Problem 194 - Intermediate Algebra

Question

In the following exercises, simplify. $$ \frac{8 \cdot 9-7 \cdot 6}{5 \cdot 6-9 \cdot 2} $$

Step-by-Step Solution

Answer

\( \frac{5}{2} \)

Step 1: Simplify the Numerator

First, simplify the numerator of the fraction. The numerator is given by the expression \(8 \times 9 - 7 \times 6\). Calculate each multiplication separately and then subtract. \(8 \times 9 = 72\) \(7 \times 6 = 42\) Now subtract: \(72 - 42 = 30\) So, the simplified numerator is 30.

Step 2: Simplify the Denominator

Next, simplify the denominator of the fraction. The denominator is given by the expression \(5 \times 6 - 9 \times 2\). Calculate each multiplication separately and then subtract. \(5 \times 6 = 30\) \(9 \times 2 = 18\) Now subtract: \(30 - 18 = 12\) So, the simplified denominator is 12.

Step 3: Divide Numerator by Denominator

Finally, divide the simplified numerator by the simplified denominator to get the final simplified form of the fraction. \( \frac{30}{12} \) This fraction can be further simplified by finding the greatest common divisor of 30 and 12, which is 6. \( \frac{30 \div 6}{12 \div 6} = \frac{5}{2} \). So, the final simplified form of the fraction is \( \frac{5}{2} \).

Key Concepts

Numerator Simplification

To start simplifying a fraction, begin by looking at the numerator ─ the top part of the fraction. The goal is to reduce any complex operations to a single number. Here, the numerator is given by the expression: \[8 \times 9 - 7 \times 6\].
Follow these steps:

• First, handle each multiplication separately:
  \[8 \times 9 = 72\]
  \[7 \times 6 = 42\]
  
• Afterward, subtract the results:
  \[72 - 42 = 30\]
  

Therefore, the numerator simplifies to 30.

Denominator Simplification

Next, move to the denominator, which is the bottom part of the fraction. Similar to the numerator, break down the given expression and simplify it step by step. The denominator expression is: \[5 \times 6 - 9 \times 2\].
Here’s how to simplify it:

• Calculate each multiplication individually:
  \[5 \times 6 = 30\]
  \[9 \times 2 = 18\]
  
• Then, subtract the results:
  \[30 - 18 = 12\]
  

Thus, the denominator simplifies to 12.

Greatest Common Divisor

After simplifying both the numerator and the denominator, you can further reduce the fraction. This involves finding the greatest common divisor (GCD) of the two numbers. The GCD is the largest number that divides both the numerator and the denominator without leaving a remainder.
For \[30\text{ and } 12\]:

• List the divisors of both numbers:
  Divisors of 30: 1, 2, 3, 5, 6, 10, 15, 30
  Divisors of 12: 1, 2, 3, 4, 6, 12
  
• Identify the greatest common divisor:
  The largest number common to both sets is 6.
  

Hence, the GCD of 30 and 12 is 6.

Fraction Reduction

With the GCD in hand, you can now reduce the fraction. Fraction reduction is the process of dividing the numerator and denominator by their GCD to obtain the simplest form of the fraction. Here, the original reduced fraction is: \[ \frac{30}{12} \text{ and the GCD is 6}\].
Apply the GCD to reduce the fraction:

• Divide both the numerator and denominator by the GCD:
  \[ \frac{30 \text{ ÷ } 6}{12 \text{ ÷ } 6} = \frac{5}{2}\]
  

Consequently, the fraction \[ \frac{30}{12} \] simplifies to \[ \frac{5}{2} \]. This is the simplest form of the fraction.