Problem 10 - Calculus Early Transcendentals: Pearson New International Edition

Question

, simplify as much as possible. Be sure to remove all parentheses and reduce all fractions. $$ \left(\frac{2}{7}-5\right) /\left(1-\frac{1}{7}\right) $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{-11}{2} \).

Step 1: Simplify the Expression Inside the Numerator

We start with the numerator of the given expression, which is \( \frac{2}{7} - 5 \). To simplify this, convert \( 5 \) into a fraction with a denominator of 7: \( \frac{35}{7} \). Now, the subtraction becomes: \( \frac{2}{7} - \frac{35}{7} \). Subtract the numerators to get \( \frac{2 - 35}{7} = \frac{-33}{7} \).

Step 2: Simplify the Expression Inside the Denominator

Next, simplify the denominator: \( 1 - \frac{1}{7} \). Convert \( 1 \) to \( \frac{7}{7} \) to have a common denominator: \( \frac{7}{7} - \frac{1}{7} \). Now, subtract the numerators: \( \frac{7 - 1}{7} = \frac{6}{7} \).

Step 3: Form the Division of Fractions

Reassemble the expression as a division of two fractions: \( \frac{\frac{-33}{7}}{\frac{6}{7}} \). Recall that dividing by a fraction is equivalent to multiplying by its reciprocal.

Step 4: Multiply by the Reciprocal

Multiply the fractions: \( \frac{-33}{7} \times \frac{7}{6} \). Simplify by canceling out the \( 7 \) in the numerator and the denominator: \( -33 \times \frac{1}{6} \).

Step 5: Simplify the Resulting Expression

Finally, perform the multiplication: \( \frac{-33}{6} \). Simplify the fraction by finding the greatest common divisor of 33 and 6, which is 3. Divide both the numerator and the denominator by 3 to get \( \frac{-11}{2} \).

Key Concepts

Numerator and Denominator

In fractions, the **numerator** is the top number, while the **denominator** is the bottom number. Understanding these terms is important for simplifying fractions effectively.
In the given exercise, we have two initial fractions to work with:

• Numerator: \( \frac{2}{7} - 5 \)
• Denominator: \(1 - \frac{1}{7}\)

Both the numerator and the denominator are key players in this fraction simplification! It helps to visualize these two parts separately for simplification.
The process begins by converting any whole numbers into fractions with a shared denominator. This adjustment allows us to work seamlessly with the expressions and prepare them for subtraction or addition.

Division of Fractions

Dividing fractions can initially seem tricky, but it becomes easier once you know the magic rule: dividing by a fraction is the same as multiplying by its **reciprocal**! To find a reciprocal, simply swap the numerator and denominator.
In our exercise, the division we need to handle is:

• \( \frac{\frac{-33}{7}}{\frac{6}{7}} \)

This expression represents the division of two fractions. To simplify, we multiply the first fraction by the reciprocal of the second fraction.
So the division turns into a multiplication: \( \frac{-33}{7} \times \frac{7}{6} \). This transformation makes it easier to work with and allows us to focus on simplifying further.

Simplification Steps

Simplifying fractions involves several key steps:

• **First**, convert whole numbers involved into fractions. This means turning integers into fractions with the same denominator as the other parts of the expression.
• **Second**, when subtracting or adding fractions, make sure they have the same denominator. Simplify by performing the arithmetic on the numerators alone, keeping the denominator unchanged.
• **Third**, when dividing fractions, remember to multiply by the reciprocal. In this way, the problem transitions from division to multiplication.
• **Finally**, simplify the result by canceling common factors in the numerator and denominator.

Following these steps can help break down complex expressions into simpler, more manageable calculations.

Greatest Common Divisor

Finding the **Greatest Common Divisor (GCD)** is crucial in simplifying fractions to their lowest terms. The GCD of two numbers is the largest number that divides both without leaving any remainder.
In our example, after simplifying the fractions through subtraction and reciprocal multiplication, we arrive at the fraction \( \frac{-33}{6} \).
To simplify further, we look for the GCD of 33 and 6. The common factors of these numbers are:

• 33: 1, 3, 11, 33
• 6: 1, 2, 3, 6

The largest common factor here is 3. Thus, divide both the numerator and denominator by this GCD:
\( \frac{-33}{3} \) and \( \frac{6}{3} \), simplifying to \( \frac{-11}{2} \).
This ensures our fraction is as simple as possible, making it easier to understand and use.