The Shopkeeper Receives More Biscuits

Oct 13, 2025 by windailys.com 114 views

Introduction

In the world of mathematics, algebraic expressions are used to represent complex relationships between variables. In this article, we will explore how a shopkeeper uses algebraic expressions to keep track of his stock. We will delve into the world of variables, equations, and algebraic manipulations to solve a real-world problem.

The Initial Situation

Let's assume that a shopkeeper has x packets of biscuits and y packets of chips in his store. The total number of items in the shop can be represented by the algebraic expression 3x + 4y. This expression takes into account the number of biscuits and chips, and their respective multipliers.

The Algebraic Expression

The algebraic expression 3x + 4y can be broken down into two parts:

3x represents the number of biscuits, where x is the number of packets.
4y represents the number of chips, where y is the number of packets.

By combining these two parts, we get the total number of items in the shop: 3x + 4y.

The Shopkeeper Receives More Biscuits

The shopkeeper receives 5 more packets of biscuits. This means that the number of biscuits increases by 5. We can represent this increase by adding 5 to the original number of biscuits, x.

The new number of biscuits can be represented by the expression x + 5.

The Shopkeeper Sells Some Chips

The shopkeeper sells 2 packets of chips. This means that the number of chips decreases by 2. We can represent this decrease by subtracting 2 from the original number of chips, y.

The new number of chips can be represented by the expression y - 2.

The New Total

After receiving more biscuits and selling some chips, the shopkeeper needs to update the total number of items in the shop. We can do this by substituting the new expressions for biscuits and chips into the original algebraic expression.

The new total can be represented by the expression:

3(x + 5) + 4(y - 2)

Simplifying the Expression

To simplify the expression, we can use the distributive property of multiplication over addition. This means that we can multiply the terms inside the parentheses by the corresponding coefficients.

The expression 3(x + 5) can be simplified to 3x + 15.

The expression 4(y - 2) can be simplified to 4y - 8.

The Final Expression

By substituting the simplified expressions back into the original expression, we get:

3x + 15 + 4y - 8

Combining Like Terms

We can combine like terms by adding or subtracting the coefficients of the same variables.

The expression 3x + 15 can be combined with 4y - 8 to get:

3x + 4y + 7

Conclusion

In this article, we used algebraic expressions to solve a real-world problem. We started with an initial situation, where a shopkeeper had x packets of biscuits and y packets of chips. We then received more biscuits and sold some chips, and updated the total number of items in the shop. By using algebraic manipulations, we simplified the expression and arrived at the final answer.

The Importance of Algebraic Expressions

Algebraic expressions are a powerful tool for solving complex problems. By representing relationships between variables, we can analyze and manipulate data to arrive at meaningful conclusions. In the world of mathematics, algebraic expressions are used to model real-world situations, and to solve problems that involve variables and equations.

Real-World Applications

Algebraic expressions have numerous real-world applications. In business, they are used to model sales, revenue, and expenses. In science, they are used to model physical systems, such as motion and energy. In engineering, they are used to design and optimize systems, such as bridges and buildings.

Tips for Solving Algebraic Expressions

When solving algebraic expressions, it's essential to follow these tips:

Start by identifying the variables and their coefficients.
Use the distributive property to simplify expressions.
Combine like terms to simplify the expression.
Check your work by plugging in values for the variables.

By following these tips, you can master the art of solving algebraic expressions and apply it to real-world problems.

Common Mistakes to Avoid

When solving algebraic expressions, it's essential to avoid these common mistakes:

Not identifying the variables and their coefficients.
Not using the distributive property to simplify expressions.
Not combining like terms to simplify the expression.
Not checking your work by plugging in values for the variables.

By avoiding these mistakes, you can ensure that your solutions are accurate and reliable.

Conclusion

Introduction

In our previous article, we explored how a shopkeeper uses algebraic expressions to keep track of his stock. We delved into the world of variables, equations, and algebraic manipulations to solve a real-world problem. In this article, we will answer some frequently asked questions (FAQs) related to algebraic expressions and their applications in real-world scenarios.

Q&A

Q: What is an algebraic expression?

A: An algebraic expression is a mathematical expression that consists of variables, constants, and mathematical operations. It is used to represent a relationship between variables and constants.

Q: What are the types of algebraic expressions?

A: There are two main types of algebraic expressions:

Polynomial expressions: These are algebraic expressions that consist of variables and constants, and are used to represent relationships between variables and constants.
Rational expressions: These are algebraic expressions that consist of variables and constants, and are used to represent relationships between variables and constants.

Q: How do I simplify an algebraic expression?

A: To simplify an algebraic expression, you can use the following steps:

Combine like terms: Combine the terms that have the same variable and coefficient.
Use the distributive property: Use the distributive property to simplify expressions.
Check your work: Check your work by plugging in values for the variables.

Q: What is the distributive property?

A: The distributive property is a mathematical property that states that a single term can be distributed to multiple terms. It is used to simplify expressions and is represented by the following formula:

a(b + c) = ab + ac

Q: How do I solve an algebraic equation?

A: To solve an algebraic equation, you can use the following steps:

Isolate the variable: Isolate the variable by moving all the terms that contain the variable to one side of the equation.
Use inverse operations: Use inverse operations to isolate the variable.
Check your work: Check your work by plugging in values for the variables.

Q: What is the difference between an algebraic expression and an algebraic equation?

A: An algebraic expression is a mathematical expression that consists of variables, constants, and mathematical operations. An algebraic equation is a mathematical equation that consists of an algebraic expression set equal to a value or another expression.

Q: How do I use algebraic expressions in real-world scenarios?

A: Algebraic expressions are used in a variety of real-world scenarios, including:

Business: Algebraic expressions are used to model sales, revenue, and expenses.
Science: Algebraic expressions are used to model physical systems, such as motion and energy.
Engineering: Algebraic expressions are used to design and optimize systems, such as bridges and buildings.

Q: What are some common mistakes to avoid when working with algebraic expressions?

A: Some common mistakes to avoid when working with algebraic expressions include:

Not identifying the variables and their coefficients: Make sure to identify the variables and their coefficients in the algebraic expression.
Not using the distributive property: Use the distributive property to simplify expressions.
Not combining like terms: Combine like terms to simplify the expression.
Not checking your work: Check your work by plugging in values for the variables.

Conclusion

In conclusion, algebraic expressions are a powerful tool for solving complex problems. By representing relationships between variables and constants, we can analyze and manipulate data to arrive at meaningful conclusions. In the world of mathematics, algebraic expressions are used to model real-world situations, and to solve problems that involve variables and equations. By following the tips and avoiding common mistakes, you can master the art of solving algebraic expressions and apply it to real-world problems.

Additional Resources

For more information on algebraic expressions and their applications, check out the following resources:

Algebraic Expression Calculator: A calculator that can be used to simplify and solve algebraic expressions.
Algebraic Equation Solver: A solver that can be used to solve algebraic equations.
Mathematics Textbooks: A collection of textbooks that cover algebraic expressions and their applications.

Final Thoughts

Algebraic expressions are a powerful tool for solving complex problems. By representing relationships between variables and constants, we can analyze and manipulate data to arrive at meaningful conclusions. In the world of mathematics, algebraic expressions are used to model real-world situations, and to solve problems that involve variables and equations. By following the tips and avoiding common mistakes, you can master the art of solving algebraic expressions and apply it to real-world problems.