Evaluate The Expression: ${ -4.8 \div \left| 2 \frac{1}{5} + \left(-3 \frac{7}{10}\right) \right| }$

by ADMIN 102 views

===========================================================

Introduction

In mathematics, evaluating expressions is a crucial skill that involves simplifying complex mathematical operations. It requires a deep understanding of mathematical concepts, including fractions, absolute values, and division. In this article, we will evaluate the expression: ${ -4.8 \div \left| 2 \frac{1}{5} + \left(-3 \frac{7}{10}\right) \right| }$ and provide a step-by-step guide on how to solve it.

Understanding the Expression

The given expression involves several mathematical operations, including addition, absolute value, and division. To evaluate this expression, we need to follow the order of operations (PEMDAS):

  1. Parentheses: Evaluate the expressions inside the parentheses.
  2. Exponents: Evaluate any exponential expressions.
  3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
  4. Addition and Subtraction: Evaluate any addition and subtraction operations from left to right.

Step 1: Convert Mixed Numbers to Improper Fractions

The expression contains two mixed numbers: 2152 \frac{1}{5} and −3710-3 \frac{7}{10}. To evaluate the expression, we need to convert these mixed numbers to improper fractions.

215=(2×5)+15=10+15=1152 \frac{1}{5} = \frac{(2 \times 5) + 1}{5} = \frac{10 + 1}{5} = \frac{11}{5}

−3710=(−3×10)+710=−30+710=−2310-3 \frac{7}{10} = \frac{(-3 \times 10) + 7}{10} = \frac{-30 + 7}{10} = \frac{-23}{10}

Step 2: Add the Improper Fractions

Now that we have converted the mixed numbers to improper fractions, we can add them together.

115+−2310=2210+−2310=22−2310=−110\frac{11}{5} + \frac{-23}{10} = \frac{22}{10} + \frac{-23}{10} = \frac{22 - 23}{10} = \frac{-1}{10}

Step 3: Evaluate the Absolute Value

The expression contains an absolute value operation: ∣−110∣\left| \frac{-1}{10} \right|. To evaluate this expression, we need to take the absolute value of the fraction.

∣−110∣=110\left| \frac{-1}{10} \right| = \frac{1}{10}

Step 4: Divide -4.8 by the Absolute Value

Finally, we can divide -4.8 by the absolute value of the fraction.

−4.8÷110=−4.8×101=−48-4.8 \div \frac{1}{10} = -4.8 \times \frac{10}{1} = -48

Conclusion

In this article, we evaluated the expression: ${ -4.8 \div \left| 2 \frac{1}{5} + \left(-3 \frac{7}{10}\right) \right| }$ and provided a step-by-step guide on how to solve it. We converted mixed numbers to improper fractions, added the improper fractions, evaluated the absolute value, and finally divided -4.8 by the absolute value. The final answer is -48.

Frequently Asked Questions

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

A: The order of operations (PEMDAS) is a mnemonic device that helps us remember the order in which to evaluate mathematical expressions. It stands for:

  1. Parentheses
  2. Exponents
  3. Multiplication and Division
  4. Addition and Subtraction

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 by the denominator and add the numerator. Then, write the result as a fraction with the denominator as the new numerator.

Q: What is the absolute value of a fraction?

A: The absolute value of a fraction is the distance of the fraction from zero on the number line. It is always positive, regardless of the sign of the fraction.

Q: How do I divide a decimal by a fraction?

A: To divide a decimal by a fraction, multiply the decimal by the reciprocal of the fraction.

===========================================================

Introduction

In mathematics, evaluating expressions is a crucial skill that involves simplifying complex mathematical operations. It requires a deep understanding of mathematical concepts, including fractions, absolute values, and division. In this article, we will evaluate the expression: ${ -4.8 \div \left| 2 \frac{1}{5} + \left(-3 \frac{7}{10}\right) \right| }$ and provide a step-by-step guide on how to solve it.

