Problem 215 - Intermediate Algebra

Question

In the following exercises, simplify. $$ \frac{11}{12 a} \cdot \frac{9 a}{16} $$

Step-by-Step Solution

Answer

\(\frac{33}{64}\)

Step 1: Multiply the numerators

First, multiply the numerators of the two fractions together: \( 11 \times 9a = 99a \)

Step 2: Multiply the denominators

Next, multiply the denominators of the two fractions together: \( 12a \times 16 = 192a \)

Step 3: Form the new fraction

Now, create a new fraction using the results of the previous two steps: \( \frac{99a}{192a} \)

Step 4: Simplify the fraction

Cancel the common factor, which in this case is \(a\), in the numerator and the denominator: \( \frac{99a}{192a} = \frac{99}{192} \)

Step 5: Simplify further if possible

Finally, simplify \( \frac{99}{192} \). The greatest common divisor (GCD) of 99 and 192 is 3, so divide both the numerator and denominator by 3: \( \frac{99 \/ 3}{192 \/ 3} = \frac{33}{64} \)

Key Concepts

Multiplying Fractions

One of the foundational concepts in fraction operations is multiplying fractions. When you multiply two fractions, you simply multiply the numerators together to get the new numerator and multiply the denominators together to get the new denominator.

In the exercise, we had two fractions: \[ \frac{11}{12a} \] and \[\frac{9a}{16}\].

To multiply these fractions:

• Multiply the numerators: \[11 \times 9a = 99a\]
• Multiply the denominators: \[12a \times 16 = 192a\]

After multiplying, we form a new fraction: \[\frac{99a}{192a}\].

This new fraction can be simplified further by canceling out common factors.

Greatest Common Divisor

Simplification is a key step in handling fractions, and it often involves finding the Greatest Common Divisor (GCD). The GCD of two numbers is the largest number that divides both without leaving a remainder.

In our fraction, \[\frac{99a}{192a}\], the variable '\(a\)' is present in both the numerator and the denominator. Cancel these common factors to simplify:

• Firstly, remove the variable \(a\): \[\frac{99a}{192a} = \frac{99}{192}\]


• Next, find the GCD of 99 and 192, which is 3


• Lastly, divide both the numerator and the denominator by 3
\[\frac{99 \/ 3}{192 \/ 3} = \frac{33}{64}\]

Always check whether the fraction can be simplified further by looking for any more common factors.

Algebraic Expressions

Algebraic expressions often include variables (like \(a\)), in addition to constants. When working with such expressions in fractions, it's crucial to remember that the rules of arithmetic operations apply equally to the variables and the constants.

In our exercise, we dealt with algebraic fractions. Here's a brief rundown:

• First, we multiplied constants and variables: \[ 11 \times 9a = 99a\] and \[12a \times 16 = 192a\].
• After multiplication, we had \[\frac{99a}{192a}\]
.

To simplify algebraic fractions:

• Cancel out common variables from numerator and denominator
• Simplify constants using the GCD.

In our example, the variable '\(a\)' canceled out, and we were left to simplify \[\frac{99}{192}\] by finding their GCD. This reduced the fraction to \[\frac{33}{64}\].

Understanding these principles will enhance your ability to manipulate algebraic fractions efficiently.