Problem 87 - Precalculus

Question

Astronomers have determined the number of stars in a small region of the universe to be 2,880,968 classified as red dwarfs, yellow, and blue stars. For every blue star there are 120 red dwarfs; for every red dwarf there are 3000 yellow stars. Determine the number of stars by type in that region of the universe.

Step-by-Step Solution

Answer

There are 8 blue stars, 960 red dwarfs, and 2,880,000 yellow stars.

Step 1: Understand the Ratios

Let's interpret the relationships given: - There is 1 blue star. - For each blue star, there are 120 red dwarfs. - For each red dwarf, there are 3000 yellow stars.

Step 2: Set Up Equations

Define variables: - Let the number of blue stars be denoted by \( b \).- Then, the number of red dwarfs is \( 120b \).- The number of yellow stars is \( 120b \times 3000 = 360,000b \).

Step 3: Formulate Total Equation

The total number of stars is the sum of blue stars, red dwarfs, and yellow stars. Based on the problem:\[ b + 120b + 360,000b = 2,880,968 \]

Step 4: Simplify the Equation

Combine like terms in the equation:\[ (1 + 120 + 360,000)b = 2,880,968 \]\[ 360,121b = 2,880,968 \]

Step 5: Solve for Blue Stars

Divide both sides of the equation by 360,121:\[ b = \frac{2,880,968}{360,121} \approx 8 \]Thus, there are approximately 8 blue stars.

Step 6: Calculate Red Dwarfs

With 8 blue stars, calculate the number of red dwarfs:\[ 120 imes 8 = 960 \]There are 960 red dwarfs.

Step 7: Calculate Yellow Stars

Next, find the number of yellow stars:\[ 960 imes 3000 = 2,880,000 \]So, there are 2,880,000 yellow stars.

Key Concepts

Ratios in Astronomy

In astronomy, and especially when classifying various types of stars, ratios become extremely useful. They allow scientists to compare different classes or groups by setting a consistent relationship.

In our original exercise, these relationships or ratios are:

• 1 blue star to 120 red dwarfs
• 1 red dwarf to 3000 yellow stars

Ratios help us understand the relative quantities of different star types in a given region. This comparison is crucial in analyzing vast, complex data without needing precise measurements for each star.

By understanding these ratios, we can easily scale the numbers up or down. For instance, if one knows how many blue stars exist, they can quickly calculate how many red dwarfs and yellow stars follow, just by using the ratios.

Solving Equations

Equations are a fundamental tool in mathematics that helps us find unknown values by expressing known relationships. When faced with the problem of determining the number of each type of star, forming an equation from the ratios allows you to assign definitive numbers.

Start by defining variables:

• Let the number of blue stars be represented by \( b \).
• The number of red dwarfs is then \( 120b \).
• The number of yellow stars is \( 120b \times 3000 \).

These variables and equations represent the direct relationships and can be summed up as:
\[ b + 120b + 360,000b = 2,880,968 \]
This step is crucial as it forms the bridge between the known and the unknown, letting us proceed with calculating actual numbers through simplification.

Number Theory

Number theory often deals with the properties and relationships of numbers, especially integers. In our exercise, we're trying to find integer solutions for the numbers of blue, red, and yellow stars in a region of the universe.

This problem involves ensuring that the ratios derived amongst the stars are maintained correctly through integer values. We aim to find an integer solution where the values of \( b \), \( 120b \), and \( 360,000b \) not only respect the given total number but also maintain practical relationships—ensuring actual counts of each star type.

By performing operations such as division and multiplication, we preserve these integer relationships, ultimately achieving a precise count of how these stars populate the universe, all while working exclusively with whole numbers.

Algebraic Expressions

Algebraic expressions are the mathematical phrases that can contain numbers, variables, and arithmetic operators. In the context of our problem, they help simplify relationships through manipulation.

Consider the expression \( 360,121b = 2,880,968 \) formed during the solution process. Here, algebraic manipulation is used to isolate \( b \). Once \( b \) is determined, we use it to find the other quantities. This is done by switching, combining, or factoring terms, depending on what's necessary.

The power in this approach is its broad application. Once set, you can easily substitute values, providing flexibility. By solving \( 360,121b = 2,880,968 \) to find \( b \), we directly gain insights into the quantities of red and yellow stars through simple multiplication using our earlier established expressions. The elegance of algebra lies in its methodical approach, which remains consistent across different problems.