Question

Monthly sales of a particular brand of smartphone are inversely proportional to the number of occurrences of exploding cellphone batteries during that month. If 88 million phones were sold when there were 2 exploding batteries, how many cell phones would be sold during a month in which 22 batteries exploded? (A) 4 million (B) 8 million (C) 44 million (D) 176 million

Short answer

During the month when 22 batteries exploded, 8 million cell phones would be sold. The correct answer is (B) 8 million.

Step-by-step solution

Understand the inverse proportionality relationship

In this case, the number of smartphones sold (S) is inversely proportional to the number of exploding batteries (B). Mathematically, this can be represented as: S ∝ \(\frac{1}{B}\)

Use the data given to find the constant of proportionality

We are given that 88 million phones were sold when there were 2 exploding batteries. So, we can write the equation as: S = k\(\frac{1}{B}\) or \(S = \frac{k}{B}\) Now, using the given data, we have: 88 million = \(\frac{k}{2}\) Let's solve for k: k = 2 * 88 million = 176 million

Plug the value of k in the equation for the other month

We need to find the sales during a month in which 22 batteries exploded. We plug the value of k found in step 2 and B = 22 into our equation to get: S = \(\frac{176 \text{ million}}{22}\)

Solve for the sales (S)

Now, let's compute the value of S: S = \(\frac{176 \text{ million}}{22}\) = 8 million Hence, during the month when 22 batteries exploded, 8 million cell phones would be sold. The correct answer is (B) 8 million.

Key concepts

Proportional Relationships

In mathematics, understanding proportional relationships is fundamental, especially when dealing with real-world problems.

Proportional relationships express how one quantity changes in a direct or inverse proportion to another. In a directly proportional relationship, as one quantity increases, the other also increases at a consistent rate, represented by the formula: \( y = kx \), where \( y \) and \( x \) are the two variables, and \( k \) is the constant of proportionality.

In contrast, an inverse proportional relationship, which is the focus of the textbook exercise, shows that as one quantity increases, another decreases at a rate that maintains a constant product. This relationship can be symbolized by the equation: \( y = \frac{k}{x} \). An everyday example of this could be speed and travel time: the faster you travel (increasing speed), the less time it takes to reach a destination (decreasing time). The constant \( k \) is what links the two inversely related quantities. Students practicing for their GED math tests will benefit from recognizing and solving problems involving these kinds of relationships.

GED Math Practice

When preparing for the GED math exam, it's crucial to practice a diverse set of problems. The GED math test covers various topics, including algebra, geometry, and number operations.

As part of this preparation, students should be comfortable with concepts like inverse proportionality, which can appear in questions like the smartphone sales problem from the exercise. Understanding how to manipulate equations involving inversely proportional relationships is a valuable skill.

Practice Tips

• Familiarize yourself with identifying proportional relationships in problems.
• Work on converting word problems into mathematical equations.
• Remember to find the constant of proportionality and use it to solve for unknown quantities.
• Balance your study sessions with a mix of problem solving, concept review, and timed practice tests.

By integrating these strategies into study routines, students are better equipped to tackle the GED math section with confidence.

Constant of Proportionality

The constant of proportionality is the link that maintains the proportional relationship between two quantities.

In the provided exercise, the constant of proportionality (\( k \)) represents the product of the phones sold (\( S \)) and the occurrences of exploding batteries (\( B \)) in their inverse relationship. It is the fixed value that determines how one variable affects the other and it remains constant regardless of the values of the variables. You can find \( k \) by arranging the equation to isolate it, which in this case is shown by multiplying the number of exploding batteries with the quantity of phones sold.

Finding the Constant

To calculate \( k \) for inverse proportionality, take the initial known values, as seen in the smartphone problem: \( k = S \times B \). Once determined, this constant allows us to predict other values within the inverse relationship, thus providing a powerful tool for solving problems with variables that are inversely related. Moreover, an understanding of how to identify and use this constant is an important part of algebra and will prove invaluable not just for academic purposes, but in real-world applications as well.