Problem 70 - Intermediate Algebra

Question

Use a proportion to solve each problem. Telephones. In \(2009,\) the number of mobile telephone lines used by residents of the United Arab Emirates reached a record high of 191 lines per 100 people. If the Emirates' population was about \(4,599,000\) at that time, how many mobile telephone lines were being used?

Step-by-Step Solution

Answer

The Emirates had approximately 8,785,290 mobile telephone lines in 2009.

Step 1: Understand the Proportion

The problem provides a ratio of mobile telephone lines to people, which is 191 lines for every 100 people. This can be set up as the proportion \( \frac{191}{100} \).

Step 2: Set up the Proportion Equation

We need to find the number of mobile telephone lines for the given population of 4,599,000. Set up the proportion: \( \frac{191}{100} = \frac{x}{4,599,000} \), where \( x \) is the number of mobile lines to find.

Step 3: Solve for x

To find \( x \), use cross multiplication. The equation becomes \( 191 \times 4,599,000 = 100 \times x \). Simplify and solve for \( x \): \( x = \frac{191 \times 4,599,000}{100} \).

Step 4: Calculate the Solution

Calculate \( x \) using the values: \( x = \frac{191 \times 4,599,000}{100} = 8,785,290 \).

Step 5: Conclusion

Therefore, the number of mobile telephone lines being used was approximately 8,785,290.

Key Concepts

Understanding Ratios

Ratios are a way to compare two quantities, showing the relationship between them. In the original problem, we are given a ratio of mobile telephone lines to people: 191 lines for every 100 people.
Ratios can be expressed in several ways:

• In words, like "191 line per 100 people."
• Using a colon, like "191:100."
• As a fraction, like \( \frac{191}{100} \).

When using ratios, it's crucial to maintain the same order, so they reflect the given relationship correctly. In problem-solving, knowing how to interpret and use ratios gives you a powerful tool for understanding relationships in data.

Mastering Cross Multiplication

Cross multiplication is an essential tool for solving proportions. A proportion is simply an equation that states two ratios are equal. The original problem is a great example where this technique comes in handy. We set up our proportion as \( \frac{191}{100} = \frac{x}{4,599,000} \), aiming to find the unknown \( x \).
The process of cross multiplication involves multiplying the numerator of one fraction by the denominator of the other. Here's how it works:

• Multiply 191 by 4,599,000.
• Multiply 100 by \( x \) (our unknown).

This gives us the equation:
\[ 191 \times 4,599,000 = 100 \times x \]
By simplifying this equation, we isolate and solve for \( x \). Cross multiplication is a fantastic shortcut that simplifies the method of finding unknowns in ratios.

Applying Problem Solving Techniques

Problem solving is all about breaking down complex tasks into manageable steps. In our exercise, the solution involved solving for the number of mobile lines using a series of logical steps. Here’s a recap of the approach:

• Understand the problem statement and identify the ratio.
• Set up the proportion that needs solving.
• Use cross multiplication to form an equation.
• Solve the equation to find the unknown.
• Verify that the solution makes sense by reviewing the context.

By following these steps, even complex problems can be handled systematically. Each step builds upon the previous one. This approach not only ensures accuracy but also makes the problem-solving process much less daunting.

Building Mathematical Reasoning Skills

Mathematical reasoning is about making logical connections. It helps us understand why certain methods work, which leads to better problem-solving skills. In our exercise, reasoning is applied in several ways:

• Recognizing the given ratio and what it means in real-world terms.
• Understanding how to transform a proportion into a solvable equation.
• Integrating cross multiplication to handle the calculated values.

Mathematical reasoning also helps you evaluate whether your solution is realistic. For instance, in our calculation of the mobile lines, you would ask, "Does this number make sense given the population?"
By cultivating this reasoning ability, you enhance your capacity to solve not only textbook problems but also everyday mathematical challenges effectively.