Problem 62 - Algebra 1

Question

MARBLES In Exercises \(61-63\), consider a bag containing 12 marbles that are either red or blue. A marble is drawn at random. There are three times as many red marbles as there are blue marbles in the bag. How many red marbles are in the bag?

Step-by-Step Solution

Answer

There are 9 red marbles in the bag.

Step 1: Understanding the Information Given

The total number of marbles in the bag is 12. The number of red marbles is three times that of the blue ones. Let's denote the number of blue marbles as X. Therefore, the number of red marbles will be 3X.

Step 2: Setting up the equation

Based on the given information and our understanding in Step 1, an equation can be set up to represent the total number of marbles. The total marbles, 12, is equal to sum of the blue marbles (X) and the red marbles (3X). That is, 12 = X + 3X.

Step 3: Solving the equation

Combine like terms on the right-hand side of the equation to find: 12 = 4X. Then, to find the value for X, divide both sides by 4. This results in X = 3. With X being the number of blue marbles, it means there are 3 blue marbles.

Step 4: Calculating the number of red marbles

We know the number of red marbles is three times that of the blue marbles. So, multiply the number of blue marbles by 3. That is, 3 * 3 = 9. So, there are 9 red marbles in the bag.

Key Concepts

Algebraic Representation

When dealing with algebraic representation, it's all about translating the words of a problem into mathematical expressions. This is your first step in solving any algebra-related task. In our exercise, we begin by interpreting the statement about the number of marbles. We're told there are red and blue marbles, with the former being three times the latter.
The problem asks us to find out how many red marbles there are. First, we use a variable to represent an unknown number. Let's assign the letter \( X \) to the number of blue marbles. This is an algebraic representation because \( X \) can now be used to create expressions or equations that describe the relationship between the marbles.

• Red marbles = three times the number of blue marbles
• Expression: \( 3X \) for red marbles and \( X \) for blue marbles

Creating such expressions simplifies the process of solving problems because it turns words into numbers we can manipulate with mathematical operations.

Equation Solving

After setting up an algebraic expression, the next critical step is solving the equation to find the numerical values. Here, we sum up the marbles using the expressions identified earlier to form an equation.
The given problem provides the total number of marbles as 12, thus our equation becomes: \( 12 = X + 3X \).
This equation represents that the total count of blue and red marbles equals 12. By solving this equation, we can find out how many blue marbles are present, and thus the red ones too. The process involves combining like terms on the right-hand side, giving us the equation \( 12 = 4X \).

• Combine like terms: \( X + 3X = 4X \)
• Divide both sides by 4: \( X = \frac{12}{4} = 3 \)

So, we find that there are 3 blue marbles. This straightforward equation-solving step is crucial because it provides the unknown variable's value, enabling us to understand other connected values.

Word Problems

Word problems in mathematics require translating real-life scenarios into mathematical models. Our example with marbles is a typical word problem where we have been given a real-world situation to interpret mathematically.
Successfully solving word problems means understanding the scenario and creating equations that correspond to the relationships and constraints discussed in the problem.
Here are steps to follow when tackling word problems:

• Identify what you are asked to find. In this case, the number of red marbles.
• Determine what information you are given. We know the total number of marbles and the relational ratio between the red and blue marbles.
• Formulate a mathematical equation using variables. Define any unknown quantity with a variable, like \( X \) for blue marbles.
• Solve the equation for the unknown and use the outcome to address the question. Knowing \( X = 3 \), calculate red marbles as \( 3 \times 3 = 9 \).

Once you turn a word problem into a mathematical equation, it becomes much simpler to work through, using basic algebra skills to find the answer.