Problem 89 - Algebra 1: Concepts and Skills

Question

The distance from the sun to Earth is approximately \(1.5 \times 10^{8} \mathrm{km} .\) The distance from the sun to the planet Neptune is approximately \(4.5 \times 10^{9} \mathrm{km} .\) What is the ratio of Earth's distance from the sun to Neptune's distance from the sun?

Step-by-Step Solution

Answer

The ratio of Earth's distance from the sun to Neptune's distance from the sun is \(\frac{1}{30}\)

Step 1: Understanding the Problem

The exercise gives two distances: the distance from the sun to Earth which is approximately \(1.5 \times 10^{8}\) km, and the distance from the sun to Neptune which is approximately \(4.5 \times 10^{9}\) km. The task is to calculate the ratio of Earth's distance from the sun to Neptune's distance from the sun.

Step 2: Writing the Ratio

The ratio of Earth's distance from the sun to Neptune's distance from the sun can be written as follows: \(\frac{Earth's Distance}{Neptune's Distance} = \frac{1.5 \times 10^{8}}{4.5 \times 10^{9}}\)

Step 3: Simplifying the Ratio

To simplify the ratio, divide the numerator by the denominator: \(\frac{1.5 \times 10^{8}}{4.5 \times 10^{9}} = 3.33 \times 10^{-2}\)

Step 4: Convert the Ratio to Simplest Form

Convert the decimal \(3.33 \times 10^{-2}\) to a fraction in its simplest form. This is equal to \(\frac{1}{30}\).

Key Concepts

Simplifying Ratios

When we talk about simplifying ratios, we are referring to the process of making a ratio easier to understand by eliminating any common factors between the numbers involved. Simplifying ratios is vital in comparing quantities in mathematics and in real-life situations.

For instance, if we have a ratio like \( \frac{8}{12} \), we can simplify it by dividing both the numerator and the denominator by their greatest common divisor, which is 4 in this case. This results in the simplified ratio of \( \frac{2}{3} \). This process is analogous to reducing fractions in arithmetic.

This simplification is useful especially when dealing with large numbers since it can offer a clearer understanding of the relationship between the quantities. In the exercise given, simplifying the ratio of distances between Earth and Neptune helps us grasp the scale of astronomical distances without getting overwhelmed by the large digits.

Scientific Notation

Scientific notation is a method of expressing very large or very small numbers in a compact form. In this system, numbers are written as the product of two factors: a decimal number greater than or equal to 1 but less than 10, and a power of 10.

For example, \( 3.2 \times 10^{6} \) is scientific notation for 3,200,000. It's incredibly useful in fields such as science and engineering where one often encounters quantities that span many orders of magnitude.

In our exercise, the distances from the sun to Earth and Neptune were given in scientific notation to make the figures more manageable. When you work with these numbers in calculations, such as finding ratios, you maintain the clarity and precision of the numbers without getting lost in the zeros.

Astronomical Distances

Astronomical distances are measures used to describe the vast expanses between celestial bodies in the universe. These distances are so immense that they are challenging to comprehend using regular units like kilometers or miles.

For context, the average distance from the Earth to the sun, also known as an Astronomical Unit (AU), is about \(1.5 \times 10^{8}\) km. Meanwhile, the distance to Neptune is much larger and utilizes the same unit to maintain consistency and avoid excessive use of zeros.

Understanding these distances in terms of ratios, as shown in the exercise, can provide a sense of scale and the relative proximity of different celestial bodies. This understanding is crucial for astronomers and astrophysicists when plotting the courses of spacecraft or observing the movement of objects in our solar system.

Mathematical Problem-Solving

Mathematical problem-solving encompasses understanding the problem, devising a plan, carrying out that plan, and lastly evaluating the solution for correctness. It's the process that brings us from a question to a solution in a systematic manner.

In the given exercise, the problem-solving process began with interpreting the provided astronomical distances, then moved on to express those distances in the form of a ratio. Next, the ratio was simplified to its most elementary form to find the answer. This step-by-step strategy is a key approach in tackling mathematical problems effectively, ensuring that complex issues are broken down into simpler, more manageable parts. Moreover, it encourages thoroughness and precision at each stage, helping students not only reach an answer but also to understand the logic and reasoning behind it.