The Larger Of Two Numbers Is Seven Less Than Three Times The Smaller Number. If The Sum Is 61, Find The

The Larger Of Two Numbers Is Seven Less Than Three Times The Smaller Number. If The Sum Is 61, Find The

Understanding how to approach and solve word problems involving relationships between numbers is a fundamental skill in mathematics. These problems often require translating written descriptions into algebraic expressions, setting up equations, and solving those equations systematically. The problem statement "The larger of two numbers is seven less than three times the smaller number. If the sum is 61, find the" is an excellent example of such an algebraic word problem.

In this article, we will explore how to interpret this problem, translate it into a mathematical model, and then solve it step by step. We will also discuss common pitfalls and tips for solving similar problems efficiently. Whether you're a student preparing for exams or someone interested in sharpening your problem-solving skills, this comprehensive guide will help you understand the process thoroughly.

Understanding the Problem Statement

Before jumping into calculations, it's essential to understand what the problem is asking and identify the key information given:

• Relationship between the two numbers: The larger number is seven less than three times the smaller number.


• Sum of the two numbers: The total of the larger and smaller numbers is 61.


• Objective: Find the two numbers based on this information.

Let's analyze these statements carefully.

Breaking Down the Key Components

1. The larger number: Let's denote the smaller number as \( x \).
2. Relationship statement: "The larger of two numbers is seven less than three times the smaller number" can be expressed as:

\[ \text{Larger number} = 3x - 7 \]

1. Sum of the numbers: The sum of the smaller and larger numbers is 61:

\[ x + (3x - 7) = 61 \]

Once the algebraic expressions are set, the problem reduces to solving for \( x \), the smaller number.

Formulating the Mathematical Model

Based on the above analysis, the main equation becomes:

\[
x + (3x - 7) = 61
\]

Simplify this equation:

\[
x + 3x - 7 = 61
\]

Combine like terms:

\[
4x - 7 = 61
\]

Now, solve for \( x \):

\[
4x = 61 + 7
\]

\[
4x = 68
\]

\[
x = \frac{68}{4} = 17
\]

So, the smaller number is 17.

Next, find the larger number using the relationship:

\[
\text{Larger number} = 3x - 7 = 3 \times 17 - 7 = 51 - 7 = 44
\]

Final answer:

• Smaller number: 17


• Larger number: 44

This completes the solution process.

Step-by-Step Solution Summary

To summarize, here are the steps taken to solve the problem:

1. Assign variables to the unknowns (e.g., \( x \) for the smaller number).


2. Translate the problem statement into an algebraic equation.


3. Simplify and solve the algebraic equation.


4. Use the solution for \( x \) to find the other number.


5. Verify the solution by checking the conditions given in the problem.

Let's verify:

• The larger number is \( 44 \).


• The smaller number is \( 17 \).


• Sum: \( 17 + 44 = 61 \), which matches the given total.


• Relationship check: \( 3 \times 17 - 7 = 51 - 7 = 44 \), which matches the larger number.

Since all conditions are satisfied, the solution is correct.

Additional Tips for Solving Similar Word Problems

Solving algebraic word problems requires a strategic approach. Here are some tips:

1. Read Carefully and Highlight Key Information

• Identify the quantities involved.
• Note relationships and constraints.
• Highlight or underline important data points.

2. Assign Clear Variables

• Use simple, memorable variables like \( x \) and \( y \).
• Clearly define what each variable represents.

3. Translate Words into Equations

• Convert descriptive relationships into algebraic expressions.
• Be precise with signs (+, -).

4. Set Up the Main Equation

• Combine all information into a single equation.
• Keep the equation as simple as possible.

5. Solve Step-by-Step

• Simplify algebraic expressions.
• Perform inverse operations carefully.
• Check your solution at each step.

6. Verify the Solution

• Substitute the found values back into the original conditions.
• Confirm that all conditions are satisfied.

Common Mistakes to Avoid

• Mixing up the variables or their meanings.
• Forgetting to translate all parts of the problem into equations.
• Making algebraic errors like incorrect simplification.
• Forgetting to verify the solution with the original conditions.

Practice Problems for Mastery

To reinforce your understanding, try solving these similar problems:

1. The sum of two numbers is 50. The larger number is twice the smaller. Find the numbers.


2. A number is 4 less than twice another number. Their sum is 30. Find the numbers.


3. The difference between two numbers is 8. The larger number is three times the smaller. Find the numbers.

Working through these problems using the same systematic approach will strengthen your problem-solving skills.

Conclusion

Solving word problems involving relationships between numbers requires careful reading, proper translation into algebraic equations, and systematic solving. In the example discussed, "The larger of two numbers is seven less than three times the smaller number. If the sum is 61, find the numbers," we demonstrated how to set up the problem, solve the equations, and verify the solution.

Mastering these techniques will not only help you excel in mathematics exams but also develop critical thinking and analytical skills useful in many real-world scenarios. Remember always to approach such problems methodically, verify your answers, and practice regularly to build confidence and proficiency.

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