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Chapter 0: Problem 24

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

Short Answer

Expert verified
The simplified result of the fraction division operation is \(\frac{x + 5}{4x + 20}\).

Step by step solution

01

Understanding the Problem

The problem presents the operation of division between two fractions, \(\frac{x+5}{7}\) and \(\frac{4x+20}{9}\), and asks for the result of this operation. The rule to divide fractions is to multiply the first fraction by the inverse of the second one.
02

Finding the Reciprocal

Begin by rewriting the division problem as a multiplication problem by taking the reciprocal of the second fraction. The reciprocal of a fraction is obtained by simply swapping the numerator and the denominator. Therefore, the reciprocal of \(\frac{4x+20}{9}\) is \(\frac{9}{4x+20}\). Hence, the problem now becomes: \(\frac{x+5}{7} \cdot \frac{9}{4x+20}\)
03

Performing the multiplication operation

Multiply the two fractions normally as: multiply the numerators together and the denominators together. This gives \(\frac{(x+5) \cdot 9}{7 \cdot (4x+20)}\).
04

Simplifying the expression

With the distributive law, the operation yields \(\frac{9x + 45}{28x + 140}\). Considering that both the numerator and the denominator can be divided by 9, the expression simplifies to \(\frac{x + 5}{4x + 20}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Reciprocal
To divide one fraction by another, we use a magic trick called the **reciprocal**. It's like flipping the second fraction upside down. In other words, we swap its numerator and denominator. This turns the division process into multiplication, which is much easier to handle.

For example, in the problem \( \frac{x+5}{7} \div \frac{4x+20}{9} \), we take the reciprocal of \( \frac{4x+20}{9} \), changing it to \( \frac{9}{4x+20} \). This changes the whole division process into \( \frac{x+5}{7} \times \frac{9}{4x+20} \). Using reciprocals makes it straightforward to solve such problems and sets the stage for multiplying fractions.
Multiplication of Fractions
Once we're set up to multiply, things flow smoothly. **Multiplication of fractions** is simple; just straight across! You multiply the numerators of the fractions together to form a new numerator. Then, you do the same with the denominators.

In this example, we multiply \( (x+5) \times 9 \) as the new numerator, and \( 7 \times (4x+20) \) as the new denominator. So, the product becomes \( \frac{9(x+5)}{7(4x+20)} \). This direct approach keeps things neat and manageable.
Simplification of Algebraic Expressions
After multiplying fractions, the next step is to **simplify the algebraic expression**. Simplification makes the expression cleaner and easier to read.

First, we distribute the terms: \( 9(x+5) \) becomes \( 9x + 45 \), and \( 7(4x+20) \) becomes \( 28x + 140 \). These expressions can be overwhelming, but the trick is to find common factors.
  • Numerator: \( 9x + 45 \) simplifies to \( 3(3x + 15) \).
  • Denominator: \( 28x + 140 \) simplifies to \( 4(7x + 35) \).
By factoring out common elements and reducing, the entire expression simplifies beautifully to \( \frac{x+5}{4x+20} \). This simplification not only makes the result neater but also prepares it for further mathematical calculations, if needed.

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Most popular questions from this chapter

Will help you prepare for the material covered in the first section of the next chapter. If \(y=4-x,\) find the value of \(y\) that corresponds to values of \(x\) for each integer starting with \(-3\) and ending with 3

Will help you prepare for the material covered in the next section. Multiply and simplify: \(12\left(\frac{x+2}{4}-\frac{x-1}{3}\right)\)

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The equation \((2 x-3)^{2}=25\) is equivalent to \(2 x-3=5\)

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I'll win the contest if I can complete the crossword puzzle in 20 minutes plus or minus 5 minutes, so my winning time, \(x,\) is modeled by \(|x-20| \leq 5\)

Explain how to solve \(x^{2}+6 x+8=0\) using the quadratic formula.
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