Evaluate The Expression: 12 25 ⋅ 15 16 = □ \frac{12}{25} \cdot \frac{15}{16} = \square 25 12 ​ ⋅ 16 15 ​ = □

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Introduction

In mathematics, evaluating expressions is a fundamental concept that involves simplifying complex mathematical statements to obtain a final value. When dealing with fractions, multiplication is a common operation that requires careful consideration of the numerator and denominator. In this article, we will evaluate the expression $\frac{12}{25} \cdot \frac{15}{16}$ and provide a step-by-step solution to obtain the final result.

Understanding the Expression

The given expression involves the multiplication of two fractions: $\frac{12}{25}$ and $\frac{15}{16}$. To evaluate this expression, we need to multiply the numerators and denominators separately. The numerator of the first fraction is 12, and the denominator is 25. The numerator of the second fraction is 15, and the denominator is 16.

Multiplying the Numerators and Denominators

To multiply the fractions, we multiply the numerators (12 and 15) and denominators (25 and 16) separately. The product of the numerators is 12 × 15 = 180, and the product of the denominators is 25 × 16 = 400.

Simplifying the Expression

Now that we have the product of the numerators and denominators, we can simplify the expression by dividing the product of the numerators by the product of the denominators. This can be written as:

$\frac{12}{25} \cdot \frac{15}{16} = \frac{12 \times 15}{25 \times 16} = \frac{180}{400}$

Reducing the Fraction

The fraction $\frac{180}{400}$ can be reduced by dividing both the numerator and denominator by their greatest common divisor (GCD). To find the GCD of 180 and 400, we can use the Euclidean algorithm or list the factors of each number. The factors of 180 are 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, and 180. The factors of 400 are 1, 2, 4, 5, 8, 10, 16, 20, 25, 40, 50, 80, 100, 200, and 400. The greatest common divisor of 180 and 400 is 20.

Final Result

Now that we have the GCD, we can reduce the fraction by dividing both the numerator and denominator by 20. This gives us:

$\frac{180}{400} = \frac{180 ÷ 20}{400 ÷ 20} = \frac{9}{20}$

Therefore, the final result of the expression $\frac{12}{25} \cdot \frac{15}{16}$ is $\frac{9}{20}$.

Conclusion

Evaluating expressions involving fractions requires careful consideration of the numerator and denominator. By multiplying the numerators and denominators separately and simplifying the resulting fraction, we can obtain the final result. In this article, we evaluated the expression $\frac{12}{25} \cdot \frac{15}{16}$ and obtained the final result of $\frac{9}{20}$. This demonstrates the importance of understanding the rules of fraction multiplication and simplification in mathematics.

Frequently Asked Questions

• Q: What is the product of the numerators and denominators in the expression $\frac{12}{25} \cdot \frac{15}{16}$? A: The product of the numerators is 12 × 15 = 180, and the product of the denominators is 25 × 16 = 400.
• Q: How do we simplify the expression $\frac{180}{400}$? A: We can simplify the expression by dividing both the numerator and denominator by their greatest common divisor (GCD), which is 20.
• Q: What is the final result of the expression $\frac{12}{25} \cdot \frac{15}{16}$? A: The final result of the expression is $\frac{9}{20}$.

Step-by-Step Solution

1. Multiply the numerators (12 and 15) and denominators (25 and 16) separately.
2. Simplify the expression by dividing the product of the numerators by the product of the denominators.
3. Reduce the fraction by dividing both the numerator and denominator by their greatest common divisor (GCD).
4. Obtain the final result by simplifying the fraction.

Related Topics

• Multiplication of fractions
• Simplification of fractions
• Greatest common divisor (GCD)
• Euclidean algorithm

References

• [1] "Multiplication of Fractions" by Math Open Reference
• [2] "Simplification of Fractions" by Khan Academy
• [3] "Greatest Common Divisor (GCD)" by Wolfram MathWorld
• [4] "Euclidean Algorithm" by Brilliant.org
  

Introduction

Evaluating expressions involving fractions can be a challenging task, especially for those who are new to mathematics. In this article, we will address some of the most frequently asked questions related to evaluating expressions with fractions. Whether you are a student, teacher, or simply someone who wants to improve their math skills, this article will provide you with the answers you need to succeed.

Q: What is the difference between multiplying and dividing fractions?

A: When multiplying fractions, we multiply the numerators and denominators separately. When dividing fractions, we invert the second fraction (i.e., flip the numerator and denominator) and then multiply.

Q: How do I simplify a fraction?

A: To simplify a fraction, we need to find the greatest common divisor (GCD) of the numerator and denominator. We then divide both the numerator and denominator by the GCD to obtain the simplified fraction.

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

A: The greatest common divisor (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, because 6 is the largest number that divides both 12 and 18 without leaving a remainder.

Q: How do I find the GCD of two numbers?

A: There are several ways to find the GCD of two numbers. One way is to use the Euclidean algorithm, which involves repeatedly dividing the larger number by the smaller number and taking the remainder. Another way is to list the factors of each number and find the largest common factor.

Q: What is the Euclidean algorithm?

A: The Euclidean algorithm is a method for finding the greatest common divisor (GCD) of two numbers. It involves repeatedly dividing the larger number by the smaller number and taking the remainder. The process is repeated until the remainder is zero, at which point the GCD is the last non-zero remainder.

Q: How do I multiply fractions with different denominators?

A: To multiply fractions with different denominators, we need to find the least common multiple (LCM) of the denominators. We then multiply the numerators and denominators separately, using the LCM as the new denominator.

Q: What is the least common multiple (LCM)?

