If The Sum Of Two Numbers Is 112, And When The Larger Number Is Divided By The Smaller Number, The Quotient Is 3 And The Remainder Is 4, Find The Larger Number.

Jun 18, 2025 by ADMIN 161 views

Finding the Larger Number Two Numbers Sum to 112 with Quotient 3 and Remainder 4

In the realm of mathematics, problem-solving often involves deciphering relationships between numbers and applying fundamental principles to arrive at a solution. One such problem presents us with two numbers that, when added together, yield a sum of 112. Additionally, when the larger number is divided by the smaller number, the quotient is 3, and the remainder is 4. Our objective is to determine the larger of these two numbers.

Understanding the Problem

To effectively tackle this problem, it's crucial to grasp the core concepts involved. We are dealing with two unknown numbers, let's call them x and y, where x is the larger number and y is the smaller number. The problem provides us with two key pieces of information:

The sum of the two numbers is 112: This can be expressed as the equation x + y = 112.
When the larger number (x) is divided by the smaller number (y), the quotient is 3 and the remainder is 4: This can be expressed as the equation x = 3y + 4.

These two equations form a system of linear equations, which we can solve to find the values of x and y. The goal is to find the value of x, which represents the larger number.

Solving the System of Equations

There are several methods to solve a system of linear equations, such as substitution, elimination, or matrix methods. In this case, the substitution method seems most straightforward. We can substitute the expression for x from the second equation into the first equation:

(3y + 4) + y = 112

Now, we can simplify and solve for y:

4y + 4 = 112

4y = 108

y = 27

We have now found the value of the smaller number, y, which is 27. To find the larger number, x, we can substitute the value of y back into either of the original equations. Let's use the second equation:

x = 3(27) + 4

x = 81 + 4

x = 85

Therefore, the larger number, x, is 85.

Verifying the Solution

It's always a good practice to verify the solution to ensure its accuracy. Let's check if our values for x and y satisfy the given conditions:

Sum of the numbers: 85 + 27 = 112 (This condition is satisfied).
Division with quotient and remainder: 85 divided by 27 gives a quotient of 3 and a remainder of 4 (This condition is also satisfied).

Since both conditions are met, our solution is correct. The larger number is indeed 85.

Alternative Approaches

While the substitution method provided a clear and efficient solution, let's briefly explore other approaches that could be used to solve this problem.

Elimination Method

We could rearrange the equations to align the variables and then use the elimination method. First, rewrite the second equation as:

x - 3y = 4

Now we have the following system:

x + y = 112

x - 3y = 4

Subtracting the second equation from the first eliminates x:

4y = 108

Which leads us to the same value for y, y = 27, and subsequently x = 85.

Graphical Method

Although less practical for precise solutions, we could graph the two equations on a coordinate plane. The point of intersection would represent the solution to the system. However, this method is more suitable for visualization and may not yield exact numerical answers without careful graphing.

Key Concepts and Applications

This problem highlights the application of several key mathematical concepts:

System of Linear Equations: The ability to represent real-world problems using multiple equations and solve them simultaneously is a fundamental skill in algebra.
Substitution Method: This technique allows us to solve for one variable in terms of another, simplifying the system and making it easier to find a solution.
Division Algorithm: The concept of quotient and remainder is essential in understanding division and its relationship to multiplication.

These concepts are widely applicable in various fields, including:

Engineering: Solving systems of equations is crucial in circuit analysis, structural design, and control systems.
Economics: Modeling supply and demand curves often involves solving systems of linear equations.
Computer Science: Many algorithms and data structures rely on the principles of solving equations and manipulating numerical relationships.

Real-World Scenarios

Problems involving systems of equations often arise in real-world scenarios. For instance, consider the following:

Mixture Problems: Determining the amounts of different ingredients needed to create a specific mixture with desired properties often involves setting up and solving systems of equations.
Financial Planning: Calculating investment returns, loan payments, or budget allocations can involve solving systems of equations.
Distance, Rate, and Time Problems: These classic problems often require setting up equations based on the relationship between distance, rate, and time, and then solving them simultaneously.

Conclusion

In summary, we successfully determined the larger of the two numbers by translating the given information into a system of linear equations and solving it using the substitution method. The larger number is 85. This problem demonstrates the power of algebraic techniques in solving mathematical puzzles and real-world problems. Understanding the underlying concepts and exploring different solution methods enhances problem-solving skills and provides a deeper appreciation for the elegance and applicability of mathematics.

