Question

, simplify as much as possible. Be sure to remove all parentheses and reduce all fractions. $$ \frac{\frac{11}{7}-\frac{12}{21}}{\frac{11}{7}+\frac{12}{21}} $$

Short answer

\(\frac{7}{15}\)

Step-by-step solution

Simplify the fractions within the numerator and denominator

First, simplify the fractions \( \frac{12}{21} \). We can do this by finding the greatest common divisor of 12 and 21, which is 3. Thus, \( \frac{12}{21} = \frac{4}{7} \). So the expression becomes:\[\frac{\frac{11}{7} - \frac{4}{7}}{\frac{11}{7} + \frac{4}{7}}\]

Subtract and add fractions in the numerator and the denominator

Next, perform the operations in the numerator and the denominator:For the numerator: \( \frac{11}{7} - \frac{4}{7} = \frac{11 - 4}{7} = \frac{7}{7} = 1 \).For the denominator: \( \frac{11}{7} + \frac{4}{7} = \frac{11 + 4}{7} = \frac{15}{7} \).

Simplify the division

Now, simplify the expression by dividing the simplified numerator by the denominator:\[\frac{1}{\frac{15}{7}} = 1 \times \frac{7}{15} = \frac{7}{15}\]

Confirm the answer and present in simplest form

Finally, confirm that the fraction is in its simplest form: \( \frac{7}{15} \).No further simplification can be done since 7 and 15 have no common factors other than 1.

Key concepts

Understanding Fractions

Fractions are essential in mathematics for representing parts of a whole. A fraction consists of two numbers: the numerator and the denominator. The numerator is the top number and indicates how many parts of the whole are taken or considered, while the denominator, the bottom number, shows the total number of equal parts in the whole.
For example, in the fraction \( \frac{11}{7} \), 11 is the numerator meaning we have 11 parts, and 7 is the denominator which shows that the whole is divided into 7 equal parts. Always remember, a fraction like \( \frac{4}{7} \) means we are considering 4 parts out of a total 7 parts.
Fractions can often seem tricky, especially when they are part of more complex equations, but they play a critical role in algebraic simplification and understanding of ratios, proportions, and real-world measurements.

Step-by-Step Solutions for Simplifying Fractions

Simplifying fractions involves a few steps that need to be done in sequence to make the process easier. Performing these steps in order ensures that we efficiently simplify the expression. Let's break it down.

• First, look at the fractions involved in the problem. In our case, you need to simplify \( \frac{12}{21} \) because 12 and 21 have a common factor.
• Next, use the greatest common divisor (GCD) to reduce the fraction to its simplest form. For \( \frac{12}{21} \), the GCD is 3, so dividing both the numerator (12) and the denominator (21) by their GCD gives \( \frac{4}{7} \).
• Once fractions inside parentheses are simplified, handle operations like addition or subtraction between fractions by ensuring they have common denominators. Here, our fractions already had a common denominator of 7, which simplifies calculations.

By following these steps, you streamline the process and reduce the chance of errors.

Numerator and Denominator in Fractions

In any fraction, it is crucial to understand the roles of the numerator and denominator. They have distinct functions that impact how fractions are manipulated and simplified.
The numerator tells you how many parts of the whole you are focusing on. For instance, in the exercise where \( \frac{11}{7} - \frac{4}{7} \), the subtraction occurs in the numerator: 11 minus 4 gives 7. This operation changes the focus to just 7 parts out of the original 11 parts considered.
Meanwhile, the denominator provides a uniform base, showing how many parts should make a complete whole. When we add \( \frac{11}{7} + \frac{4}{7} \), we maintain a denominator of 7 to ensure we're talking about parts from the same whole.
This separation of roles between numerator and denominator is the foundation for solving problems that require adding, subtracting, multiplying, or dividing fractions.

Greatest Common Divisor: A Simplification Tool

The greatest common divisor (GCD) is an essential concept in mathematics, especially when simplifying fractions. The GCD is the largest number that can exactly divide both the numerator and the denominator of a fraction without leaving a remainder.
To find the GCD of numbers, list factors of each number. For example, to simplify \( \frac{12}{21} \), find the factors: 12 (1, 2, 3, 4, 6, 12) and 21 (1, 3, 7, 21). The largest common factor is 3. Therefore, dividing both 12 and 21 by 3 reduces the fraction to \( \frac{4}{7} \).
Using the GCD to simplify fractions makes expressions more manageable and facilitates easier arithmetic operations later on. In expressions with fractions, finding and applying the GCD can significantly impact the clarity and simplicity of the solution.