The Sum Of Three Number Is 99. The Second Number Is 9 More Than The First The Third Number Is 4 Times
Understanding how to solve algebraic word problems involving multiple variables is a fundamental skill in mathematics. One such intriguing problem involves finding three numbers based on given relationships and their sum. This article explores the problem statement: "The sum of three numbers is 99. The second number is 9 more than the first. The third number is 4 times the first." We will delve into how to approach such problems, formulate equations, and find solutions step-by-step. Whether you're a student looking to improve your algebra skills or a math enthusiast interested in problem-solving techniques, this comprehensive guide will help you grasp the concepts thoroughly.
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Understanding the Problem Statement
Before jumping into solving the problem, it's essential to understand what is being asked and identify the key information provided:
- The sum of three numbers is 99.
- The second number is 9 more than the first number.
- The third number is 4 times the first number.
This problem involves three unknowns, typically labeled as variables, and relationships between them expressed algebraically.
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Defining Variables and Setting Up Equations
The first step in solving such problems is to assign variables to the unknown quantities. Let's define:
- Let x be the first number.
- Since the second number is 9 more than the first, let the second number be x + 9.
- The third number is 4 times the first number, so let the third number be 4x.
Now, based on the problem, the sum of these three numbers is 99. This gives us the primary equation:
\[ x + (x + 9) + 4x = 99 \]
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Formulating the Mathematical Equation
By simplifying the above equation, we can find the value of x:
\[
\begin{aligned}
x + x + 9 + 4x &= 99 \\
(1x + 1x + 4x) + 9 &= 99 \\
6x + 9 &= 99
\end{aligned}
\]
Now, we will solve for x.
Solving for the First Number
Subtract 9 from both sides:
\[
6x + 9 - 9 = 99 - 9
\]
\[
6x = 90
\]
Divide both sides by 6:
\[
x = \frac{90}{6} = 15
\]
With x = 15, we can now find the other two numbers.
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Calculating the Other Two Numbers
- Second number: since it's 9 more than the first,
- Third number: which is 4 times the first,
Summary of the three numbers:
| Number | Calculation | Result |
| --- | --- | --- |
| First number | x | 15 |
| Second number | x + 9 | 24 |
| Third number | 4x | 60 |
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Verifying the Solution
It's crucial to verify whether these numbers satisfy the original problem conditions.
- Sum check:
\[
15 + 24 + 60 = 99
\]
- Relationship checks:
- Second number is 9 more than the first?
\[
24 - 15 = 9 \quad \checkmark
\]
- Third number is 4 times the first?
\[
4 \times 15 = 60 \quad \checkmark
\]
Since all conditions are satisfied, the solution is correct.
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Practical Applications of Such Problems
Solving problems involving sums and relationships between numbers is not just an academic exercise; it has real-world applications in various fields:
Financial Planning and Budgeting
- Calculating allocations where certain expenses are proportional or have fixed differences.
Engineering and Physics
- Determining quantities that relate through linear or multiplicative relationships.
Data Analysis
- Modeling datasets with known relationships among variables.
Common Mistakes and Tips for Solving Similar Problems
While solving algebraic problems involving multiple variables, some common pitfalls include:
- Incorrect Variable Assignments: Always clearly define variables based on problem statements.
- Misinterpreting Relationships: Carefully read the problem to understand whether relationships are additive, multiplicative, or involves other operations.
- Arithmetic Errors: Double-check calculations, especially during simplification and solving equations.
- Ignoring Constraints: Ensure solutions satisfy all conditions, not just the primary sum or relationships.
Tips to avoid these mistakes:
- Write down all known relationships explicitly.
- Verify each step.
- Substitute the found values back into the original conditions.
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Other Similar Problems and Practice Exercises
To strengthen your understanding, try solving the following problems:
- The sum of three numbers is 120. The second is 12 more than the first, and the third is 3 times the first. Find the three numbers.
- Three numbers add up to 150. The second is 10 less than the first, and the third is twice the first. Determine the numbers.
- The sum of three positive integers is 90. The second is 7 more than the first, and the third is 5 times the first. Find the numbers.
Practice Tip: For each problem, define variables, set up equations based on relationships, and solve systematically.
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Conclusion
Solving algebraic problems involving multiple numbers and their relationships requires a clear understanding of variables, setting up correct equations, and methodical solving. The problem where the sum of three numbers is 99, with the second being 9 more than the first and the third being four times the first, exemplifies how to approach such tasks. By defining variables, forming equations, simplifying, and verifying solutions, you can accurately find the unknowns. Mastery of these techniques enhances your problem-solving skills and prepares you for more complex mathematical challenges, whether in academics or real-world applications.
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Additional Resources for Learning Algebra
- Online Algebra Courses: Websites like Khan Academy and Coursera offer free courses on algebra fundamentals.
- Math Practice Websites: Platforms such as IXL and Mathway provide practice problems and step-by-step solutions.
- Algebra Textbooks: Standard textbooks like "Elementary Algebra" by Harold R. Jacobs provide comprehensive explanations and exercises.
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Remember: Mathematics is about logical thinking and systematic problem-solving. Keep practicing, and you'll find that complex problems become manageable with patience and methodical approaches.
