One Number Is 2 More Than 3 Times Another

One Number is 2 More Than 3 Times Another: A Deep Dive into Problem Solving

This seemingly simple statement, "One number is 2 more than 3 times another," opens the door to a fascinating exploration of algebra, problem-solving strategies, and the power of mathematical modeling. While the core concept might appear elementary, understanding how to translate this sentence into a mathematical equation and subsequently solve it forms the bedrock of more complex mathematical reasoning. This article will delve into various approaches to solving this problem, discuss its applications in real-world scenarios, and explore extensions that demonstrate its broader implications.

Understanding the Problem Statement

The phrase "One number is 2 more than 3 times another" is a word problem, a common type found in algebra. The key to solving it lies in correctly translating the verbal description into a mathematical equation. Let's break it down:

• "One number": This represents an unknown quantity, which we can label as a variable, typically 'x' or 'y'.
• "is": This translates to the equals sign (=) in an equation.
• "2 more than": This indicates adding 2 to another quantity.
• "3 times another": This signifies multiplying another unknown quantity (let's call it 'y') by 3.

Therefore, the complete mathematical translation is: x = 3y + 2

Solving the Equation: A Multifaceted Approach

The equation x = 3y + 2, however, isn't immediately solvable. We need more information. This is where the importance of context comes in. Word problems often imply constraints or additional information that isn't explicitly stated. Let's explore different scenarios and their solutions:

Scenario 1: The Sum of the Numbers

Suppose the problem adds, "And their sum is 26." This provides the necessary second equation:

x + y = 26

Now we have a system of two linear equations with two unknowns:

1. x = 3y + 2
2. x + y = 26

We can solve this system using several methods:

• Substitution: Substitute the expression for 'x' from equation (1) into equation (2):

  (3y + 2) + y = 26

  4y + 2 = 26

  4y = 24

  y = 6

  Now, substitute the value of 'y' back into either equation (1) or (2) to solve for 'x':

  x = 3(6) + 2

  x = 20

• Elimination: Although less efficient in this case, the elimination method could also be used. We would manipulate the equations to eliminate one variable and solve for the other.

Therefore, in this scenario, the two numbers are 20 and 6.

Scenario 2: The Difference of the Numbers

Let's change the constraint. Suppose the problem states, "And their difference is 14." This leads to a new equation:

x - y = 14

Again, we have a system of two equations:

1. x = 3y + 2
2. x - y = 14

Using substitution:

(3y + 2) - y = 14

2y + 2 = 14

2y = 12

y = 6

Substituting back into equation (1) or (2):

x = 3(6) + 2

x = 20

Interestingly, even with a different constraint (difference instead of sum), we arrive at the same solution. This highlights the importance of defining the problem constraints clearly. However, it doesn't always lead to the same results. Let's explore a different scenario.

Scenario 3: The Product of the Numbers

Now, imagine the problem states, "And their product is 120". This yields the equation:

xy = 120

Substituting x = 3y + 2:

(3y + 2)y = 120

3y² + 2y - 120 = 0

This is a quadratic equation. We can solve it using the quadratic formula:

y = [-b ± √(b² - 4ac)] / 2a

where a = 3, b = 2, and c = -120.

Solving this gives two possible values for y, and subsequently, two pairs of solutions for x and y. This illustrates that changing the relationship between the numbers drastically alters the solution set. The problem becomes more complex and requires a more sophisticated solution method.

Real-World Applications

The seemingly simple problem, "One number is 2 more than 3 times another," finds applications in various real-world scenarios:

• Geometry: Calculating the dimensions of rectangles, given relationships between length and width. For instance, a rectangle's length might be 2 more than 3 times its width.
• Finance: Modeling investment growth or comparing different investment options. One investment's return might be described as being a certain amount more than a multiple of another investment's return.
• Physics: Representing relationships between physical quantities. The speed of an object might be defined in relation to another object's speed.
• Everyday problem-solving: Many everyday situations can be modeled using this type of relationship. For instance, dividing tasks or resources based on proportional relationships.

Expanding the Concept

The core concept can be extended to more complex scenarios involving three or more numbers, or using different mathematical relationships (e.g., one number is 5 less than twice another). The principles remain the same:

1. Translate the word problem into mathematical equations. This requires careful attention to the meaning of each phrase.
2. Solve the system of equations. This may involve techniques such as substitution, elimination, or the quadratic formula, depending on the complexity of the equations.
3. Interpret the solution in the context of the original problem. This ensures that the mathematical solution makes sense in the real-world scenario.

Conclusion: The Power of Mathematical Modeling

"One number is 2 more than 3 times another" might seem like a simple problem, but its solution demonstrates the power of mathematical modeling. The ability to translate a verbal description into a mathematical equation and solve it is a fundamental skill in mathematics and its numerous applications across diverse fields. By understanding various approaches to solving such problems, we build a strong foundation for tackling more complex mathematical challenges and enhancing our problem-solving skills. The different scenarios explored in this article highlight the importance of precise problem definition and selecting the appropriate solution strategy. Remember, the key lies in carefully translating the words into mathematical symbols and systematically applying the relevant algebraic techniques.