Understanding the Expression

The given expression involves several mathematical operations, including addition, absolute value, and division. To evaluate this expression, we need to follow the order of operations (PEMDAS):

  1. Parentheses: Evaluate the expressions inside the parentheses.
  2. Exponents: Evaluate any exponential expressions.
  3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
  4. Addition and Subtraction: Evaluate any addition and subtraction operations from left to right.

Step 1: Convert Mixed Numbers to Improper Fractions

The expression contains two mixed numbers: 2152 \frac{1}{5} and −3710-3 \frac{7}{10}. To evaluate the expression, we need to convert these mixed numbers to improper fractions.

215=(2×5)+15=10+15=1152 \frac{1}{5} = \frac{(2 \times 5) + 1}{5} = \frac{10 + 1}{5} = \frac{11}{5}

−3710=(−3×10)+710=−30+710=−2310-3 \frac{7}{10} = \frac{(-3 \times 10) + 7}{10} = \frac{-30 + 7}{10} = \frac{-23}{10}

Step 2: Add the Improper Fractions

Now that we have converted the mixed numbers to improper fractions, we can add them together.

115+−2310=2210+−2310=22−2310=−110\frac{11}{5} + \frac{-23}{10} = \frac{22}{10} + \frac{-23}{10} = \frac{22 - 23}{10} = \frac{-1}{10}

Step 3: Evaluate the Absolute Value

The expression contains an absolute value operation: ∣−110∣\left| \frac{-1}{10} \right|. To evaluate this expression, we need to take the absolute value of the fraction.

∣−110∣=110\left| \frac{-1}{10} \right| = \frac{1}{10}

Step 4: Divide -4.8 by the Absolute Value

Finally, we can divide -4.8 by the absolute value of the fraction.

−4.8÷110=−4.8×101=−48-4.8 \div \frac{1}{10} = -4.8 \times \frac{10}{1} = -48

Conclusion

In this article, we evaluated the expression: ${ -4.8 \div \left| 2 \frac{1}{5} + \left(-3 \frac{7}{10}\right) \right| }$ and provided a step-by-step guide on how to solve it. We converted mixed numbers to improper fractions, added the improper fractions, evaluated the absolute value, and finally divided -4.8 by the absolute value. The final answer is -48.

Frequently Asked Questions

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

A: The order of operations (PEMDAS) is a mnemonic device that helps us remember the order in which to evaluate mathematical expressions. It stands for:

  1. Parentheses
  2. Exponents
  3. Multiplication and Division
  4. Addition and Subtraction

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 by the denominator and add the numerator. Then, write the result as a fraction with the denominator as the new numerator.

Q: What is the absolute value of a fraction?

A: The absolute value of a fraction is the distance of the fraction from zero on the number line. It is always positive, regardless of the sign of the fraction.

Q: How do I divide a decimal by a fraction?

A: To divide a decimal by a fraction, multiply the decimal by the reciprocal of the fraction.

Q: What is the difference between a fraction and a decimal?

A: A fraction is a way of expressing a part of a whole, while a decimal is a way of expressing a part of a whole using a base-10 system. For example, the fraction 12\frac{1}{2} is equal to the decimal 0.5.

Q: How do I add fractions with different denominators?

A: To add fractions with different denominators, find the least common multiple (LCM) of the denominators and convert each fraction to have that LCM as the denominator. Then, add the fractions.

Q: How do I subtract fractions with different denominators?

A: To subtract fractions with different denominators, find the least common multiple (LCM) of the denominators and convert each fraction to have that LCM as the denominator. Then, subtract the fractions.

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, while an irrational number is a number that cannot be expressed as a fraction.

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

A: To evaluate an expression with multiple operations, follow the order of operations (PEMDAS) and evaluate each operation from left to right.

Additional Resources

Conclusion

In this article, we evaluated the expression: ${ -4.8 \div \left| 2 \frac{1}{5} + \left(-3 \frac{7}{10}\right) \right| }$ and provided a step-by-step guide on how to solve it. We also answered frequently asked questions about evaluating expressions, converting mixed numbers to improper fractions, and more.