A: The least common multiple (LCM) of two numbers is the smallest number that is a multiple of both numbers. For example, the LCM of 4 and 6 is 12, because 12 is the smallest number that is a multiple of both 4 and 6.

Q: How do I find the LCM of two numbers?

A: There are several ways to find the LCM of two numbers. One way is to list the multiples of each number and find the smallest common multiple. Another way is to use the prime factorization method, which involves finding the prime factors of each number and multiplying them together.

Q: What is the prime factorization method?

A: The prime factorization method is a way of finding the prime factors of a number. It involves breaking down the number into its prime factors, which are the smallest numbers that multiply together to give the original number.

Q: How do I add and subtract fractions with different denominators?

A: To add and subtract fractions with different denominators, we need to find the least common multiple (LCM) of the denominators. We then convert each fraction to have the LCM as the denominator, and then add or subtract the numerators.

Q: What is the difference between adding and subtracting fractions?

A: When adding fractions, we add the numerators and keep the same denominator. When subtracting fractions, we subtract the numerators and keep the same denominator.

Q: How do I convert a fraction to a decimal?

A: To convert a fraction to a decimal, we divide the numerator by the denominator.

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 number as a sum of powers of 10.

Q: How do I convert a decimal to a fraction?

A: To convert a decimal to a fraction, we can use the method of converting a decimal to a fraction by writing it as a sum of powers of 10.

Q: What is the method of converting a decimal to a fraction?

A: The method of converting a decimal to a fraction involves writing the decimal as a sum of powers of 10, and then simplifying the resulting fraction.

Q: How do I simplify a fraction?

A: To simplify a fraction, we need to find the greatest common divisor (GCD) of the numerator and denominator. We then divide both the numerator and denominator by the GCD to obtain the simplified fraction.

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

A: Simplifying a fraction involves finding the greatest common divisor (GCD) of the numerator and denominator and dividing both by the GCD. Reducing a fraction involves finding the least common multiple (LCM) of the numerator and denominator and dividing both by the LCM.

Q: How do I reduce a fraction?

A: To reduce a fraction, we need to find the least common multiple (LCM) of the numerator and denominator. We then divide both the numerator and denominator by the LCM to obtain the reduced fraction.

Q: What is the difference between a simplified fraction and a reduced fraction?

A: A simplified fraction is a fraction that has been simplified by dividing both the numerator and denominator by their greatest common divisor (GCD). A reduced fraction is a fraction that has been reduced by dividing both the numerator and denominator by their least common multiple (LCM).

Q: How do I add and subtract fractions with the same denominator?

A: To add and subtract fractions with the same denominator, we simply add or subtract the numerators.

Q: What is the difference between adding and subtracting fractions with the same denominator?

A: When adding fractions with the same denominator, we add the numerators. When subtracting fractions with the same denominator, we subtract the numerators.

Q: How do I multiply fractions with the same denominator?

A: To multiply fractions with the same denominator, we simply multiply the numerators.

Q: What is the difference between multiplying and dividing fractions with the same denominator?

A: When multiplying fractions with the same denominator, we multiply the numerators. When dividing fractions with the same denominator, we invert the second fraction and then multiply.

Q: How do I convert a fraction to a percentage?

A: To convert a fraction to a percentage, we divide the numerator by the denominator and then multiply by 100.

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

A: A fraction is a way of expressing a part of a whole, while a percentage is a way of expressing a number as a proportion of 100.

Q: How do I convert a percentage to a fraction?

A: To convert a percentage to a fraction, we divide the percentage by 100 and then simplify the resulting fraction.

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

A: Converting a fraction to a decimal involves dividing the numerator by the denominator, while converting a fraction to a percentage involves dividing the numerator by the denominator and then multiplying by 100.

Q: How do I add and subtract mixed numbers?

A: To add and subtract mixed numbers, we need to convert the mixed numbers to improper fractions and then add or subtract the fractions.

Q: What is the difference between adding and subtracting mixed numbers?

A: When adding mixed numbers, we add the whole numbers and then add the fractions. When subtracting mixed numbers, we subtract the whole numbers and then subtract the fractions.

Q: How do I multiply and divide mixed numbers?

A: To multiply and divide mixed numbers, we need to convert the mixed numbers to improper fractions and then multiply or divide the fractions.

Q: What is the difference between multiplying and dividing mixed numbers?

A: When multiplying mixed numbers, we multiply the whole numbers and then multiply the fractions. When dividing mixed numbers, we invert the second fraction and then multiply.

Q: How do I convert a mixed number to an improper fraction?

A: To convert a mixed number to an improper fraction, we multiply the whole number by the denominator and then add the numerator.

Q: What is the difference between a mixed number and an improper fraction?

A: A mixed number is a way of expressing a number as a whole number and a fraction, while an improper fraction is a way of expressing a number as a single fraction.

Q: How do I simplify an improper fraction?

A: To simplify an improper fraction, we need to find the greatest common divisor (GCD) of the numerator and denominator and divide both by the GCD.

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

A: Simplifying an improper fraction involves finding the greatest common divisor (GCD) of the numerator and denominator and dividing both by the GCD. Reducing an improper fraction involves finding the least common multiple (LCM) of the numerator and denominator and dividing both by the LCM.

Q: How do I reduce an improper fraction?

A: To reduce an improper fraction, we need to find the least common multiple (LCM) of the numerator and denominator and divide both by the LCM.

Q: What is the difference between a simplified improper fraction and a reduced improper fraction?

A: A simplified improper fraction is a fraction that has been simplified by dividing both the numerator and denominator by their greatest common divisor (GCD). A reduced improper fraction is a fraction that has been reduced by dividing both the numerator and denominator by their