Simplify The Expression:$\frac{18}{21} \div \frac{19}{20}$

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Introduction

In mathematics, simplifying expressions is an essential skill that helps us solve problems efficiently. When dealing with fractions, we often need to simplify them to make calculations easier. In this article, we will focus on simplifying the expression $\frac{18}{21} \div \frac{19}{20}$ using various techniques.

Understanding the Problem

The given expression involves two fractions: $\frac{18}{21}$ and $\frac{19}{20}$. We are asked to simplify the expression by dividing the two fractions. To do this, we need to understand the concept of division with fractions.

Division with Fractions

When we divide one fraction by another, we can multiply the first fraction by the reciprocal of the second fraction. In other words, $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$.

Simplifying the Expression

To simplify the expression $\frac{18}{21} \div \frac{19}{20}$, we can use the concept of division with fractions. We will multiply the first fraction by the reciprocal of the second fraction.

$\frac{18}{21} \div \frac{19}{20} = \frac{18}{21} \times \frac{20}{19}$

Finding the Least Common Multiple (LCM)

To simplify the expression further, we need to find the least common multiple (LCM) of the denominators, which are 21 and 19. However, since 21 and 19 are relatively prime (i.e., they have no common factors), the LCM is simply their product.

Simplifying the Expression Using LCM

Now that we have found the LCM, we can simplify the expression by multiplying the numerators and denominators by the LCM.

$\frac{18}{21} \times \frac{20}{19} = \frac{18 \times 20}{21 \times 19}$

Canceling Common Factors

To simplify the expression further, we can cancel out any common factors between the numerator and denominator.

$\frac{18 \times 20}{21 \times 19} = \frac{360}{399}$

Reducing the Fraction

To reduce the fraction, we need to find the greatest common divisor (GCD) of the numerator and denominator. The GCD of 360 and 399 is 3.

Simplifying the Fraction

Now that we have found the GCD, we can simplify the fraction by dividing both the numerator and denominator by the GCD.

$\frac{360}{399} = \frac{360 \div 3}{399 \div 3} = \frac{120}{133}$

Conclusion

In this article, we simplified the expression $\frac{18}{21} \div \frac{19}{20}$ using various techniques, including division with fractions, finding the least common multiple (LCM), canceling common factors, and reducing the fraction. We found that the simplified expression is $\frac{120}{133}$.

Final Answer

The final answer is $\boxed{\frac{120}{133}}$.

Related Topics

• Simplifying fractions
• Division with fractions
• Least common multiple (LCM)
• Greatest common divisor (GCD)
• Reducing fractions

Further Reading

• Simplifying Fractions
• Division with Fractions
• Least Common Multiple (LCM)
• Greatest Common Divisor (GCD)
• Reducing Fractions
  

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Introduction

In our previous article, we simplified the expression $\frac{18}{21} \div \frac{19}{20}$ using various techniques, including division with fractions, finding the least common multiple (LCM), canceling common factors, and reducing the fraction. In this article, we will answer some frequently asked questions related to simplifying the expression.

Q&A

Q: What is the concept of division with fractions?

A: Division with fractions involves multiplying the first fraction by the reciprocal of the second fraction. In other words, $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$.

Q: How do I find the least common multiple (LCM) of two numbers?

A: To find the LCM of two numbers, you can list the multiples of each number and find the smallest multiple that appears in both lists. Alternatively, you can use the formula: LCM(a, b) = (a × b) / GCD(a, b), where GCD is the greatest common divisor.

Q: What is the greatest common divisor (GCD)?

A: The GCD of two numbers is the largest number that divides both numbers without leaving a remainder. For example, the GCD of 12 and 18 is 6.

Q: How do I reduce a fraction?

A: To reduce a fraction, you need to find the greatest common divisor (GCD) of the numerator and denominator and divide both numbers by the GCD.

Q: What is the difference between simplifying and reducing a fraction?

A: Simplifying a fraction involves finding an equivalent fraction with a smaller numerator and denominator, while reducing a fraction involves finding the simplest form of a fraction by dividing both the numerator and denominator by their greatest common divisor.

Q: Can I simplify a fraction with a negative numerator or denominator?

A: Yes, you can simplify a fraction with a negative numerator or denominator by following the same steps as simplifying a fraction with positive numbers.

Q: How do I simplify a fraction with a decimal numerator or denominator?

A: To simplify a fraction with a decimal numerator or denominator, you can convert the decimal to a fraction and then simplify the fraction.

Q: Can I use a calculator to simplify a fraction?

A: Yes, you can use a calculator to simplify a fraction by dividing the numerator by the denominator and then simplifying the result.

Conclusion

In this article, we answered some frequently asked questions related to simplifying the expression $\frac{18}{21} \div \frac{19}{20}$. We hope that this article has provided you with a better understanding of the concepts involved in simplifying fractions and has helped you to simplify fractions more efficiently.

Final Answer

The final answer is $\boxed{\frac{120}{133}}$.

Related Topics

• Simplifying fractions
• Division with fractions
• Least common multiple (LCM)
• Greatest common divisor (GCD)
• Reducing fractions

Further Reading

• Simplifying Fractions
• Division with Fractions
• Least Common Multiple (LCM)
• Greatest Common Divisor (GCD)
• Reducing Fractions