Problem 61 - Prealgebra

Question

The Eiffel Tower is \(1,060\) feet tall and the Stratosphere Tower in Las Vegas is \(1,150\) feet tall. Write the height of the Eiffel tower over the height of the Stratosphere Tower and then reduce to lowest terms.

Step-by-Step Solution

Answer

The reduced ratio is \( \frac{106}{115} \).

Step 1: Write the Ratio

The first step is to express the given data as a ratio. The height of the Eiffel Tower is 1,060 feet, and the height of the Stratosphere Tower is 1,150 feet. Thus, the ratio is written as \( \frac{1060}{1150} \).

Step 2: Find the Greatest Common Divisor (GCD)

To reduce the fraction, we need to find the greatest common divisor of 1,060 and 1,150. Using the Euclidean algorithm or a GCD calculator, we find that the GCD is 10.

Step 3: Divide Both Numerator and Denominator by the GCD

Now, divide both the numerator and the denominator of the ratio by the GCD (which is 10). So, \( \frac{1060}{1150} = \frac{1060 \div 10}{1150 \div 10} = \frac{106}{115} \).

Step 4: Check for Further Reduction

Check if the resulting fraction \( \frac{106}{115} \) can be reduced further by finding the GCD of 106 and 115. The GCD of 106 and 115 is 1, which means the fraction is already in its lowest terms.

Key Concepts

Greatest Common Divisor

The greatest common divisor, often abbreviated as GCD, is used to simplify fractions. It is the largest number that divides two or more integers without leaving a remainder. For example, let's consider the numbers 1,060 and 1,150 in our exercise. To find the GCD:

• List the factors of 1,060 and 1,150.
• Identify the common factors shared by these two numbers.
• Select the largest of these common factors.

For this exercise, the GCD of 1,060 and 1,150 is determined to be 10 using the Euclidean algorithm. This algorithm involves repetitive division steps until a remainder of zero is achieved. Understanding and applying the GCD is crucial for the next concept: reducing fractions.

Reducing Fractions

Reducing fractions involves dividing both the numerator and the denominator by their greatest common divisor until no common factors remain. This process simplifies the fraction to its smallest form, making it easier to work with or interpret.In our specific exercise, we start with the ratio of the Eiffel Tower's height to the Stratosphere Tower's height, which is expressed as:\[ \frac{1060}{1150} \]Using the GCD of 10, we reduce this fraction by dividing both terms by 10:\[ \frac{1060 \div 10}{1150 \div 10} = \frac{106}{115} \]Then, we check if the new fraction can be simplified further. Since the GCD of 106 and 115 is 1, the fraction is already at its lowest terms. It's essential for solving problems efficiently and accurately.

Comparison of Heights

Comparison of ratios in practical situations, like comparing heights, reveals relative sizes or magnitudes in a straightforward way. By using ratios, complex comparisons become simple.The ratio we worked with is a straightforward method to compare the height of the Eiffel Tower to that of the Stratosphere Tower. After reduction, it remains:\[ \frac{106}{115} \]This fraction effectively tells us that for every 115 feet of the Stratosphere Tower, the Eiffel Tower stands 106 feet tall. This process is particularly helpful in various real-world applications:

• Scaling models accurately.
• Understanding proportional differences.
• Making data-driven decisions based on size comparisons.

Understanding and using ratios and proportions allows for elegant and manageable representations of real-world data.