Add The Following Fractions And Simplify To The Lowest Terms. Enter The Numerator Of The Simplified Fraction.\[$\frac{1}{6} + \frac{6}{11}\$\]

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Introduction

Fractions are a fundamental concept in mathematics, and simplifying them is an essential skill to master. In this article, we will explore the process of adding fractions and simplifying them to their lowest terms. We will use the given problem, 16+611\frac{1}{6} + \frac{6}{11}, as a case study to demonstrate the steps involved.

Understanding the Problem

The problem requires us to add two fractions, 16\frac{1}{6} and 611\frac{6}{11}, and simplify the result to its lowest terms. To do this, we need to find a common denominator for the two fractions.

Finding a Common Denominator

A common denominator is the least common multiple (LCM) of the denominators of the two fractions. In this case, the denominators are 6 and 11. To find the LCM, we can list the multiples of each denominator:

  • Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, ...
  • Multiples of 11: 11, 22, 33, 44, 55, 66, ...

As we can see, the first common multiple is 66. Therefore, the common denominator is 66.

Adding the Fractions

Now that we have a common denominator, we can add the fractions:

16+611=1×116×11+6×611×6=1166+3666\frac{1}{6} + \frac{6}{11} = \frac{1 \times 11}{6 \times 11} + \frac{6 \times 6}{11 \times 6} = \frac{11}{66} + \frac{36}{66}

We can now add the numerators:

1166+3666=11+3666=4766\frac{11}{66} + \frac{36}{66} = \frac{11 + 36}{66} = \frac{47}{66}

Simplifying the Result

The result, 4766\frac{47}{66}, is already in its simplest form. However, we can simplify it further by dividing both the numerator and the denominator by their greatest common divisor (GCD).

Finding the GCD

To find the GCD of 47 and 66, we can use the Euclidean algorithm:

  • 66 = 1 × 47 + 19
  • 47 = 2 × 19 + 9
  • 19 = 2 × 9 + 1
  • 9 = 9 × 1 + 0

The last non-zero remainder is 1, so the GCD of 47 and 66 is 1.

Simplifying the Fraction

Since the GCD of 47 and 66 is 1, we cannot simplify the fraction further. The result, 4766\frac{47}{66}, is already in its simplest form.

Conclusion

In this article, we have demonstrated the process of adding fractions and simplifying them to their lowest terms. We used the given problem, 16+611\frac{1}{6} + \frac{6}{11}, as a case study to illustrate the steps involved. By finding a common denominator, adding the fractions, and simplifying the result, we arrived at the final answer, 4766\frac{47}{66}. This result is already in its simplest form, and we cannot simplify it further.

Tips and Tricks

  • When adding fractions, make sure to find a common denominator.
  • Use the Euclidean algorithm to find the GCD of two numbers.
  • Simplify the result by dividing both the numerator and the denominator by their GCD.

Practice Problems

  • Add the fractions 38+512\frac{3}{8} + \frac{5}{12} and simplify the result to its lowest terms.
  • Add the fractions 29+715\frac{2}{9} + \frac{7}{15} and simplify the result to its lowest terms.

Real-World Applications

  • Fractions are used in cooking to measure ingredients.
  • Fractions are used in music to measure time and rhythm.
  • Fractions are used in science to measure quantities and ratios.

Common Mistakes

  • Failing to find a common denominator when adding fractions.
  • Not simplifying the result to its lowest terms.
  • Using the wrong GCD when simplifying a fraction.

Conclusion

Q: What is the first step in simplifying a fraction?

A: The first step in simplifying a fraction is to find a common denominator. This is the least common multiple (LCM) of the denominators of the two fractions.

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

A: To find the LCM of two numbers, you can list the multiples of each number and find the first common multiple. Alternatively, you can use the Euclidean algorithm to find the greatest common divisor (GCD) of the two numbers, and then divide the product of the two numbers by the GCD.

Q: What is the Euclidean algorithm?

A: The Euclidean algorithm is a method for finding the greatest common divisor (GCD) of two numbers. It works by repeatedly dividing the larger number by the smaller number and taking the remainder.

Q: How do I add fractions with different denominators?

A: To add fractions with different denominators, you need to find a common denominator. Once you have a common denominator, you can add the fractions by adding the numerators and keeping the common denominator.

Q: What is the difference between a numerator and a denominator?

A: The numerator is the top number of a fraction, and the denominator is the bottom number. The numerator represents the number of equal parts, and the denominator represents the total number of parts.

Q: Can I simplify a fraction by dividing both the numerator and the denominator by a number?

A: Yes, you can simplify a fraction by dividing both the numerator and the denominator by a number. However, you need to make sure that the number you are dividing by is a factor of both the numerator and the denominator.

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

A: The greatest common divisor (GCD) is the largest number that divides both the numerator and the denominator of a fraction.

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

A: To find the GCD of two numbers, you can use the Euclidean algorithm or list the factors of each number and find the largest common factor.

Q: Can I simplify a fraction by multiplying both the numerator and the denominator by a number?

A: No, you cannot simplify a fraction by multiplying both the numerator and the denominator by a number. This will only make the fraction more complicated.

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

A: The least common multiple (LCM) is the smallest number that is a multiple of both the numerator and the denominator of a fraction.

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

A: To find the LCM of two numbers, you can list the multiples of each number and find the smallest common multiple. Alternatively, you can use the Euclidean algorithm to find the GCD of the two numbers, and then divide the product of the two numbers by the GCD.

Q: Can I simplify a fraction by canceling out common factors?

A: Yes, you can simplify a fraction by canceling out common factors. This is a quick and easy way to simplify a fraction.

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

A: Simplifying a fraction means expressing it in its simplest form, while reducing a fraction means expressing it in a simpler form by canceling out common factors.

Q: Can I simplify a fraction with a variable in the numerator or denominator?

A: Yes, you can simplify a fraction with a variable in the numerator or denominator. However, you need to make sure that the variable is a factor of both the numerator and the denominator.

Q: How do I simplify a fraction with a variable in the numerator or denominator?

A: To simplify a fraction with a variable in the numerator or denominator, you need to factor out the variable and simplify the resulting fraction.

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

A: Yes, you can simplify a fraction with a negative number in the numerator or denominator. However, you need to make sure that the negative number is a factor of both the numerator and the denominator.

Q: How do I simplify a fraction with a negative number in the numerator or denominator?

A: To simplify a fraction with a negative number in the numerator or denominator, you need to factor out the negative number and simplify the resulting fraction.

Conclusion

In conclusion, simplifying fractions is an essential skill in mathematics. By following the steps outlined in this article, you can master this skill and apply it to real-world problems. Remember to find a common denominator, add the fractions, and simplify the result to its lowest terms. With practice and patience, you will become proficient in simplifying fractions and solving problems involving fractions.