Evaluate The Expression:$\[ -\frac{2}{5} \div \frac{4}{25} \\]

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Introduction

In mathematics, division is a fundamental operation that involves finding the quotient of two numbers. When dealing with fractions, division can be a bit more complex, but it can be simplified using the concept of inverting and multiplying. In this article, we will evaluate the expression $-\frac{2}{5} \div \frac{4}{25}$ and explore the steps involved in simplifying it.

Understanding the Concept of Division with Fractions

Division with fractions involves inverting the second fraction and multiplying it with the first fraction. This concept is based on the fact that division is the inverse operation of multiplication. When we divide a number by another number, we are essentially finding the number that, when multiplied by the divisor, gives us the dividend.

Evaluating the Expression

To evaluate the expression $-\frac{2}{5} \div \frac{4}{25}$, we need to follow the steps outlined above. The first step is to invert the second fraction, which means flipping the numerator and denominator. So, $\frac{4}{25}$ becomes $\frac{25}{4}$.

Step 1: Invert the Second Fraction

$\frac{4}{25} \rightarrow \frac{25}{4}$

Step 2: Multiply the First Fraction with the Inverted Second Fraction

Now that we have inverted the second fraction, we can multiply it with the first fraction. To do this, we need to multiply the numerators and denominators separately.

$-\frac{2}{5} \times \frac{25}{4} = -\frac{2 \times 25}{5 \times 4}$

Step 3: Simplify the Expression

Now that we have multiplied the fractions, we can simplify the expression by canceling out any common factors in the numerator and denominator.

$-\frac{2 \times 25}{5 \times 4} = -\frac{50}{20}$

Step 4: Reduce the Fraction to Its Lowest Terms

To reduce the fraction to its lowest terms, we need to find the greatest common divisor (GCD) of the numerator and denominator and divide both numbers by it.

$-\frac{50}{20} = -\frac{50 \div 10}{20 \div 10} = -\frac{5}{2}$

Conclusion

In conclusion, the expression $-\frac{2}{5} \div \frac{4}{25}$ can be evaluated by inverting the second fraction and multiplying it with the first fraction. By following the steps outlined above, we can simplify the expression and reduce it to its lowest terms.

Frequently Asked Questions

• What is the concept of division with fractions?
• How do we invert a fraction?
• What is the greatest common divisor (GCD) and how do we use it to reduce a fraction to its lowest terms?

Final Answer

The final answer to the expression $-\frac{2}{5} \div \frac{4}{25}$ is $-\frac{5}{2}$.

Additional Resources

For more information on division with fractions, you can refer to the following resources:

• Khan Academy: Division with Fractions
• Mathway: Division with Fractions
• Wolfram Alpha: Division with Fractions

Step-by-Step Solution

Here is the step-by-step solution to the expression $-\frac{2}{5} \div \frac{4}{25}$:

1. Invert the second fraction: $\frac{4}{25} \rightarrow \frac{25}{4}$
2. Multiply the first fraction with the inverted second fraction: $-\frac{2}{5} \times \frac{25}{4} = -\frac{2 \times 25}{5 \times 4}$
3. Simplify the expression: $-\frac{2 \times 25}{5 \times 4} = -\frac{50}{20}$
4. Reduce the fraction to its lowest terms: $-\frac{50}{20} = -\frac{50 \div 10}{20 \div 10} = -\frac{5}{2}$
   

Introduction

Division with fractions can be a bit complex, but with the right approach, it can be simplified. In this article, we will answer some frequently asked questions about division with fractions and provide step-by-step solutions to common problems.

Q&A

Q: What is the concept of division with fractions?

A: Division with fractions involves inverting the second fraction and multiplying it with the first fraction. This concept is based on the fact that division is the inverse operation of multiplication.

Q: How do we invert a fraction?

A: To invert a fraction, we need to flip the numerator and denominator. For example, if we have the fraction $\frac{4}{25}$, we can invert it by flipping the numerator and denominator to get $\frac{25}{4}$.

