Problem 192 - Elementary Algebra

Question

Mario invested \(\$ 475\) in \(\$ 45\) and \(\$ 25\) stock shares. The number of \(\$ 25\) shares was five less than three times the number of \(\$ 45\) shares. How many of each type of share did he buy?

Step-by-Step Solution

Answer

Mario bought 5 shares of \$ 45 and 10 shares of \$ 25.

Step 1: Define Variables

Let the number of \$ 45 shares be \( x \). The number of \$ 25 shares can then be expressed as \( 3x - 5 \).

Step 2: Set Up the Equation

Write an equation based on the total investment: \[ 45x + 25(3x - 5) = 475 \]

Step 3: Simplify the Equation

Distribute and combine like terms: \[ 45x + 75x - 125 = 475 \] Simplify further: \[ 120x - 125 = 475 \]

Step 4: Solve for x

Add 125 to both sides of the equation: \[ 120x = 600 \] Divide by 120: \[ x = 5 \]

Step 5: Determine the Number of Shares

Calculate the number of \$ 45 shares: \( x = 5 \). Calculate the number of \$ 25 shares: \[ 3x - 5 = 3(5) - 5 = 15 - 5 = 10 \]

Key Concepts

Variable Definition

In algebra, a variable is a symbol used to represent an unknown value. This symbol is usually a letter, such as x or y. For this exercise, we need to define a variable to express the number of \(45 shares Mario bought. Assume the number of \)45 shares Mario bought is represented as x. Once we have this, we can express other unknown quantities in terms of x. For instance, the number of \(25 shares is given as five less than three times the number of \)45 shares. Therefore, we can define it as 3x - 5.

Linear Equations

Linear equations are equations of the first degree, meaning they have no exponents higher than one. They typically look like ax + b = c. In our problem, we need to set up a linear equation to relate the total amounts invested in the \(45 and \)25 shares to the total money Mario invested. Let's set this up: The investment for \(45 shares is 45x dollars. The investment for \)25 shares is 25(3x - 5) dollars. So, our linear equation becomes: \ 45x + 25(3x - 5) = 475.

Problem-Solving Steps

• First, define your variables. Here, x represents the number of \(45 shares.
• Then, relate the other unknowns to your variable. We used 3x - 5 for the \)25 shares.
• Next, write an equation based on the problem's conditions. In this case, the total investment is \(475.
• After that, simplify and solve the equation step-by-step. Distribute and combine like terms: 45x + 75x - 125 = 475, then simplify to 120x - 125 = 475. Add 125 to both sides: 120x = 600, then divide by 120: x = 5.
• Finally, use the solution to find the specific number of other shares. Here, \)45 shares were 5, and $25 shares were 3(5) - 5 = 10.

Investment Problems

Investment problems often involve determining how to allocate resources to maximize returns or stay within budget constraints. Here, Mario's investment in stock shares required us to figure out the number of shares of each type he could buy within his budget. The key is translating the problem into a mathematical model using variables and equations and then solving those equations. By carefully defining variables and understanding how different parts of the problem relate to each other, investment problems become more manageable. In this problem, we connected the cost of each type of share and the total investment to create a solvable equation.