Calculate The Following Expression:${ \frac{50 \times 33 + 25 \times 50}{10 \times 50 - 50 \times 19} }$

Mar 9, 2025 by ADMIN 106 views

**Solving Complex Mathematical Expressions: A Step-by-Step Guide**

Introduction

Mathematics is a vast and complex subject that involves solving various types of problems, including algebraic expressions. In this article, we will focus on calculating a specific mathematical expression that involves multiplication, addition, and subtraction. We will break down the expression into smaller parts, simplify it, and finally arrive at the solution.

The Expression

The given expression is:

${ \frac{50 \times 33 + 25 \times 50}{10 \times 50 - 50 \times 19} }$

This expression involves multiplication, addition, and subtraction, making it a complex mathematical problem.

Step 1: Multiply the Numbers

The first step in solving this expression is to multiply the numbers involved in the numerator and denominator.

${ 50 \times 33 = 1650 }$ ${ 25 \times 50 = 1250 }$ ${ 10 \times 50 = 500 }$ ${ 50 \times 19 = 950 }$

Step 2: Add and Subtract the Results

Now that we have the results of the multiplication, we can add and subtract them to simplify the expression.

${ 1650 + 1250 = 2900 }$ ${ 500 - 950 = -450 }$

Step 3: Simplify the Expression

Now that we have the results of the addition and subtraction, we can simplify the expression by dividing the numerator by the denominator.

${ \frac{2900}{-450} = -\frac{2900}{450} }$

Step 4: Simplify the Fraction

To simplify the fraction, we can divide both the numerator and denominator by their greatest common divisor (GCD).

${ \text{GCD}(2900, 450) = 50 }$

${ -\frac{2900}{450} = -\frac{2900 \div 50}{450 \div 50} }$

${ -\frac{58}{9} }$

Conclusion

In this article, we solved a complex mathematical expression that involved multiplication, addition, and subtraction. We broke down the expression into smaller parts, simplified it, and finally arrived at the solution. The final answer is:

${ -\frac{58}{9} }$

Real-World Applications

Solving complex mathematical expressions like this one has many real-world applications. For example, in finance, it can be used to calculate interest rates, investment returns, and other financial metrics. In science, it can be used to model complex systems, such as population growth, chemical reactions, and other phenomena.

Tips and Tricks

When solving complex mathematical expressions, it's essential to break them down into smaller parts, simplify them, and finally arrive at the solution. Here are some tips and tricks to help you solve complex mathematical expressions:

Use the order of operations: When solving mathematical expressions, it's essential to follow the order of operations (PEMDAS): Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction.
Simplify the expression: Before solving the expression, simplify it by combining like terms, canceling out common factors, and other techniques.
Use algebraic manipulations: Algebraic manipulations, such as factoring, expanding, and canceling, can help simplify complex mathematical expressions.
Check your work: Finally, it's essential to check your work by plugging the solution back into the original expression to ensure that it's correct.

Common Mistakes

When solving complex mathematical expressions, it's easy to make mistakes. Here are some common mistakes to avoid:

Incorrect order of operations: Failing to follow the order of operations (PEMDAS) can lead to incorrect solutions.
Insufficient simplification: Failing to simplify the expression can lead to complex and difficult-to-solve equations.
Incorrect algebraic manipulations: Failing to use algebraic manipulations correctly can lead to incorrect solutions.
Failure to check work: Failing to check your work can lead to incorrect solutions.

Conclusion

Q: What is the order of operations?

A: The order of operations is a set of rules that dictates the order in which mathematical operations should be performed when there are multiple operations in an expression. The order of operations is often remembered using the acronym PEMDAS, which stands for:

Parentheses: Evaluate expressions inside parentheses first.
Exponents: Evaluate any exponential expressions next (e.g., 2^3).
Multiplication and Division: Evaluate any multiplication and division operations from left to right.
Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I simplify a complex mathematical expression?

A: Simplifying a complex mathematical expression involves breaking it down into smaller parts, combining like terms, and canceling out common factors. Here are some steps you can follow:

Combine like terms: Combine any terms that have the same variable and coefficient.
Cancel out common factors: Cancel out any common factors between the numerator and denominator.
Simplify fractions: Simplify any fractions by dividing both the numerator and denominator by their greatest common divisor (GCD).
Use algebraic manipulations: Use algebraic manipulations, such as factoring, expanding, and canceling, to simplify the expression.

Q: What are some common algebraic manipulations?

A: Some common algebraic manipulations include:

Factoring: Factoring involves expressing an expression as a product of simpler expressions.
Expanding: Expanding involves expressing an expression as a sum of simpler expressions.
Canceling: Canceling involves canceling out common factors between the numerator and denominator.
Simplifying fractions: Simplifying fractions involves dividing both the numerator and denominator by their greatest common divisor (GCD).

Q: How do I check my work when solving a complex mathematical expression?

A: Checking your work when solving a complex mathematical expression involves plugging the solution back into the original expression to ensure that it's correct. Here are some steps you can follow:

Plug the solution back into the original expression: Plug the solution back into the original expression to ensure that it's correct.
Simplify the expression: Simplify the expression to ensure that it's correct.
Check for errors: Check for any errors in the solution, such as incorrect signs or values.

Q: What are some common mistakes to avoid when solving complex mathematical expressions?

A: Some common mistakes to avoid when solving complex mathematical expressions include:

Incorrect order of operations: Failing to follow the order of operations (PEMDAS) can lead to incorrect solutions.
Insufficient simplification: Failing to simplify the expression can lead to complex and difficult-to-solve equations.
Incorrect algebraic manipulations: Failing to use algebraic manipulations correctly can lead to incorrect solutions.
Failure to check work: Failing to check your work can lead to incorrect solutions.

Q: How can I practice solving complex mathematical expressions?

A: Practicing solving complex mathematical expressions involves working through a variety of problems and exercises. Here are some tips to help you practice:

Work through a variety of problems: Work through a variety of problems and exercises to help you practice solving complex mathematical expressions.
Use online resources: Use online resources, such as math websites and apps, to help you practice solving complex mathematical expressions.
Join a study group: Join a study group or find a study partner to help you practice solving complex mathematical expressions.
Seek help when needed: Seek help when needed, such as from a teacher or tutor, to help you practice solving complex mathematical expressions.

Conclusion

Solving complex mathematical expressions requires patience, persistence, and practice. By following the order of operations, simplifying the expression, using algebraic manipulations, and checking your work, you can solve even the most complex mathematical problems. Remember to avoid common mistakes, such as incorrect order of operations, insufficient simplification, incorrect algebraic manipulations, and failure to check work. With practice and dedication, you can become proficient in solving complex mathematical expressions.