Let's delve into the intriguing problem of finding two numbers when their sum and the quotient with a remainder after division are known. This is a classic mathematical puzzle that requires a blend of algebraic thinking and problem-solving skills. In this comprehensive guide, we will dissect the problem, explore different approaches to solving it, and understand the underlying mathematical concepts.

Problem Statement

The core of the problem lies in unraveling the relationship between two unknown numbers based on two crucial pieces of information:

Their sum is a given value (in our case, 112).
When the larger number is divided by the smaller number, the quotient and remainder are known (in our case, the quotient is 3, and the remainder is 4).

Our mission is to determine the value of the larger of these two numbers. This seemingly simple problem unveils the beauty and power of algebraic equations in representing and solving real-world scenarios. Let's embark on this mathematical journey together.

Defining Variables and Setting Up Equations

The first step towards solving any mathematical problem is to translate the given information into a mathematical framework. In this case, we introduce two variables to represent the unknown numbers:

Let x represent the larger number.
Let y represent the smaller number.

Now, we can express the given information as algebraic equations:

The sum of the two numbers is 112: This translates to the equation x + y = 112. This equation represents a linear relationship between the two numbers, and it forms the foundation of our solution.
When the larger number (x) is divided by the smaller number (y), the quotient is 3, and the remainder is 4: This translates to the equation x = 3y + 4. This equation encapsulates the division relationship between the two numbers, incorporating the quotient and remainder.

With these two equations, we have successfully transformed the word problem into a system of linear equations. The next step involves solving this system to find the values of x and y. Understanding how to set up these equations is crucial, as it lays the groundwork for the entire solution process.

Method 1: Substitution Method - A Detailed Walkthrough

The substitution method is a powerful technique for solving systems of equations. It involves expressing one variable in terms of the other and substituting that expression into the other equation. This reduces the system to a single equation with one variable, which can then be solved easily. Let's apply this method to our problem:

Express one variable in terms of the other: We already have an equation where x is expressed in terms of y: x = 3y + 4. This equation is perfectly suited for substitution.
Substitute the expression into the other equation: Substitute the expression for x (3y + 4) into the first equation (x + y = 112):

(3y + 4) + y = 112

This step is crucial as it eliminates x and leaves us with an equation involving only y.
Simplify and solve for y: Combine like terms and solve for y:

4y + 4 = 112

4y = 108

y = 27

We have successfully found the value of y, the smaller number. This is a significant milestone in our solution process.
Substitute the value of y back into either equation to find x: Let's use the equation x = 3y + 4:

x = 3(27) + 4

x = 81 + 4

x = 85

We have now found the value of x, the larger number. This completes our solution using the substitution method.

The substitution method provides a systematic approach to solving systems of equations. By carefully following each step, we can unravel the relationships between variables and find their values.

Method 2: Elimination Method - An Alternative Approach

The elimination method offers an alternative strategy for solving systems of equations. This method involves manipulating the equations so that the coefficients of one variable are opposites, and then adding the equations together to eliminate that variable. Let's apply this method to our problem:

Rearrange the equations to align variables: Rewrite the equations so that the x and y terms are aligned:

x + y = 112

x - 3y = 4 (Subtract 3y from both sides of the second original equation)

This step is essential for the elimination method to work effectively.
Multiply one or both equations by a constant so that the coefficients of one variable are opposites: Notice that the coefficients of x are already the same (1). To eliminate x, we can multiply the second equation by -1:

x + y = 112

-x + 3y = -4

Now, the coefficients of x are opposites (1 and -1).
Add the equations together to eliminate one variable: Add the two equations:

( x + y ) + ( -x + 3y ) = 112 + (-4)

4y = 108

x is eliminated, leaving us with an equation in y.
Solve for the remaining variable: Solve for y:

y = 27

We have found the value of y, the smaller number, using the elimination method.
Substitute the value of y back into either equation to find x: As before, let's use the equation x + y = 112:

x + 27 = 112

x = 85

We have now found the value of x, the larger number, using the elimination method.

The elimination method provides a different perspective on solving systems of equations. By strategically manipulating the equations, we can eliminate variables and simplify the problem.

Solution and Verification

Regardless of the method used (substitution or elimination), we arrive at the same solution:

x = 85 (The larger number)
y = 27 (The smaller number)

However, a solution is not complete without verification. We must ensure that our values for x and y satisfy the original conditions of the problem:

Sum of the numbers: 85 + 27 = 112 (This condition is satisfied).
Division with quotient and remainder: When 85 is divided by 27, the quotient is 3, and the remainder is 4 (This condition is also satisfied).

