Calculate The Following Expression:$ \frac{8}{75} + \frac{13}{50} - \frac{1}{6} $

Mar 13, 2025 by ADMIN 82 views

$**Simplifying Complex Fractions: A Step-by-Step Guide to Calculating the Expression $ \frac{8}{75} + \frac{13}{50} - \frac{1}{6} $**$

Understanding the Problem

When dealing with complex fractions, it's essential to simplify them before performing any operations. In this case, we're given the expression $ \frac{8}{75} + \frac{13}{50} - \frac{1}{6} $, and our goal is to calculate its value. To do this, we need to find a common denominator for the fractions and then add and subtract them accordingly.

Finding a Common Denominator

To find a common denominator, we need to identify the least common multiple (LCM) of the denominators. In this case, the denominators are 75, 50, and 6. To find the LCM, we can list the multiples of each denominator:

75: 75, 150, 225, 300, 375, 450, 525, 600, ...
50: 50, 100, 150, 200, 250, 300, 350, 400, ...
6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, ...

As we can see, the first number that appears in all three lists is 300. Therefore, the LCM of 75, 50, and 6 is 300.

Simplifying the Fractions

Now that we have a common denominator, we can simplify each fraction by multiplying the numerator and denominator by the necessary factors:

$ \frac{8}{75} = \frac{8 \times 4}{75 \times 4} = \frac{32}{300} $
$ \frac{13}{50} = \frac{13 \times 6}{50 \times 6} = \frac{78}{300} $
$ \frac{1}{6} = \frac{1 \times 50}{6 \times 50} = \frac{50}{300} $

Adding and Subtracting the Fractions

Now that we have simplified the fractions, we can add and subtract them:

$ \frac{32}{300} + \frac{78}{300} - \frac{50}{300} = \frac{60}{300} $

Reducing the Fraction

To reduce the fraction, we can divide both the numerator and denominator by their greatest common divisor (GCD). In this case, the GCD of 60 and 300 is 60. Therefore, we can divide both the numerator and denominator by 60:

$ \frac{60}{300} = \frac{60 \div 60}{300 \div 60} = \frac{1}{5} $

Conclusion

In conclusion, the value of the expression $ \frac{8}{75} + \frac{13}{50} - \frac{1}{6} $ is $ \frac{1}{5} $. This is a simplified fraction that cannot be reduced further.

Real-World Applications

Simplifying complex fractions is an essential skill in many real-world applications, including:

Finance: When working with financial data, it's often necessary to simplify complex fractions to make calculations easier.
Science: In scientific calculations, complex fractions can arise when working with measurements and conversions.
Engineering: Engineers often need to simplify complex fractions when working with mathematical models and simulations.

Tips and Tricks

Here are some tips and tricks for simplifying complex fractions:

Use a common denominator: When adding or subtracting fractions, it's essential to use a common denominator.
Simplify the fractions: Before adding or subtracting fractions, simplify them by dividing the numerator and denominator by their greatest common divisor.
Reduce the fraction: After adding or subtracting fractions, reduce the resulting fraction by dividing both the numerator and denominator by their greatest common divisor.

Common Mistakes

Here are some common mistakes to avoid when simplifying complex fractions:

Not using a common denominator: Failing to use a common denominator can lead to incorrect results.
Not simplifying the fractions: Failing to simplify the fractions before adding or subtracting them can lead to incorrect results.
Not reducing the fraction: Failing to reduce the fraction after adding or subtracting it can lead to incorrect results.

Conclusion

In conclusion, simplifying complex fractions is an essential skill that requires attention to detail and a solid understanding of mathematical concepts. By following the steps outlined in this article, you can simplify complex fractions with ease and accuracy. Remember to use a common denominator, simplify the fractions, and reduce the fraction to get the correct result.

Frequently Asked Questions

Q: What is a complex fraction?

A: A complex fraction is a fraction that contains one or more fractions in its numerator or denominator.

Q: Why is it important to simplify complex fractions?

A: Simplifying complex fractions is essential because it makes calculations easier and more accurate. By simplifying complex fractions, you can avoid errors and get the correct result.

Q: How do I find a common denominator for complex fractions?

A: To find a common denominator, you need to identify the least common multiple (LCM) of the denominators. You can list the multiples of each denominator and find the first number that appears in all lists.

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

A: The least common multiple (LCM) is the smallest number that is a multiple of two or more numbers.

Q: How do I simplify a complex fraction?

A: To simplify a complex fraction, you need to multiply the numerator and denominator by the necessary factors to get a common denominator.

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

A: The greatest common divisor (GCD) is the largest number that divides two or more numbers without leaving a remainder.

Q: How do I reduce a fraction?

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

Q: What are some common mistakes to avoid when simplifying complex fractions?

A: Some common mistakes to avoid when simplifying complex fractions include:

Not using a common denominator
Not simplifying the fractions
Not reducing the fraction

Q: How do I apply simplifying complex fractions in real-world scenarios?

A: Simplifying complex fractions is essential in many real-world applications, including finance, science, and engineering. By simplifying complex fractions, you can make calculations easier and more accurate.

Q: What are some tips and tricks for simplifying complex fractions?

A: Some tips and tricks for simplifying complex fractions include:

Using a common denominator
Simplifying the fractions
Reducing the fraction

Q: Can I use a calculator to simplify complex fractions?

A: Yes, you can use a calculator to simplify complex fractions. However, it's essential to understand the mathematical concepts behind simplifying complex fractions to avoid errors.

Q: How do I check my work when simplifying complex fractions?

A: To check your work, you can plug the simplified fraction into the original equation and verify that it's true.

Q: What are some common applications of simplifying complex fractions?

A: Some common applications of simplifying complex fractions include:

Finance: Simplifying complex fractions is essential when working with financial data.
Science: Simplifying complex fractions is essential when working with measurements and conversions.
Engineering: Simplifying complex fractions is essential when working with mathematical models and simulations.

Q: Can I simplify complex fractions with negative numbers?

A: Yes, you can simplify complex fractions with negative numbers. However, it's essential to follow the rules of arithmetic when working with negative numbers.

Q: How do I simplify complex fractions with decimals?

A: To simplify complex fractions with decimals, you need to convert the decimals to fractions and then simplify the fraction.

Q: Can I simplify complex fractions with variables?

A: Yes, you can simplify complex fractions with variables. However, it's essential to follow the rules of algebra when working with variables.

Q: How do I simplify complex fractions with exponents?

A: To simplify complex fractions with exponents, you need to follow the rules of exponentiation and then simplify the fraction.

Q: What are some common errors to avoid when simplifying complex fractions?

A: Some common errors to avoid when simplifying complex fractions include:

Not using a common denominator
Not simplifying the fractions
Not reducing the fraction
Not following the rules of arithmetic
Not following the rules of algebra

Q: How do I practice simplifying complex fractions?

A: To practice simplifying complex fractions, you can try solving problems and exercises that involve simplifying complex fractions. You can also use online resources and calculators to help you practice.