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Chapter 7: Problem 2

For exercises 1-66, simplify. $$ \frac{240}{540} $$

Short Answer

Expert verified
\( \frac{4}{9} \)

Step by step solution

01

Identify the greatest common divisor (GCD)

To simplify \(\frac{240}{540}\), first find the greatest common divisor (GCD) of 240 and 540. The GCD of 240 and 540 is 60.
02

Divide the numerator and denominator by the GCD

Next, divide both the numerator (240) and the denominator (540) by the GCD (60): \(\frac{240 \div 60}{540 \div 60} = \frac{4}{9}\).
03

Simplified fraction

After dividing, the simplified form of \(\frac{240}{540}\) is \(\frac{4}{9}\).

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Key Concepts

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

greatest common divisor
When simplifying fractions, the greatest common divisor (GCD) plays a crucial role. The GCD is the largest number that can evenly divide both the numerator and the denominator of the fraction.

For instance, to simplify the fraction \(\frac{240}{540}\), we start by finding the GCD of 240 and 540. The GCD of these numbers is 60.

To find the GCD, you can list the factors of each number and select the highest one that appears in both lists, or you can use the Euclidean algorithm, an efficient method for finding the GCD. In either case, for 240 and 540, the GCD is 60.
numerator and denominator
Understanding the numerator and denominator is key to working with fractions. The numerator is the top number in a fraction and it represents how many parts of the whole are being considered. The denominator is the bottom number and it shows into how many parts the whole is divided.

In the fraction \(\frac{240}{540}\), 240 is the numerator and 540 is the denominator. Once you've identified the GCD, you can divide both the numerator and the denominator by it. This process helps in simplifying the fraction.
simplified fraction
A simplified fraction is a fraction where the numerator and the denominator have no common factors other than 1. Simplification makes fractions easier to understand and work with.

Taking our example \(\frac{240}{540}\), once we identify that the GCD is 60, we divide both the numerator and the denominator by 60:

- The numerator: \(\frac{240}{60} = 4\)
- The denominator: \(\frac{540}{60} = 9\)

Thus, \(\frac{240}{540}\) simplifies to \(\frac{4}{9}\). This is the simplified fraction, as 4 and 9 have no common factors other than 1.

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Most popular questions from this chapter

For exercises 1-10, (a) solve. (b) check. $$ \frac{1}{6} w+\frac{23}{8}=-3 $$

For exercises 61-64, the completed problem has one mistake. (a) Describe the mistake in words or copy down the whole problem and highlight or circle the mistake. (b) Do the problem correctly. Problem: The relationship of the number of gallons of gas, \(x\), and the total cost of the gas, \(y\), is a direct variation. If 8 gallons of gas costs \(\$ 24\), find the constant of proportionality. Incorrect Answer: $$ \begin{aligned} &k=x y \\ &k=(8 \mathrm{gal})(\$ 24) \\ &k=\$ 192 \mathrm{gal} \end{aligned} $$

If both sides of the equation \(\frac{1}{x-1}+\frac{2}{x}=\frac{x}{x-1}\) are multiplied by \(x(x-1)\), the simplified equation is \(1 x+2(x-1)=x^{2}\). Rewriting in standard form and factoring, the equation is \((x-2)(x-1)=0\) and its solutions are \(x=1\) or \(x=2\). Explain why the solution \(x=1\) is extraneous.

For exercises \(45-48\), the formula \(R=\frac{U F}{P}\) describes the glomular filtration rate by a kidney \(R\). Is the relationship of the given variables a direct variation or an inverse variation? $$ R \text { and } P \text { are constant; the relationship of } U \text { and } F \text {. } $$

When the radiation is constant, the relationship of the current in an X-ray tube, \(x\), and the exposure time, \(y\), is an inverse variation. When the current is 600 milliamp, the exposure time is \(0.2 \mathrm{~s}\). Write an equation that represents this variation. Include the units.
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