11. One Number Is Twelve More Than Another. The Sum Of The Two Numbers Is The Same As Fourtimes The Smaller.
This statement introduces a classic algebraic problem that involves understanding relationships between two numbers and translating those relationships into equations. Such problems are common in mathematics, especially in algebra, and they serve as excellent exercises for developing problem-solving skills. In this article, we will explore how to approach this problem step-by-step, derive the necessary equations, and find the solution. Additionally, we will provide insights into similar problems and tips for solving them efficiently.
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Understanding the Problem
Before jumping into equations and calculations, it's vital to understand what the problem is asking. Let's break down the statement:
- One number is twelve more than another: This indicates a relationship between two numbers, say x and y, where one is larger by 12.
- The sum of the two numbers is the same as four times the smaller: The total of x and y equates to four times the smaller number.
By analyzing these clues, we can set up equations that represent these relationships.
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Defining Variables
The first step in solving algebraic problems is to assign variables to the unknowns. Here, we are dealing with two numbers, so:
- Let x = the smaller number.
- Let y = the larger number.
Based on the problem statement, we have:
- y is twelve more than x:
y = x + 12
\]
- The sum of the two numbers equals four times the smaller:
x + y = 4x
\]
Our goal is to find the values of x and y that satisfy these equations.
---
Formulating the Equations
With the variables defined, the next step is to write the equations explicitly:
- Relationship between the two numbers:
y = x + 12
\]
- Sum of the numbers equals four times the smaller:
x + y = 4x
\]
Substituting the first into the second allows us to solve for x.
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Solving the Equations Step-by-Step
Let's proceed with solving for x and y.
Step 1: Substitute the expression for y into the sum equation
\[
x + (x + 12) = 4x
\]
Step 2: Simplify the equation
\[
x + x + 12 = 4x
\]
\[
2x + 12 = 4x
\]
Step 3: Isolate x
Subtract 2x from both sides:
\[
12 = 4x - 2x
\]
\[
12 = 2x
\]
Step 4: Solve for x
\[
x = \frac{12}{2} = 6
\]
Step 5: Find y
Recall that y = x + 12:
\[
y = 6 + 12 = 18
\]
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Final Answer and Verification
The two numbers are:
- Smaller number (x): 6
- Larger number (y): 18
Let's verify the solution:
- Is y twelve more than x?
18 = 6 + 12 \quad \checkmark
\]
- Is the sum of the numbers equal to four times the smaller?
6 + 18 = 24 \quad \text{and} \quad 4 \times 6 = 24 \quad \checkmark
\]
Both conditions are satisfied, confirming the correctness of the solution.
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Understanding the Key Concepts
This problem illustrates several important concepts in algebra:
1. Translating Word Problems into Equations
Identifying the relationships described in words and converting them into algebraic expressions is fundamental.2. Substitution Method
Using one equation to replace a variable in another simplifies the problem, making it easier to solve.3. Verification of Solutions
Always check your solutions against the original conditions to ensure accuracy.---
Additional Practice Problems
To deepen understanding, consider practicing with similar problems:
- One number is three less than twice another. The sum of the two numbers is 20. Find the numbers.
- The difference between two numbers is 8. The larger number is twice the smaller. Find both numbers.
- Three more than a number is equal to twice another number. Their sum is 30. Find the numbers.
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Tips for Solving Similar Problems
- Clearly define variables: Assign meaningful letters to unknowns.
- Translate relationships carefully: Pay close attention to words like "more than," "less than," "twice," etc.
- Write equations systematically: Use the relationships described to formulate equations.
- Check units and conditions: Make sure the solutions satisfy all parts of the problem.
- Verify your answer: Substitute back into original conditions to confirm.
Conclusion
The problem "11. One Number Is Twelve More Than Another. The Sum Of The Two Numbers Is The Same As Fourtimes The Smaller." demonstrates how algebra can be used to solve real-world-like problems involving relationships between numbers. By carefully defining variables, translating the problem into equations, and solving systematically, we can find the unknown numbers efficiently and accurately. Mastering such techniques not only enhances problem-solving skills but also prepares you for more complex algebraic challenges. Practice regularly with similar problems to build confidence and proficiency in algebra.
