Problem 193 - Intermediate Algebra

Question

In the following exercises, simplify. $$ \frac{5 \cdot 6-3 \cdot 4}{4 \cdot 5-2 \cdot 3} $$

Step-by-Step Solution

Answer

\( \frac{9}{7} \)

Step 1 - Simplify the Numerator

Simplify the expression in the numerator: \(5 \times 6 - 3 \times 4\). Calculate each multiplication: \(5 \times 6 = 30\) \(3 \times 4 = 12\) Then subtract: \(30 - 12 = 18\)

Step 2 - Simplify the Denominator

Simplify the expression in the denominator: \(4 \times 5 - 2 \times 3\). Calculate each multiplication: \(4 \times 5 = 20\) \(2 \times 3 = 6\) Then subtract: \(20 - 6 = 14\)

Step 3 - Formulate the Simplified Fraction

Combine the simplified numerator and denominator to form the fraction: \( \frac{18}{14} \)

Step 4 - Reduce the Fraction

Reduce the fraction \( \frac{18}{14} \) by dividing both the numerator and denominator by their greatest common divisor (GCD). The GCD of 18 and 14 is 2. Thus, \( \frac{18}{14} = \frac{18 \div 2}{14 \div 2} = \frac{9}{7} \)

Key Concepts

Numerator and Denominator

To grasp the concept of simplifying fractions, it’s essential to understand the parts of a fraction: the numerator and the denominator. The numerator is the top part of the fraction, representing how many parts we have. The denominator is the bottom part, indicating the total number of equal parts. For the fraction involved in our exercise, the numerator is derived by simplifying the expression **5 * 6 - 3 * 4** and the denominator is from **4 * 5 - 2 * 3**. Breaking down these components will help us proceed to more complex mathematical operations.

Multiplication and Subtraction

Multiplication and subtraction are fundamental operations needed for simplifying our fraction. Let’s start with the numerator. Firstly, perform the multiplications in the expression: 5 * 6 = 30 and 3 * 4 = 12. Next, subtract these two products: 30 - 12 = 18. Repeat these steps for the denominator: 4 * 5 = 20 and 2 * 3 = 6. Then subtract: 20 - 6 = 14. Now, our fraction has been simplified to \(\frac{18}{14}\) and is ready for reduction.

Greatest Common Divisor (GCD)

Reducing a fraction requires understanding the Greatest Common Divisor (GCD). The GCD is the largest number that divides both the numerator and denominator without a remainder. For instance, in our fraction \(\frac{18}{14}\), the GCD of 18 and 14 is 2. On breaking down the numbers: 18 = 2 * 9 and 14 = 2 * 7. The highest common factor is 2. By dividing both the numerator and the denominator by this GCD, we simplify the fraction further.

Fraction Reduction

Finally, reducing the fraction to its simplest form is essential. We've identified our GCD as 2. Now, divide both the numerator and the denominator of \(\frac{18}{14}\) by 2: \(\frac{18 \div 2}{14 \div 2} = \frac{9}{7}\). The fraction \(\frac{9}{7}\) is now in its simplest form since there are no common divisors other than 1 between 9 and 7. Hence, simplifying fractions to their lowest terms makes them easier to understand and work with in future calculations.