Problem 42 - Prealgebra

Question

Simplify each complex fraction as much as possible. [Examples 4–7] $$\frac{9-\frac{3}{2}}{\frac{7}{4}}$$

Step-by-Step Solution

Answer

The simplified form of the complex fraction is \( \frac{30}{7} \).

Step 1: Simplify the Complex Fraction's Numerator

Firstly, address the expression in the numerator which is \( 9 - \frac{3}{2} \). To simplify this, convert the whole number 9 into a fraction with a denominator of 2: \( \frac{18}{2} \). Then subtract: \( \frac{18}{2} - \frac{3}{2} = \frac{15}{2} \).

Step 2: Divide the Simplified Numerator by the Denominator

Now take the simplified numerator \( \frac{15}{2} \) and divide it by the denominator \( \frac{7}{4} \). To divide by a fraction, multiply by its reciprocal. The division \( \frac{15}{2} \div \frac{7}{4} \) is the same as multiplying \( \frac{15}{2} \) by the reciprocal of \( \frac{7}{4} \), which is \( \frac{4}{7} \).

Step 3: Simplify the Expression by Cross-Multiplying

Multiply the numerators and the denominators: \( \frac{15 \times 4}{2 \times 7} = \frac{60}{14} \).

Step 4: Simplify the Fraction Result

The fraction \( \frac{60}{14} \) can be simplified. Find the greatest common divisor (GCD) of 60 and 14, which is 2. Divide both numerator and denominator by 2: \( \frac{60 \div 2}{14 \div 2} = \frac{30}{7} \).

Step 5: Observe the Result

The simplified form of the complex fraction \( \frac{9-\frac{3}{2}}{\frac{7}{4}} \) is \( \frac{30}{7} \). No further simplification is necessary as 30 and 7 share no common factors other than 1.

Key Concepts

Simplifying Fractions

Simplifying fractions is an essential part of understanding and working with complex fractions. It involves reducing a fraction to its simplest form by ensuring that both the numerator and the denominator have no common divisors except 1.

To simplify a fraction, follow these steps:

• Identify the Greatest Common Divisor (GCD) of the numerator and denominator.
• Divide both the numerator and denominator by their GCD.
• Rewrite the fraction in its simplest form.

This process is crucial because it helps us express fractions in the most concise way, making them easier to interpret and work with in further calculations. For example, starting with the fraction \( \frac{60}{14} \), we simplify by dividing both the numerator and denominator by their GCD, which is 2. This results in \( \frac{30}{7} \). Now, \( \frac{30}{7} \) is simplified, as 30 and 7 do not have any common factors besides 1, meaning no further reduction is possible.

Fraction Division

Fraction division involves a straightforward method: multiplying by the reciprocal. When you divide one fraction by another, you actually multiply by the inverse of the second fraction.

Here's how to divide fractions step-by-step:

• Identify the reciprocal of the divisor (the second fraction).
• Multiply the first fraction by this reciprocal.
• Multiply straight across the numerators and the denominators.
• Simplify the resulting fraction if possible.

For instance, in the exercise, dividing \( \frac{15}{2} \) by \( \frac{7}{4} \) is achieved by multiplying \( \frac{15}{2} \) by the reciprocal of \( \frac{7}{4} \), which is \( \frac{4}{7} \). This operation yields \( \frac{15 \times 4}{2 \times 7} = \frac{60}{14} \), which you can then simplify further.

Greatest Common Divisor (GCD)

The Greatest Common Divisor (GCD) plays a vital role in simplifying fractions. It is the largest number that divides both the numerator and the denominator without leaving a remainder.

To find the GCD:

• List all factors of each number.
• Identify the largest common factor shared by both numbers.

Knowing how to find the GCD helps streamline the simplification process. For example, if we take the fraction \( \frac{60}{14} \), we need to find the GCD of 60 and 14. Both numbers share the factors 1 and 2, with 2 being the greatest. Thus, dividing both 60 and 14 by 2 yields \( \frac{30}{7} \), the simplest form of the fraction. This skill is essential not only in simplifying fractions but also in solving more complex problems.