Problem 53 - Prealgebra

Question

Pyramids. The Luxor Hotel in Las Vegas is \(\frac{5}{7}\) the original height of the Great Pyramid of Giza. If the hotel is 350 feet tall, what was the original height of the Great Pyramid of Giza?

Step-by-Step Solution

Answer

The original height of the Great Pyramid of Giza was 490 feet.

Step 1: Understanding the Relationship

We are given that the height of the Luxor Hotel is \( \frac{5}{7} \) of the height of the Great Pyramid of Giza. We need to find the original height of the Great Pyramid of Giza (denoted as \( H \)). The Luxor Hotel's height is given as 350 feet.

Step 2: Setting Up the Equation

The relationship between the Luxor Hotel's height and the Great Pyramid can be expressed as an equation: \( \frac{5}{7}H = 350 \). This equation is derived from the fact that the Luxor Hotel's height is \( \frac{5}{7} \) of the Great Pyramid's height.

Step 3: Solving the Equation for H

To find \( H \), we need to isolate \( H \) in the equation \( \frac{5}{7}H = 350 \). Multiply both sides of the equation by \( \frac{7}{5} \) to cancel out the fraction on the left: \[ H = 350 \times \frac{7}{5} \]

Step 4: Calculating the Original Height

Now, perform the multiplication:\[ H = 350 \times \frac{7}{5} = 350 \times 1.4 = 490 \]So, the original height \( H \) of the Great Pyramid of Giza was 490 feet.

Key Concepts

Understanding Fractions

Fractions are a way of representing a part of a whole. They are composed of two numbers: a numerator and a denominator. The numerator, the top number, signifies how many equal parts are being considered. The denominator, the bottom number, indicates into how many parts the whole is divided. For instance, in the fraction \(\frac{5}{7}\), the number 5 is the numerator, and 7 is the denominator. This means we have 5 out of 7 equal parts.

In the context of proportion problems, like the one we have here, fractions help describe the relative size or quantity compared to the whole. By understanding that \(\frac{5}{7}\) of something, like the height of the Great Pyramid, equals a specific value (like 350 feet, the height of the Luxor Hotel), we can use fractions to find unknown quantities. It's a crucial skill in math, especially when dealing with partial measurements or proportions.

Solving Equations

Equation solving is a fundamental part of mathematics that involves finding the value of unknown quantities. In problems involving proportions, equations often include fractions. Solving these equations requires understanding how to manipulate fractions and operations to isolate the unknown.

In this exercise, the equation \(\frac{5}{7}H = 350\) represents the relationship between the Luxor Hotel and the Great Pyramid of Giza. We solve for \(H\), the height of the Great Pyramid, by removing the fraction. This is done by multiplying both sides of the equation by the reciprocal of the fraction, \(\frac{7}{5}\), to cancel out \(\frac{5}{7}\).

This demonstrates how to effectively "undo" operations. By performing the same operation on both sides of an equation, we maintain its balance. Solving these proportion problems with real-world applications is not only useful but enhances logical thinking skills.

Real-World Applications of Proportions

Proportion problems have vast real-world applications. They often help in estimating quantities, scaling objects, or comparing dimensions, just like comparing the heights of two pyramids in this exercise. When we say that the Luxor Hotel is \(\frac{5}{7}\) the height of the Great Pyramid, we use our understanding of proportions to find the actual dimensions.

In architecture, proportions can be used to create models that are smaller or larger than life but maintain the true essence and scale. Engineers use proportions to understand and predict the forces at play in large structures. In many everyday situations, from cooking using recipe adjustments to converting between measurement systems, proportions are incredibly valuable.

By learning to solve proportion problems, like finding the Great Pyramid's height based on the Luxor Hotel's, students can better appreciate these connections to real-world tasks and decisions.