Problem 38 - Understanding Elementary Algebra with Geometry

Question

Solve each of the following verbal problems algebraically. You may use either a one or a two-variable approach. A certain mutual fund contains 100 stocks. On a certain day all of the stocks changed price. Three times the number of stocks that went up is 14 more than 8 times the number of stocks that went down. Find how many stocks went up and how many went down.

Step-by-Step Solution

Answer

74 stocks went up, 26 stocks went down.

Step 1: Define the Variables

Let the number of stocks that went up be represented by \( x \), and the number of stocks that went down be represented by \( y \).

Step 2: Set Up the Equation for Total Stocks

We know the total number of stocks is 100. This gives us the equation:\[ x + y = 100 \]

Step 3: Set Up the Relationship Between Stocks

Three times the number of stocks that went up is 14 more than 8 times the number of stocks that went down. This can be written as:\[ 3x = 8y + 14 \]

Step 4: Solve the System of Equations

We have two equations: 1. \( x + y = 100 \) 2. \( 3x = 8y + 14 \)First, solve the first equation for \( y \):\[ y = 100 - x \]Now, substitute \( y = 100 - x \) into the second equation:\[ 3x = 8(100 - x) + 14 \]

Step 5: Simplify and Solve for x

Distribute and simplify the equation:\[ 3x = 800 - 8x + 14 \]Combine like terms:\[ 3x + 8x = 814 \]\[ 11x = 814 \]Solve for \( x \):\[ x = \frac{814}{11} \]\[ x = 74 \]

Step 6: Find the Value of y

Substitute \( x = 74 \) back into the equation \( y = 100 - x \):\[ y = 100 - 74 \]\[ y = 26 \]

Step 7: Conclusion

The number of stocks that went up is 74, and the number of stocks that went down is 26.

Key Concepts

System of Equations

A system of equations involves solving two or more equations that share common variables. When dealing with algebra word problems, these systems help us find the relationship between different quantities. In our example, the total number of stocks and how they changed in price are related.

• The first equation tells us the sum of two variables: the number of stocks that went up and the number of stocks that went down.
• The second equation tells us how these variables relate in another way: three times the number of stocks that went up is equal to 14 more than eight times the number of stocks that went down.

This combination of equations forms a system. To solve it, we often use methods like substitution or elimination. These methods simplify the system, helping us find the values of our variables step-by-step.

Variable Definition

Defining variables is one of the core steps in solving algebra word problems. It involves assigning symbols (usually letters) to represent unknown quantities. In the given exercise, we used two variables:

• Let the number of stocks that went up be represented by \( x \).
• Let the number of stocks that went down be represented by \( y \).

These variables allow us to convert the word problem into mathematical equations that can be solved. By clearly defining variables, we make the process of setting up and solving the equations straightforward. Always remember: proper definition of variables can simplify complex problems significantly.

Solving Equations Step-by-Step

To solve the system of equations, it is essential to follow a sequence of steps. Here’s a breakdown:

• Step 1: Define the Variables - Set \( x \) for stocks that went up, and \( y \) for stocks that went down.
• Step 2: Establish Equations - From the problem, we got: \( x + y = 100 \) and \( 3x = 8y + 14 \).
• Step 3: Solve One Equation for One Variable - Using \( x + y = 100 \): \( y = 100 - x \).
• Step 4: Substitute - Insert \( y = 100 - x \) into the second equation: \( 3x = 8(100 - x) + 14 \). Simplify: \( 3x = 800 - 8x + 14 \).
• Step 5: Combine Like Terms - \( 3x + 8x = 814 \). Hence, \( 11x = 814 \).
• Step 6: Solve for One Variable - \( x = \frac{814}{11} \). Therefore, \( x = 74 \).
• Step 7: Find the Other Variable - Substitute \( x = 74 \) back into \( y = 100 - x \). We get \( y = 26 \).

This systematic approach ensures accuracy and clarity in finding the solutions. Through these steps, we determined that 74 stocks went up and 26 stocks went down.