Problem 55 - Algebra A Combined Function

Question

The area of the Sahara Desert is 7 times the area of the Gobi Desert. If the sum of their areas is 4,000,000 square miles, find the area of each desert.

Step-by-Step Solution

Answer

Gobi Desert: 500,000 sq miles; Sahara Desert: 3,500,000 sq miles.

Step 1: Define the Variables

Let's denote the area of the Gobi Desert as \( x \) square miles. Since the area of the Sahara Desert is 7 times the area of the Gobi Desert, it can be expressed as \( 7x \).

Step 2: Set Up the Equation

According to the problem, the sum of the areas of the Sahara and Gobi Deserts is 4,000,000 square miles. Thus, we can set up the equation:\[ x + 7x = 4,000,000 \]

Step 3: Simplify the Equation

Combine the terms on the left side of the equation:\[ 8x = 4,000,000 \]

Step 4: Solve for \( x \)

Divide both sides by 8 to find \( x \):\[ x = \frac{4,000,000}{8} = 500,000 \]So, the area of the Gobi Desert is 500,000 square miles.

Step 5: Find the Area of the Sahara Desert

Since the area of the Sahara Desert is 7 times the area of the Gobi Desert, multiply \( x \) by 7:\[ 7x = 7 \times 500,000 = 3,500,000 \]Thus, the area of the Sahara Desert is 3,500,000 square miles.

Key Concepts

Equation solving

Solving equations is a fundamental skill in algebra, especially when dealing with word problems like the desert areas. The goal is to find an unknown variable by establishing a mathematical equation based on the given information. In this case, the equation to solve is formed from the relationship between the Sahara and Gobi Desert's areas:

• The total is 4,000,000 square miles.
• The Sahara Desert's area is described as 7 times that of the Gobi Desert.

Using these facts, we set up an equation: \( x + 7x = 4,000,000 \). Here, solving involves combining like terms and isolating the variable \( x \). With 8x representing the total area, simplifying gives \( 8x = 4,000,000 \). Divide both sides by 8 to solve for \( x \), giving \( x = 500,000 \). Thus, solving equations helps us find each desert's area.

Variable definition

Defining variables is crucial in solving algebra word problems efficiently. A variable acts as a placeholder for unknown quantities in the problem. In this exercise, we defined the Gobi Desert's area as \( x \). This allowed us to express the Sahara Desert's area using the same variable, as it is just 7 times \( x \), or \( 7x \). This process of defining variables simplifies complex word problems. Here’s how it works:
- Identify unknown quantities in the problem.
- Assign variables to these quantities to create expressions or equations.
For the deserts, once we defined \( x \), writing the equation for the total area became straightforward. This strategic assignment of variables makes the rest of the problem-solving process more manageable and less prone to errors.

Proportional relationships

In this exercise, understanding proportional relationships is key. These relationships describe how quantities relate to each other through a constant multiplier. Here, the problem states that the Sahara Desert's area is 7 times larger than the Gobi Desert's.
This proportion is crucial as it expresses how the two quantities scale relative to each other. The Sahara to Gobi ratio is 7:1. When we defined the Gobi Desert's area as \( x \), it was intuitive to represent the Sahara Desert's area as \( 7x \).
Identifying proportional relationships allows us to set up the correct equations and work towards solutions efficiently. In real-world problems, recognizing these relationships helps navigate different scenarios, whether comparing areas, speeds, or costs. Using proportions in equations turns complicated relationships into solvable problems, as highlighted in this desert area exercise.