Q: What is the greatest common divisor (GCD) and how do we use it to reduce a fraction to its lowest terms?

A: The greatest common divisor (GCD) is the largest number that divides both the numerator and denominator of a fraction without leaving a remainder. To reduce a fraction to its lowest terms, we need to find the GCD of the numerator and denominator and divide both numbers by it.

Q: How do we simplify a fraction after dividing it by another fraction?

A: To simplify a fraction after dividing it by another fraction, we need to multiply the first fraction with the inverted second fraction and then reduce the resulting fraction to its lowest terms.

Q: What is the difference between dividing a fraction by a whole number and dividing a fraction by another fraction?

A: Dividing a fraction by a whole number involves multiplying the fraction by the reciprocal of the whole number. Dividing a fraction by another fraction involves inverting the second fraction and multiplying it with the first fraction.

Q: How do we handle negative numbers when dividing fractions?

A: When dividing fractions with negative numbers, we need to follow the same steps as dividing fractions with positive numbers. However, we need to remember that a negative number divided by a positive number is negative, and a positive number divided by a negative number is positive.

Q: Can we divide a fraction by zero?

A: No, we cannot divide a fraction by zero. Division by zero is undefined in mathematics.

Q: How do we handle fractions with different denominators when dividing?

A: When dividing fractions with different denominators, we need to find the least common multiple (LCM) of the denominators and then multiply both fractions by the LCM.

Q: Can we simplify a fraction after dividing it by another fraction?

A: Yes, we can simplify a fraction after dividing it by another fraction by reducing the resulting fraction to its lowest terms.

Step-by-Step Solutions

Here are some step-by-step solutions to common problems involving division with fractions:

Problem 1: Divide $\frac{2}{3}$ by $\frac{4}{5}$

1. Invert the second fraction: $\frac{4}{5} \rightarrow \frac{5}{4}$
2. Multiply the first fraction with the inverted second fraction: $\frac{2}{3} \times \frac{5}{4} = \frac{2 \times 5}{3 \times 4}$
3. Simplify the expression: $\frac{2 \times 5}{3 \times 4} = \frac{10}{12}$
4. Reduce the fraction to its lowest terms: $\frac{10}{12} = \frac{10 \div 2}{12 \div 2} = \frac{5}{6}$

Problem 2: Divide $\frac{3}{4}$ by $2$

1. Invert the second fraction: $2 \rightarrow \frac{1}{2}$
2. Multiply the first fraction with the inverted second fraction: $\frac{3}{4} \times \frac{1}{2} = \frac{3 \times 1}{4 \times 2}$
3. Simplify the expression: $\frac{3 \times 1}{4 \times 2} = \frac{3}{8}$
4. Reduce the fraction to its lowest terms: $\frac{3}{8}$ is already in its lowest terms.

Problem 3: Divide $\frac{5}{6}$ by $\frac{3}{4}$

1. Invert the second fraction: $\frac{3}{4} \rightarrow \frac{4}{3}$
2. Multiply the first fraction with the inverted second fraction: $\frac{5}{6} \times \frac{4}{3} = \frac{5 \times 4}{6 \times 3}$
3. Simplify the expression: $\frac{5 \times 4}{6 \times 3} = \frac{20}{18}$
4. Reduce the fraction to its lowest terms: $\frac{20}{18} = \frac{20 \div 2}{18 \div 2} = \frac{10}{9}$

Conclusion

In conclusion, division with fractions involves inverting the second fraction and multiplying it with the first fraction. By following the steps outlined above, we can simplify the expression and reduce it to its lowest terms. We can also handle negative numbers, fractions with different denominators, and simplify fractions after dividing them by another fraction.

Additional Resources

For more information on division with fractions, you can refer to the following resources:

• Khan Academy: Division with Fractions
• Mathway: Division with Fractions
• Wolfram Alpha: Division with Fractions

Final Answer

The final answer to the expression $-\frac{2}{5} \div \frac{4}{25}$ is $-\frac{5}{2}$.