Problem 24 - Algebra and Trigonometry

Question

multiply or divide as indicated. $$ \frac{x+5}{7} \div \frac{4 x+20}{9} $$

Step-by-Step Solution

Answer

The result of the operation is \(\frac{9}{28}\)

Step 1: Rewrite the division operation as multiplication

The division of fractions can be converted into multiplication by taking the reciprocal of the divisor. So, we rewrite the expression. The expression will become \[ \frac{x+5}{7} \times \frac{9}{4x+20} \]

Step 2: Simplify the expression

By simplifying the expression we observe that \(4x + 20\) can be simplified as \(4(x + 5)\). Hence, the expression will become: \[ \frac{x+5}{7} \times \frac{9}{4(x+5)} \]

Step 3: Cancel out the common terms

The term \(x + 5\) can be cancelled from the numerator of the first expression and the denominator of the second expression, which will yield: \[ \frac{1}{7} \times \frac{9}{4} \]

Step 4: Perform the multiplication

The multiplication of these two fractions will give us the final answer: \[ \frac{1 \times 9}{7 \times 4} = \frac{9}{28} \]

Key Concepts

Division of Fractions

Dividing fractions might initially seem complex, but it's made simple if you remember the key rule: divide by a fraction by multiplying with its reciprocal. In the exercise, we started with the expression \( \frac{x+5}{7} \div \frac{4x+20}{9} \). To divide these fractions:

• Take the second fraction, which is the divisor \( \frac{4x+20}{9} \), and find its reciprocal. To find the reciprocal, flip the numerator and the denominator. The reciprocal becomes \( \frac{9}{4x+20} \).
• Then, replace the division sign with a multiplication sign.

This turns our original division problem into a multiplication problem: \( \frac{x+5}{7} \times \frac{9}{4x+20} \). Changing division into multiplication in this way simplifies the entire process and sets up for the next steps.

Multiplication of Fractions

After converting division into multiplication by using the reciprocal, the expression becomes: \( \frac{x+5}{7} \times \frac{9}{4x+20} \). Now, multiplying fractions is straightforward:

• Multiply the numerators together.
• Multiply the denominators together.

But before multiplying, it's often useful to simplify. This step saves effort in the further calculations. In this case, recognize that the term \( 4x+20 \) in the denominator can be factored as \( 4(x+5) \). Thus, the expression becomes \( \frac{x+5}{7} \times \frac{9}{4(x+5)} \), allowing us to cancel \( x+5 \) from both the numerator and the denominator. Now, you just multiply: \[ \frac{1}{7} \times \frac{9}{4} = \frac{1 \times 9}{7 \times 4} \].Ultimately, simplification beforehand makes the multiplication step much easier.

Simplifying Expressions

Simplifying expressions means reducing them to their simplest form to make calculations easier. In the context of rational expressions like \( \frac{x+5}{7} \times \frac{9}{4(x+5)} \), it involves identifying and cancelling out common factors between numerators and denominators.
In our case, the term \( x+5 \) appeared in both a numerator and the denominator.

• Recognize that \( 4x+20 \) can be expressed as \( 4(x+5) \).
• This allows the \( x+5 \) term to be cancelled.

This leaves us a much simpler expression: \( \frac{1}{7} \times \frac{9}{4} \). By simplifying early, before multiplication, you avoid potential mistakes and make calculations more manageable. After simplification, the final result becomes clear and easier to derive.