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Chapter 7: Problem 8

For exercises 7-32, simplify. $$ \left(\frac{8}{5 w+10}\right)\left(\frac{5}{24}\right) $$

Short Answer

Expert verified
The simplified form is \(\frac{1}{3(w + 2)}\).

Step by step solution

01

- Factor the denominator

First, factor the denominator in the expression \(\frac{8}{5w+10}\). Notice that \(5w + 10\) can be factored as \(5(w + 2)\). Thus the expression becomes \(\frac{8}{5(w + 2)}\).
02

- Multiply the fractions

Multiply the numerators and the denominators of the fractions \(\frac{8}{5(w+2)}\) and \(\frac{5}{24}\). Therefore, we get: \(\frac{8 \times 5}{5(w+2) \times 24}\).
03

- Simplify the expression

Simplify the fraction by canceling the common factors from the numerator and the denominator. The '5' cancels out in both the numerator and denominator, so we are left with: \(\frac{8}{24(w+2)}\). Then simplify the fraction \(\frac{8}{24}\) to get \(\frac{1}{3}\).
04

- Final expression

Combine the results to get the final simplified expression: \(\frac{1}{3(w + 2)}\).

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Key Concepts

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

Factoring Denominators
Factoring denominators is a crucial step in simplifying algebraic fractions. In our exercise, we had the denominator of the first fraction as \( 5w + 10 \). To simplify this, we can look for common factors in the terms. Here, both 5 and 10 share a common factor of 5.
By factoring out the 5, we get: \[ 5w + 10 = 5(w + 2) \]
This step is essential because it helps to reveal any common factors with the numerator, which will make the simplification process more streamlined later on.
Multiplying Fractions
Once the fractions are prepared, we multiply them. Multiplying fractions involve multiplying the numerators together and the denominators together. For our example:
  • Numerators: \ 8 \ and \ 5 \ multiply to give \ 8 \times 5 = 40 \
  • Denominators: \ 5(w + 2) \ and \ 24\ multiply to give \ 5(w + 2) \times 24 \
So the fraction becomes: \[ \frac{40}{5(w + 2) \times 24} \] After simplifying common factors, this fraction becomes much easier to handle.
Simplifying Expressions
Simplifying expressions is all about reducing the fraction to its simplest form. In our exercise, after multiplying the fractions, we found: \[ \frac{40}{5(w + 2) \times 24} \] We need to look for common factors in the numerator and the denominator. Here, we can see the '5' in the denominator can be canceled out:
\[ \frac{40}{5 \times 24 \times (w+2)} = \frac{8}{24(w+2)} \]
Next, simplify \ \frac{8}{24}\ to \ \frac{1}{3} \: \[ \frac{8}{24} \rightarrow \frac{1}{3} \] This leaves us with: \[ \frac{1}{3(w + 2)} \]
Canceling Common Factors
Canceling common factors is vital for simplifying fractions. In our exercise, the common factor was '5' between the numerator and the denominator:
\[ \frac{40}{5 \times 24 \times (w+2)} = \frac{8}{24(w+2)} \]
By canceling '5' from both the numerator and denominator, we directly simplify the fraction. This makes the final multiplication and division easier. Remember, to cancel factors, they must be exactly the same in both the numerator and the denominator.
This approach helps you get to the final result quickly and correctly: \[ \frac{1}{3(w + 2)} \]

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Most popular questions from this chapter

For exercises 59-66, use the five steps. Assume that the rate of work does not change if done individually or together. The water from a garden hose turned on at full pressure fills a hot tub in \(45 \mathrm{~min}\). If the drain is open, the hot tub empties in \(62 \mathrm{~min}\). Find the amount of time to fill the hot tub with the drain open. Round to the nearest whole number.

The relationship of \(x\) and \(y\) is an inverse variation. When \(x=4, y=5\). a. Find the constant of proportionality, \(k\). b. Write an equation that represents this inverse variation. c. Find \(y\) when \(x=10\).

When a car travels a fixed distance, the relationship between the speed of the car, \(x\), and the time it travels, \(y\), is an inverse variation. When the speed is \(\frac{48 \mathrm{mi}}{1 \mathrm{hr}}\), the time is \(0.75 \mathrm{hr}\). a. Find the constant of proportionality. Include the units of measurement. b. Write an equation that represents this relationship. c. Find the time in hours to travel this distance at a speed of \(\frac{80 \mathrm{mi}}{1 \mathrm{hr}}\). d. Change the time in part \(\mathrm{c}\) to minutes.

The relationship of \(x\) and \(y\) is an inverse variation. When \(x=3, y=6\). a. Find the constant of proportionality, \(k\). b. Write an equation that represents this inverse variation. c. Find \(y\) when \(x=9\).

For exercises \(35-36, T=\frac{336 \mathrm{gm}}{R}\) represents the relationship of tire diameter, \(T\); gear ratio, \(g\); speed, \(m\); and revolutions of the tire per minute, \(R\). Is the relationship of the given variables a direct variation or an inverse variation? $$ g \text { and } R \text { are constant; the relationship of } T \text { and } m \text {. } $$
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