What Is The Value Of The Expression \left(-4 \frac{1}{4} - (-5)\right ]?

Mar 13, 2025 by ADMIN 73 views

$What is the Value of the Expression $\left(-4 \frac{1}{4} - (-5)\right)$?$

Introduction

When dealing with mathematical expressions, it's essential to understand the rules of arithmetic operations to evaluate them correctly. In this article, we will focus on the value of the expression $\left(-4 \frac{1}{4} - (-5)\right)$ . To solve this problem, we need to apply the rules of arithmetic operations, specifically the rule for subtracting negative numbers.

Understanding the Expression

The given expression is $\left(-4 \frac{1}{4} - (-5)\right)$ . To evaluate this expression, we need to follow the order of operations (PEMDAS):

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

Evaluating the Expression

Let's break down the expression into smaller parts and evaluate each part step by step:

Step 1: Convert the Mixed Number to an Improper Fraction

The first step is to convert the mixed number $-4 \frac{1}{4}$ to an improper fraction. To do this, we multiply the whole number part by the denominator and add the numerator:

-4 \frac{1}{4} = \frac{-4 \times 4 + 1}{4} = \frac{-16 + 1}{4} = \frac{-15}{4}

Step 2: Subtract the Negative Number

Now that we have converted the mixed number to an improper fraction, we can rewrite the expression as:

\left(\frac{-15}{4} - (-5)\right)

When subtracting a negative number, we can change the subtraction sign to an addition sign and change the sign of the number being subtracted:

\left(\frac{-15}{4} + 5\right)

Step 3: Add the Fractions

Now we can add the fractions:

\left(\frac{-15}{4} + \frac{5 \times 4}{4}\right) = \left(\frac{-15}{4} + \frac{20}{4}\right)

Step 4: Combine the Fractions

We can combine the fractions by adding the numerators:

\left(\frac{-15 + 20}{4}\right) = \left(\frac{5}{4}\right)

Conclusion

The value of the expression $\left(-4 \frac{1}{4} - (-5)\right)$ is $\frac{5}{4}$ .

Importance of Understanding Arithmetic Operations

Understanding arithmetic operations is crucial in mathematics. It helps us to evaluate expressions correctly and solve problems efficiently. In this article, we have seen how to evaluate the expression $\left(-4 \frac{1}{4} - (-5)\right)$ by applying the rules of arithmetic operations.

Common Mistakes to Avoid

When evaluating expressions, it's essential to avoid common mistakes. Some common mistakes to avoid include:

Not following the order of operations (PEMDAS)
Not converting mixed numbers to improper fractions
Not changing the subtraction sign to an addition sign when subtracting a negative number
Not combining fractions correctly

Real-World Applications

Understanding arithmetic operations has many real-world applications. For example, in finance, we use arithmetic operations to calculate interest rates, investment returns, and other financial metrics. In science, we use arithmetic operations to calculate distances, velocities, and other physical quantities.

Final Thoughts

In conclusion, the value of the expression $\left(-4 \frac{1}{4} - (-5)\right)$ is $\frac{5}{4}$ . Understanding arithmetic operations is crucial in mathematics, and it has many real-world applications. By following the rules of arithmetic operations and avoiding common mistakes, we can evaluate expressions correctly and solve problems efficiently.

Frequently Asked Questions

Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that tells us which operations to perform first when evaluating an expression. The rules are:

Parentheses: Evaluate expressions inside parentheses first.
Exponents: Evaluate any exponential expressions next.
Multiplication and Division: Evaluate 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 convert a mixed number to an improper fraction?

A: To convert a mixed number to an improper fraction, multiply the whole number part by the denominator and add the numerator. Then, divide the result by the denominator.

Q: What is the difference between subtracting a negative number and adding a positive number?

A: When subtracting a negative number, we can change the subtraction sign to an addition sign and change the sign of the number being subtracted. This is because subtracting a negative number is equivalent to adding a positive number.

Q: How do I combine fractions?

A: To combine fractions, we need to have the same denominator. We can then add or subtract the numerators.

References

Introduction

Arithmetic operations are the building blocks of mathematics, and understanding them is crucial for solving problems and evaluating expressions. In this article, we will answer some frequently asked questions about arithmetic operations, including the order of operations, converting mixed numbers to improper fractions, and combining fractions.

Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that tells us which operations to perform first when evaluating an expression. The rules are:

Parentheses: Evaluate expressions inside parentheses first.
Exponents: Evaluate any exponential expressions next.
Multiplication and Division: Evaluate 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 convert a mixed number to an improper fraction?

A: To convert a mixed number to an improper fraction, multiply the whole number part by the denominator and add the numerator. Then, divide the result by the denominator.

Example: Convert the mixed number $3 \frac{1}{4}$ to an improper fraction.

Multiply the whole number part by the denominator: $3 \times 4 = 12$
Add the numerator: $12 + 1 = 13$
Divide the result by the denominator: $\frac{13}{4}$

Q: What is the difference between subtracting a negative number and adding a positive number?

Example: Evaluate the expression $5 - (-3)$ .

Change the subtraction sign to an addition sign: $5 + 3$
Add the numbers: $5 + 3 = 8$

Q: How do I combine fractions?

A: To combine fractions, we need to have the same denominator. We can then add or subtract the numerators.

Example: Combine the fractions $\frac{1}{4}$ and $\frac{2}{4}$ .

Have the same denominator: $\frac{1}{4}$ and $\frac{2}{4}$
Add the numerators: $\frac{1 + 2}{4} = \frac{3}{4}$

Q: What is the difference between a rational number and an irrational number?

A: A rational number is a number that can be expressed as a fraction, where the numerator and denominator are integers. An irrational number is a number that cannot be expressed as a fraction, and its decimal representation goes on forever without repeating.

Example: The number $\frac{1}{2}$ is a rational number, while the number $\sqrt{2}$ is an irrational number.

Q: How do I evaluate an expression with multiple operations?

A: To evaluate an expression with multiple operations, we need to follow the order of operations (PEMDAS). We start with the innermost parentheses and work our way outwards.

Example: Evaluate the expression $3 \times 2 + 5 - 1$ .

Multiply the numbers: $3 \times 2 = 6$
Add the numbers: $6 + 5 = 11$
Subtract the number: $11 - 1 = 10$

Q: What is the difference between a variable and a constant?

A: A variable is a letter or symbol that represents a value that can change. A constant is a value that does not change.

Example: In the expression $x + 5$ , $x$ is a variable, while $5$ is a constant.

Q: How do I simplify an expression?

A: To simplify an expression, we need to combine like terms and eliminate any unnecessary operations.

Example: Simplify the expression $2x + 3x + 5$ .

Combine like terms: $2x + 3x = 5x$
Eliminate the unnecessary operation: $5x + 5$

Conclusion

In this article, we have answered some frequently asked questions about arithmetic operations, including the order of operations, converting mixed numbers to improper fractions, and combining fractions. We have also discussed the difference between rational and irrational numbers, and how to evaluate expressions with multiple operations. By following the rules of arithmetic operations, we can simplify expressions and solve problems efficiently.