Since both conditions are met, our solution is verified. The larger number is indeed 85.

Key Takeaways and Problem-Solving Strategies

This problem highlights several important mathematical concepts and problem-solving strategies:

Translating word problems into algebraic equations: The ability to convert real-world scenarios into mathematical equations is a fundamental skill in problem-solving.
Understanding the division algorithm: The relationship between the dividend, divisor, quotient, and remainder is crucial in solving division-related problems.
Solving systems of equations: Mastering techniques like substitution and elimination is essential for tackling problems with multiple unknowns.
Verification of solutions: Always verify your solutions to ensure accuracy and catch potential errors.

By understanding these concepts and strategies, you can approach similar problems with confidence and clarity.

Real-World Applications

Problems involving sums, quotients, and remainders often arise in various real-world contexts. Here are a few examples:

Resource allocation: Dividing a limited resource among different groups or individuals, considering specific constraints and requirements.
Financial calculations: Determining the principal and interest components of a loan or investment.
Computer programming: Implementing algorithms that involve division, remainders, and modular arithmetic.

By recognizing the underlying mathematical principles, you can apply these problem-solving skills to a wide range of practical situations.

Conclusion

In this comprehensive guide, we have successfully navigated the problem of finding two numbers given their sum and quotient with a remainder. We have explored two distinct methods for solving the system of equations, emphasized the importance of solution verification, and highlighted the real-world applications of these concepts. By mastering these problem-solving techniques, you can enhance your mathematical skills and confidently tackle similar challenges in various domains.

Number puzzles and mathematical riddles have captivated minds for centuries, challenging our logical reasoning and problem-solving abilities. One such puzzle presents us with a scenario involving two unknown numbers. We are told that the sum of these two numbers is 112, and when the larger number is divided by the smaller number, the quotient is 3, with a remainder of 4. The challenge lies in identifying the larger of these two numbers. This type of problem, while seemingly straightforward, requires a systematic approach and a solid understanding of fundamental mathematical concepts. In this article, we will dissect this puzzle, exploring different methods to arrive at the solution, and highlighting the underlying principles that make this type of problem so engaging. We will embark on a journey of mathematical exploration, unraveling the mystery of these numbers step by step.

Deconstructing the Puzzle The Importance of Careful Analysis

The first step in tackling any mathematical puzzle is to carefully analyze the information provided. In this case, we have two key pieces of information:

The sum of the two numbers is 112: This tells us that if we add the two numbers together, we will get 112. This is a direct relationship that we can express mathematically.
When the larger number is divided by the smaller number, the quotient is 3, with a remainder of 4: This statement describes the division relationship between the two numbers. It tells us that the larger number is three times the smaller number, plus an additional 4. This is a slightly more complex relationship, but it provides a crucial clue to solving the puzzle.

By carefully dissecting these statements, we can identify the core relationships between the two unknown numbers. This is a crucial step, as it allows us to translate the puzzle into a mathematical framework that we can work with. Without a clear understanding of the relationships, it would be difficult to develop a strategy for solving the puzzle. The ability to analyze information and identify key relationships is a fundamental skill in problem-solving, and this puzzle provides an excellent opportunity to practice this skill.

Translating to Equations Representing Relationships Mathematically

Once we have a clear understanding of the relationships between the numbers, the next step is to translate these relationships into mathematical equations. This allows us to express the puzzle in a symbolic form that we can manipulate and solve. Let's define our variables:

Let x represent the larger number.
Let y represent the smaller number.

Now, we can translate the given information into equations:

The sum of the two numbers is 112: This translates directly to the equation: x + y = 112
When the larger number is divided by the smaller number, the quotient is 3, with a remainder of 4: This can be expressed using the division algorithm as: x = 3y + 4

We now have a system of two equations with two unknowns. This is a classic setup for solving a mathematical puzzle. The equations represent the core relationships that we identified in the previous step, but now they are in a form that we can work with algebraically. The ability to translate verbal statements into mathematical equations is a crucial skill in mathematics and problem-solving. It allows us to move from the concrete world of words to the abstract world of symbols, where we can apply powerful tools and techniques to find solutions.

