Twice A Number Added To A Smaller Number Is 5. The Difference Of 5 Times The Smaller Number And The Larger

Understanding the Problem Statement

Introduction to the Problem

The statement "Twice A Number Added To A Smaller Number Is 5. The Difference Of 5 Times The Smaller Number And The Larger" presents a mathematical scenario involving two numbers, one larger and one smaller. This problem requires translating a verbal statement into algebraic expressions and solving for the unknowns.

Breaking Down the Statement

The statement contains two parts:

1. "Twice A Number Added To A Smaller Number Is 5"

This indicates a relationship between two numbers, where one is doubled and then added to the other, resulting in 5.

1. "The Difference Of 5 Times The Smaller Number And The Larger"

This suggests an expression involving the difference between five times the smaller number and the larger number.

Understanding these parts is essential before formulating equations.

Assigning Variables

Choosing Symbols for the Unknowns

To translate the problem into algebraic form, assign variables:

• Let x = the smaller number
• Let y = the larger number

With these variables, we can write the relationships described in the problem.

Establishing the First Equation

From "Twice A Number Added To A Smaller Number Is 5," we get:

\[ 2y + x = 5 \]

Note: Since the statement emphasizes "twice a number" and "a smaller number," and the phrase "twice a number" is typical, we need to clarify whether "twice a number" refers to the smaller or larger number. The natural interpretation here is:

• "Twice a number" refers to twice the larger number (y),


• "A smaller number" refers to the smaller number (x).

So, the first equation becomes:

\[ 2y + x = 5 \]

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Important Clarification:
The problem wording can sometimes be ambiguous. Based on common problem structures, it's more logical to interpret "Twice a number" as "twice the larger number," especially if the second part involves the smaller number. For our purposes, we'll proceed with:

• Twice the larger number: \( 2y \)


• Smaller number: \( x \)

Thus, the first equation is:

\[ 2y + x = 5 \]

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Formulating the Second Equation

Interpreting the Second Part of the Problem

"The Difference Of 5 Times The Smaller Number And The Larger" suggests an expression:

\[ \text{Difference} = 5x - y \]

But since the phrase is incomplete, it likely indicates a relationship involving this difference. Typically, in algebra problems, such phrases are completed with an equality to a certain value or another expression.

Given the structure of the problem and common algebraic problem patterns, a reasonable assumption is that:

> The difference of 5 times the smaller number and the larger number equals some constant, possibly zero or another number.

In this case, since the statement is incomplete, it is often interpreted as:

\[ 5x - y = \text{some value} \]

But the problem statement seems truncated. To proceed logically, let's assume the problem is:

• "The difference of 5 times the smaller number and the larger number is 0," meaning:

\[ 5x - y = 0 \]

This is a common pattern in algebra problems, where the difference equals zero or some specified value.

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Summary of the Equations

Based on the above interpretation, we have:

1. \( 2y + x = 5 \)
2. \( 5x - y = 0 \)

Note: If the problem has a different intended second part, adjustments can be made later.

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Solving the System of Equations

Step 1: Express \( y \) in terms of \( x \)

From the second equation:

\[ 5x - y = 0 \implies y = 5x \]

Step 2: Substitute into the first equation

Plug \( y = 5x \) into the first equation:

\[ 2(5x) + x = 5 \]
\[ 10x + x = 5 \]
\[ 11x = 5 \]
\[ x = \frac{5}{11} \]

Step 3: Find \( y \)

Using \( y = 5x \):

\[ y = 5 \times \frac{5}{11} = \frac{25}{11} \]

Summary of Solutions

• Smaller number \( x = \frac{5}{11} \)
• Larger number \( y = \frac{25}{11} \)

These are the values satisfying the assumed equations.

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Verifying the Solution

Check the First Equation

\[ 2y + x = 2 \times \frac{25}{11} + \frac{5}{11} = \frac{50}{11} + \frac{5}{11} = \frac{55}{11} = 5 \]

Correct.

Check the Second Equation

\[ 5x - y = 5 \times \frac{5}{11} - \frac{25}{11} = \frac{25}{11} - \frac{25}{11} = 0 \]

Correct.

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Alternative Interpretations and Adjustments

Possibility of Different Second Parts

If the original problem intended the second part to be different, such as:

• The difference of 5 times the smaller number and the larger number equals a particular value,
• Or the phrase was incomplete, and the actual statement was:

"Twice a number added to a smaller number is 5. The difference of 5 times the smaller number and the larger number is some value."

In such cases, the equations would adjust accordingly.

Example with Different Assumptions

Suppose the second part is "The difference of 5 times the smaller number and the larger is 3," then:

\[ 5x - y = 3 \]

The system becomes:

1. \( 2y + x = 5 \)


2. \( 5x - y = 3 \)

Solving similarly:

• From the second: \( y = 5x - 3 \)


• Substitute into the first:

\[ 2(5x - 3) + x = 5 \]
\[ 10x - 6 + x = 5 \]
\[ 11x - 6 = 5 \]
\[ 11x = 11 \]
\[ x = 1 \]
\[ y = 5(1) - 3 = 2 \]

Check:

• \( 2(2) + 1 = 4 + 1 = 5 \) (correct)


• \( 5(1) - 2 = 5 - 2 = 3 \) (matches assumption)

Solution:

• Smaller number \( x = 1 \)


• Larger number \( y = 2 \)

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Conclusion and Summary

Key Takeaways

• Carefully interpret the verbal problem to formulate accurate equations.
• Assign variables logically; typically, the smaller number is \( x \), the larger is \( y \).
• Use substitution to solve the system of equations.
• Verify solutions by substituting back into original equations.
• Recognize that ambiguous wording may lead to multiple interpretations; clarify assumptions if possible.

Final Remarks

Mathematical word problems require precision in interpretation. When faced with incomplete or ambiguous statements, it's essential to consider common problem patterns and make logical assumptions. The approach outlined here demonstrates how to translate a verbal description into algebra, solve systematically, and verify results, providing a comprehensive understanding of the problem involving two numbers related by specific algebraic expressions.

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If you encounter similar problems, remember to:

• Break down the statement into parts.


• Assign clear variables.


• Formulate equations based on the description.


• Solve the system step-by-step.


• Verify your solutions thoroughly.

This methodology ensures accurate problem-solving and enhances your algebraic reasoning skills.