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Chapter 7: Problem 36

Multiply or divide as indicated. $$ \frac{3 x+6}{20} \div \frac{4 x+8}{8} $$

Short Answer

Expert verified
The expression simplifies to \( \frac{3}{10} \).

Step by step solution

01

Interpret the Division

The problem asks us to divide two fractional expressions: \( \frac{3x+6}{20} \) by \( \frac{4x+8}{8} \). To divide fractions, we multiply the first fraction by the reciprocal of the second.
02

Find the Reciprocal

The reciprocal of \( \frac{4x+8}{8} \) is \( \frac{8}{4x+8} \). We will use this in the next step to facilitate the division.
03

Multiply by Reciprocal

Multiply \( \frac{3x+6}{20} \) by \( \frac{8}{4x+8} \). This can be set up as: \[ \frac{3x+6}{20} \times \frac{8}{4x+8} = \frac{(3x+6) \times 8}{20 \times (4x+8)} \]
04

Simplify Numerator

The numerator becomes: \( (3x+6) \times 8 = 24x + 48 \).
05

Simplify Denominator

The denominator can be simplified as: \( 20 \times (4x+8) = 80x + 160 \).
06

Combine and Simplify the Fraction

Now, the expression simplifies to: \[ \frac{24x + 48}{80x + 160} \]Factoring out common factors gives: \[ \frac{24(x + 2)}{80(x + 2)} \]This simplifies further by cancelling \( (x+2) \) in both numerator and denominator, resulting in \( \frac{24}{80} \).
07

Simplify the Reduced Fraction

Now simplify \( \frac{24}{80} \) by finding the greatest common divisor which is 8.Thus, dividing both terms, we get: \( \frac{24 \div 8}{80 \div 8} = \frac{3}{10} \).

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Key Concepts

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

Reciprocal
When we talk about dividing fractions, the concept of a *reciprocal* is essential. Simply put, a reciprocal is what you flip a fraction to. For example, the reciprocal of \( \frac{4}{5} \) is \( \frac{5}{4} \). When you multiply a number by its reciprocal, you get a product of 1. This rule is really useful in fraction division, because to divide by a fraction, you multiply by its reciprocal.
  • Start by identifying the fraction you're dividing by.
  • Flip the numerator and the denominator to find the reciprocal.
  • Use this flipped fraction to multiply with the original fraction instead of dividing.
In our problem, we had \( \frac{4x+8}{8} \) as the divisor. So, its reciprocal is \( \frac{8}{4x+8} \). Multiplying by this instead of dividing simplifies the operation immensely.
Common Factors
Understanding *common factors* helps us simplify fractions during calculations. Common factors are numbers that are a factor of two or more different numbers or expressions. When you simplify fractions, finding these shared factors can make your work easier.
  • Look for numbers that divide both the numerator and denominator completely.
  • Factor these out to reduce your fraction.
  • This will often simplify into smaller, more manageable numbers.
In our exercise, after multiplying the initial fractions, we got \( \frac{24x + 48}{80x + 160} \). Both the numerator and the denominator shared the factor \( x + 2 \), which was factored and cancelled out to bring our fraction to \( \frac{24}{80} \). Recognizing this commonality is crucial in streamlining your solution.
Simplifying Fractions
Once you've handled the division and multiplication of fractions, the next task is *simplifying fractions*. The goal is to make the fraction as simple as possible. You do this by dividing both the numerator and the denominator by their greatest common divisor (GCD).
  • Calculate or identify the greatest number that divides both the top and bottom seamlessly.
  • Divide both parts of the fraction by this GCD.
  • Your result should be the fraction in its lowest or simplest form.
In the problem, after we cancelled out the common factor of \( (x+2) \), we were left with \( \frac{24}{80} \). The GCD of 24 and 80 is 8, so dividing both the numerator and the denominator by 8 gave us \( \frac{3}{10} \). This is the finalized and simplified version of the fraction. Simplifying fractions not only cleans up your answer but also makes it easier to understand and use in further calculations.

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Most popular questions from this chapter

Find the slope of the line through each pair of points. Use the slope to determine whether the line is vertical, horizontal, or moves up. ward or downward from left to right. See Section 3.4. $$ (-2,5),(4,-3) $$

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In the study of calculus, the difference quotient \(\frac{f(a+h)-f(a)}{h}\) is often found and simplified. Find and simplify this quotient for each function \(f(x)\) by following steps a through \(d\). a. Find \((a+h)\) b. Find \(f(a)\) c. Use steps a and b to find \(\frac{f(a+h)-f(a)}{h}\) d. Simplify the result of step \(\mathbf{c}\). $$ f(x)=\frac{1}{x} $$

Does \(\frac{x}{x+5}\) simplify to \(\frac{1}{5} ?\) Why or why not?
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