Problem 27 - Algebra for College Students

Question

Complete the six problem-solving steps to solve each problem. In 2015 , the corporations securing the most U.S. patents were IBM and Samsung. Together, the two corporations secured a total of 12,368 patents, with Samsung receiving 2250 fewer patents than IBM. How many patents did each corporation secure? (Data from U.S. Patent and Trademark Office.) Step 1 Read the problem carefully. We are asked to find ______ . 2 Assign a variable. Let \(x=\) the number of patents that IBM secured. Step Then \(x-2250=\) the number of ______ . Step 3 Write an equation. ____\(+ _____ \ldots=12,368\) Step 4 Solve the equation. \(x=\) ______ . \(\begin{array}{llll}\text { Step } 5 & \text { State the answer. } & \text { IBM secured } & \text { patents. Samsung secured } & \text { patents. }\end{array}\) Step 6 Check. The number of Samsung patents was _____ fewer than the number of _____ The total number of patents was \(7309+\) _____ \(= ______\).

Step-by-Step Solution

Answer

IBM secured 7309 patents, and Samsung secured 5059 patents.

Step 1: Understand the Question

Read the problem carefully. We are asked to find the number of patents IBM and Samsung secured in total.

Step 2: Assign Variables

Let \( x \) be the number of patents that IBM secured. Then, Samsung secured \( x - 2250 \) patents, since they secured 2250 fewer than IBM.

Step 3: Formulate the Equation

The total number of patents secured by both corporations is 12,368. Therefore, the equation is \[ x + (x - 2250) = 12368 \]

Step 4: Solve the Equation

\[ x + x - 2250 = 12368 \]\[ 2x - 2250 = 12368 \]\[ 2x = 14618 \]\[ x = 7309 \]\[ x = 7309 \]

Step 5: State the Answer

IBM secured 7309 patents. Samsung secured \[ 7309 - 2250 = 5059 \] patents.

Step 6: Verify the Solution

The number of Samsung patents is 2250 fewer than the number of IBM patents. Therefore, 5059 is indeed 2250 fewer than 7309. The total number of patents is \[ 7309 + 5059 = 12368 \].

Key Concepts

Variables in Algebra

In algebra, we often use variables to represent unknown quantities. A variable is a symbol, usually a letter, that stands in for a number we do not know yet but want to find out. In the exercise, we use the variable \(x\) to represent the number of patents IBM secured. This method helps us to frame the problem in a way we can solve it mathematically.
Using variables makes complex problems easier to work with. Instead of guessing, we set up equations that clearly show the relationships between different quantities.

Solving Equations

Once we have an equation, the next step is to solve it. To solve an equation means to find the value of the variable that makes the equation true.
In our example, the equation given is: \[ x + (x - 2250) = 12368 \]
First, we simplify the equation by combining like terms: \[ 2x - 2250 = 12368 \]
Next, we isolate the variable \(x\). Move 2250 to the other side by adding 2250 to both sides: \[ 2x = 12368 + 2250 \]
After simplifying the right side, we get: \[ 2x = 14618 \]
Finally, divide by 2 to solve for \(x\): \[ x = 7309 \]

Problem-Solving Steps

Solving algebra word problems can be straightforward if you follow a structured approach. Here are the steps we followed in our exercise:

• Read the problem carefully to understand what is asked.
• Assign variables to represent the unknown quantities.
• Write an equation that models the problem using the variables.
• Simplify and solve the equation.
• State the answer clearly.
• Verify the solution to ensure it makes sense in the context of the problem.

Following these steps helps to break down the problem into manageable parts. This makes it easier to solve correctly and efficiently.