Solving Age Problems: A Step-by-Step Guide for Practice 1.9
Age problems are a common type of word problem found in math courses, particularly in algebra and pre-algebra. These problems involve determining the ages of individuals based on given relationships between their ages. While they can seem tricky at first, understanding the underlying concepts and using a systematic approach can make solving them much easier. Let's explore how to tackle these problems, using Practice 1.9 as our guide.
Understanding the Key Concepts
The foundation of solving age problems lies in understanding how ages change over time. Here are some key points to keep in mind:
- Current Age: This refers to the age of a person at the present moment.
- Past Age: This represents the age of a person at a specific point in the past.
- Future Age: This refers to the age of a person at a specific point in the future.
- Relative Ages: These are the age differences between two or more people.
The Steps to Solving Age Problems
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Read the problem carefully and identify the unknowns: Determine what ages you need to find. For instance, you might need to find the current age of a person or their age in a certain number of years.
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Define variables: Assign letters (like 'x', 'y', etc.) to represent the unknown ages. For example, you could use 'x' for the current age of one person and 'y' for the current age of another.
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Translate the problem into equations: Formulate equations based on the relationships described in the word problem. Pay close attention to how the ages are related (e.g., one person is twice as old as another, or their ages will be equal in a certain number of years).
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Solve the equations: Use algebraic techniques to solve for the unknown variables. This might involve substituting values, simplifying expressions, or using elimination methods.
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Check your answers: Ensure the solution makes sense in the context of the original problem. Are the ages plausible? Do they satisfy the conditions given in the problem?
Example: Solving an Age Problem from Practice 1.9
Let's work through an example to illustrate the process.
Problem: The sum of the ages of a father and son is 45 years. Five years ago, the father was twice as old as his son. Find their present ages.
Solution:
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Unknowns: We need to find the father's current age and the son's current age.
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Variables: Let 'x' represent the father's current age and 'y' represent the son's current age.
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Equations:
- Equation 1: x + y = 45 (The sum of their current ages is 45)
- Equation 2: x - 5 = 2(y - 5) (Five years ago, the father was twice as old as the son)
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Solving:
- Simplify Equation 2: x - 5 = 2y - 10
- Rearrange Equation 2: x = 2y - 5
- Substitute the value of 'x' from Equation 2 into Equation 1: (2y - 5) + y = 45
- Combine like terms: 3y - 5 = 45
- Solve for 'y': 3y = 50 => y = 50/3
- Substitute the value of 'y' back into Equation 1: x + (50/3) = 45
- Solve for 'x': x = 45 - (50/3) = 95/3
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Checking:
- Father's current age (x) = 95/3 ≈ 31.67 years
- Son's current age (y) = 50/3 ≈ 16.67 years
- Five years ago, the father was approximately 26.67 years old, and the son was approximately 11.67 years old. This confirms that the father was indeed twice as old as his son five years ago.
Therefore, the father's present age is approximately 31.67 years, and the son's present age is approximately 16.67 years.
Tips for Success
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Practice regularly: Solving age problems requires practice. The more problems you solve, the more comfortable you'll become with the techniques.
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Use diagrams: Visualizing the problem with a simple diagram can often help you understand the relationships between ages.
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Break down complex problems: If a problem seems overwhelming, try breaking it down into smaller, more manageable steps.
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Check for contradictions: When you find a solution, make sure it doesn't contradict any information given in the problem.
Conclusion
Solving age problems may seem intimidating at first, but with a methodical approach and consistent practice, you can master them. Practice 1.9 provides valuable examples to help you solidify your understanding of these problems. Remember to carefully read the problem, define variables, translate information into equations, solve the equations, and always verify your answers. With a little effort, you'll be able to solve age problems with confidence.