Method 1 Substitution A Direct Path to the Solution

The substitution method is a powerful technique for solving systems of equations. It involves solving one equation for one variable and then substituting that expression into the other equation. This reduces the system to a single equation with one unknown, which we can then solve directly. Let's apply the substitution method to our puzzle:

Solve one equation for one variable: We already have the second equation solved for x: x = 3y + 4. This is ideal for substitution.
Substitute into the other equation: Substitute the expression for x (3y + 4) into the first equation (x + y = 112):

(3y + 4) + y = 112

This substitution eliminates x and leaves us with an equation in terms of y only.
Simplify and solve for y: Combine like terms and solve for y:

4y + 4 = 112

4y = 108

y = 27

We have now found the value of the smaller number, y. This is a significant step forward.
Substitute back to find x: Substitute the value of y (27) back into either of the original equations to find x. Let's use x = 3y + 4:

x = 3(27) + 4

x = 81 + 4

x = 85

We have now found the value of the larger number, x. The puzzle is solved!

The substitution method provides a clear and direct path to the solution. By carefully substituting one expression into another, we were able to eliminate a variable and reduce the problem to a simpler form. This technique is a valuable tool in the problem-solver's arsenal.

Method 2 Elimination An Alternative Approach

The elimination method is another powerful technique for solving systems of equations. It involves manipulating the equations so that the coefficients of one variable are opposites, and then adding the equations together to eliminate that variable. Let's apply the elimination method to our puzzle:

Rearrange the equations: Rewrite the equations so that the x and y terms are aligned:

x + y = 112

x - 3y = 4 (Subtract 3y from both sides of the second equation)
Multiply to create opposites: Notice that the coefficients of x are already the same. To eliminate x, we can multiply the second equation by -1:

x + y = 112

-x + 3y = -4

Now, the coefficients of x are opposites.
Add the equations: Add the two equations together:

(x + y) + (-x + 3y) = 112 + (-4)

4y = 108

x is eliminated, leaving us with an equation in terms of y.
Solve for y: Solve for y:

y = 27

We have found the value of the smaller number, y, using the elimination method.
Substitute back to find x: Substitute the value of y (27) back into either of the original equations to find x. Let's use x + y = 112:

x + 27 = 112

x = 85

We have now found the value of the larger number, x. The puzzle is solved!

The elimination method provides an alternative perspective on solving systems of equations. By strategically manipulating the equations, we can eliminate variables and simplify the problem. This technique is particularly useful when the coefficients of one variable are easily made opposites.

Solution and Verification Ensuring Accuracy

Using both the substitution and elimination methods, we arrived at the same solution:

x = 85 (The larger number)
y = 27 (The smaller number)

However, it's crucial to verify our solution to ensure accuracy. Let's check if our values satisfy the original conditions:

Sum of the numbers: 85 + 27 = 112 (This condition is satisfied).
Division with quotient and remainder: When 85 is divided by 27, the quotient is 3, and the remainder is 4 (This condition is also satisfied).

Since both conditions are met, our solution is verified. The larger number is indeed 85.

Verification is an essential step in problem-solving. It allows us to catch potential errors and ensure that our solution is correct. By plugging our values back into the original conditions, we can gain confidence in our answer.

General Strategies for Solving Number Puzzles Building a Toolkit

This puzzle illustrates several general strategies that can be applied to solving number puzzles:

Careful Analysis: Thoroughly understand the information provided and identify the key relationships between the unknowns.
Translation to Equations: Express the relationships mathematically using variables and equations.
Systematic Solution: Apply techniques like substitution or elimination to solve the system of equations.
Verification: Check your solution against the original conditions to ensure accuracy.
Flexibility: Be prepared to try different approaches if one method doesn't work.

By developing these strategies, you can build a powerful toolkit for tackling number puzzles of all kinds. The more puzzles you solve, the more adept you will become at recognizing patterns and applying these strategies effectively.

Real-World Connections Where Puzzles Meet Reality

While number puzzles may seem like purely abstract exercises, they often have connections to real-world situations. Problems involving resource allocation, financial planning, and even computer programming can sometimes be modeled using systems of equations similar to the one we solved in this puzzle. The ability to translate real-world scenarios into mathematical models is a valuable skill in many fields.

Conclusion The Joy of Problem Solving

In this article, we have successfully solved a number puzzle involving two unknowns, their sum, and the quotient and remainder of their division. We explored two different methods for solving the system of equations, emphasized the importance of verification, and highlighted general strategies for tackling number puzzles. This puzzle demonstrates the power of mathematical thinking and the joy of problem-solving. By carefully analyzing the information, translating it into equations, and applying systematic techniques, we can unravel the mysteries of numbers and gain a deeper appreciation for the beauty and elegance of mathematics. Keep practicing, keep exploring, and keep enjoying the challenge of number puzzles!