Calculate The Following Expression:$\[ \frac{1}{2} + \frac{3}{4} + \frac{5}{10} = \\]

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Understanding the Problem

When dealing with fractions, it's essential to understand the concept of adding and simplifying them. In this article, we will focus on calculating the expression 12+34+510\frac{1}{2} + \frac{3}{4} + \frac{5}{10} and provide a step-by-step guide on how to simplify fractions.

What are Fractions?

Fractions are a way to represent a part of a whole. They consist of two parts: the numerator (the top number) and the denominator (the bottom number). The numerator tells us how many equal parts we have, while the denominator tells us how many parts the whole is divided into.

Adding Fractions with Different Denominators

To add fractions with different denominators, we need to find a common denominator. The common denominator is the smallest number that both denominators can divide into evenly.

Step 1: Find the Least Common Multiple (LCM)

To find the LCM of 2, 4, and 10, we need to list the multiples of each number.

  • Multiples of 2: 2, 4, 6, 8, 10, 12, ...
  • Multiples of 4: 4, 8, 12, 16, 20, ...
  • Multiples of 10: 10, 20, 30, 40, ...

The smallest number that appears in all three lists is 20. Therefore, the LCM of 2, 4, and 10 is 20.

Step 2: Convert Each Fraction to Have the Common Denominator

Now that we have the LCM, we can convert each fraction to have a denominator of 20.

  • 12=1×102×10=1020\frac{1}{2} = \frac{1 \times 10}{2 \times 10} = \frac{10}{20}
  • 34=3×54×5=1520\frac{3}{4} = \frac{3 \times 5}{4 \times 5} = \frac{15}{20}
  • 510=5×210×2=1020\frac{5}{10} = \frac{5 \times 2}{10 \times 2} = \frac{10}{20}

Step 3: Add the Fractions

Now that we have all the fractions with the same denominator, we can add them together.

1020+1520+1020=3520\frac{10}{20} + \frac{15}{20} + \frac{10}{20} = \frac{35}{20}

Step 4: Simplify the Fraction

To simplify the fraction, we need to find the greatest common divisor (GCD) of the numerator and the denominator.

The GCD of 35 and 20 is 5. Therefore, we can divide both the numerator and the denominator by 5 to simplify the fraction.

3520=35÷520÷5=74\frac{35}{20} = \frac{35 \div 5}{20 \div 5} = \frac{7}{4}

Conclusion

In this article, we calculated the expression 12+34+510\frac{1}{2} + \frac{3}{4} + \frac{5}{10} and provided a step-by-step guide on how to simplify fractions. We found the LCM of 2, 4, and 10, converted each fraction to have the common denominator, added the fractions, and simplified the resulting fraction.

Real-World Applications

Simplifying fractions is an essential skill in mathematics, and it has many real-world applications. For example, in cooking, you may need to simplify fractions to measure ingredients accurately. In science, you may need to simplify fractions to calculate the concentration of a solution.

Tips and Tricks

Here are some tips and tricks to help you simplify fractions:

  • Always find the LCM of the denominators before adding fractions.
  • Convert each fraction to have the common denominator.
  • Add the fractions together.
  • Simplify the resulting fraction by finding the GCD of the numerator and the denominator.

Common Mistakes

Here are some common mistakes to avoid when simplifying fractions:

  • Not finding the LCM of the denominators before adding fractions.
  • Not converting each fraction to have the common denominator.
  • Not simplifying the resulting fraction by finding the GCD of the numerator and the denominator.

Conclusion

Simplifying fractions is an essential skill in mathematics, and it has many real-world applications. By following the steps outlined in this article, you can simplify fractions accurately and efficiently. Remember to always find the LCM of the denominators, convert each fraction to have the common denominator, add the fractions together, and simplify the resulting fraction by finding the GCD of the numerator and the denominator.

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

A: The least common multiple (LCM) is the smallest number that two or more numbers can divide into evenly. It is used to find a common denominator when adding fractions with different denominators.

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

A: To find the LCM of two or more numbers, you can list the multiples of each number and find the smallest number that appears in all lists. Alternatively, you can use the following formula:

LCM(a, b) = (a × b) / GCD(a, b)

where GCD(a, b) is the greatest common divisor of a and b.

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

A: The greatest common divisor (GCD) is the largest number that two or more numbers can divide into evenly. It is used to simplify fractions by dividing both the numerator and the denominator by the GCD.

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

A: To find the GCD of two or more numbers, you can list the factors of each number and find the largest number that appears in all lists. Alternatively, you can use the following formula:

GCD(a, b) = (a × b) / LCM(a, b)

Q: Why is it important to simplify fractions?

A: Simplifying fractions is important because it makes it easier to compare and work with fractions. When fractions are simplified, the numerator and denominator are reduced to their simplest form, making it easier to perform calculations and make comparisons.

Q: Can I simplify fractions with decimals?

A: Yes, you can simplify fractions with decimals. To simplify a fraction with a decimal, you can convert the decimal to a fraction and then simplify the fraction.

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

A: To convert a decimal to a fraction, you can use the following steps:

  1. Write the decimal as a fraction by placing the decimal part over the place value of the last digit.
  2. Simplify the fraction by finding the GCD of the numerator and the denominator.

Q: Can I simplify fractions with negative numbers?

A: Yes, you can simplify fractions with negative numbers. To simplify a fraction with a negative number, you can follow the same steps as simplifying a fraction with positive numbers.

Q: How do I simplify a fraction with a negative numerator and a negative denominator?

A: To simplify a fraction with a negative numerator and a negative denominator, you can follow the same steps as simplifying a fraction with positive numbers. The negative signs will cancel each other out when you simplify the fraction.

Q: Can I simplify fractions with mixed numbers?

A: Yes, you can simplify fractions with mixed numbers. To simplify a fraction with a mixed number, you can convert the mixed number to an improper fraction and then simplify the fraction.

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

A: To convert a mixed number to an improper fraction, you can follow the following steps:

  1. Multiply the whole number by the denominator.
  2. Add the product to the numerator.
  3. Write the result as a fraction with the denominator.

Q: Can I simplify fractions with complex numbers?

A: Yes, you can simplify fractions with complex numbers. To simplify a fraction with a complex number, you can follow the same steps as simplifying a fraction with real numbers.

Q: How do I simplify a fraction with a complex numerator and a complex denominator?

A: To simplify a fraction with a complex numerator and a complex denominator, you can follow the same steps as simplifying a fraction with real numbers. The complex numbers will cancel each other out when you simplify the fraction.

Conclusion

Simplifying fractions is an essential skill in mathematics, and it has many real-world applications. By following the steps outlined in this article, you can simplify fractions accurately and efficiently. Remember to always find the LCM of the denominators, convert each fraction to have the common denominator, add the fractions together, and simplify the resulting fraction by finding the GCD of the numerator and the denominator.