizing that \(0.334\) of the total waste \(W\) is equal to \(85\) million tons: \[ 0.334W = 85 \]

Key Concepts

Equation Translation

Understanding how to translate a real-world problem into an algebraic equation is very important. To do this, you need to carefully analyze the problem statement and identify the mathematical relationships described.

In the provided exercise, we know that Americans recover a certain percentage of the total waste, and this recovery equals a specific quantity (85 million tons). Armed with this information, we express the relationship using variables and basic arithmetic operations.

First, let's identify the unknown quantity. We call the total amount of waste generated as the variable **W**. The relationship between the percentage recovered and the quantity recovered then has to be expressed in equation form.

In simpler terms:

• Percentage recovered (in decimal form) multiplied by total waste = recovered waste in tons.

The final equation is formed as follows: **0.334W = 85**.

Percentage Conversion

Knowing how to convert a percentage to a decimal is crucial in algebra. Percentages are often used in real-life problems to express proportions or parts of a whole in a way that is easier to understand.

Whenever you see a percentage, like **33.4%**, you can convert it to a decimal by dividing by 100. This is because 'percent' means 'per hundred'.

For example:

• 33.4% equals \(\frac{33.4}{100}\), which simplifies to 0.334.

This conversion step is necessary because equations and most mathematical operations require numbers to be in decimal form.

Therefore, converting the given percentage is a critical step in forming and solving the algebraic equation correctly.

Variable Definition

In algebra, variables are symbols used to represent unknown values or quantities that can change. Defining variables allows us to translate real-world problems into mathematical language.

In the problem given, the unknown quantity we need to find is the total amount of waste generated. Instead of using long phrases every time, we define this quantity using a single letter, which is **W** in our case.

To clearly define the variable:

• Let **W** be the total waste generated in million tons.

Defining variables simplifies the problem and makes it easier to manipulate mathematically. This step is crucial before setting up any equations as it clarifies what each symbol in the equation represents.

By using **W** for the total waste, we can set up the equation: 0.334W = 85, where **W** is the total amount of waste, 0.334 is the percentage of waste recovered expressed as a decimal, and 85 is the amount recovered in million tons.