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Chapter 7: Problem 1

For exercises 1-66, simplify. $$ \frac{180}{420} $$

Short Answer

Expert verified
\frac{3}{7}

Step by step solution

01

Identify the greatest common divisor (GCD)

To simplify the fraction \(\frac{180}{420}\), find the greatest common divisor (GCD) of 180 and 420. List out the factors of each number: \[ 180: 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180 \] \[ 420: 1, 2, 3, 4, 5, 6, 7, 10, 12, 14, 15, 20, 21, 28, 30, 35, 42, 60, 70, 84, 105, 140, 210, 420 \] The greatest common divisor is 60.
02

Divide the numerator and denominator by the GCD

Now, divide both the numerator and the denominator of \(\frac{180}{420}\) by the GCD, which is 60: \[ \frac{180 \div 60}{420 \div 60} = \frac{3}{7} \]
03

Write the simplified fraction

After dividing both the numerator and the denominator by their greatest common divisor, the fraction simplifies to \(\frac{3}{7}\).

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Key Concepts

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

greatest common divisor
The greatest common divisor (GCD) is a key concept in fraction simplification. It is the largest number that divides both the numerator and the denominator without leaving a remainder. To find the GCD of two numbers, list out all the factors of each number. For example, with 180 and 420, the factors are:
  • 180: 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180
  • 420: 1, 2, 3, 4, 5, 6, 7, 10, 12, 14, 15, 20, 21, 28, 30, 35, 42, 60, 70, 84, 105, 140, 210, 420
The greatest common factor here is 60. This number helps in reducing fractions to their simplest form.
numerator
In a fraction, the numerator is the number above the line. It represents the number of parts we have. For instance, in the fraction \(\frac{180}{420}\), 180 is the numerator. When simplifying, we divide the numerator by the GCD. So, 180 divided by 60 (GCD) gives us 3. This helps in reducing the numerator while keeping the fraction equivalent.
denominator
The denominator is the number below the line in a fraction. It shows the total number of equal parts the whole is divided into. For the fraction \(\frac{180}{420}\), 420 is the denominator. Similar to the numerator, we also divide the denominator by the GCD. Dividing 420 by 60 gives 7. This step is essential to accurately simplify the fraction.
fraction simplification
Fraction simplification is the process of making a fraction as simple as possible. This often involves dividing both the numerator and the denominator by their GCD. Doing this ensures the fraction remains equivalent to the original. Using our example: \(\frac{180}{420}\), the GCD is 60. Dividing both terms:
\(\frac{180 \div 60}{420 \div 60}\) simplifies to \(\frac{3}{7}\).
This resulting fraction, \(\frac{3}{7}\), is the simplest form of \(\frac{180}{420}\). It鈥檚 easier to understand and work with in calculations.

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Most popular questions from this chapter

For exercises 37-38, \(T=\frac{R}{A}\) represents the relationship of the asset turnover ratio, \(T\); the sales revenue of a company, \(R\); and the total revenues of a company, \(A\). Is the relationship of the given variables a direct variation or an inverse variation? $$ A \text { is constant; the relationship of } R \text { and } T \text {. } $$

For exercises 49-52, the formula \(C=\frac{P_{m} P_{i}}{T F}\) describes the cost of insurance, \(C\). Is the relationship of the given variables a direct variation or an inverse variation? $$ C, T, \text { and } F \text { are constant; the relationship of } P_{i} \text { and } P_{m} $$

For exercises 43-58, (a) solve. (b) check. $$ \frac{4}{a+6}=\frac{9}{a-4} $$

The relationship of \(x\) and \(y\) is a direct variation. When \(x=1, y=6\). a. Find the constant of proportionality, \(k\). b. Write an equation that represents this direct variation. c. Find \(y\) when \(x=4\), d. Use slope-intercept graphing to graph this equation. e. Use the graph to find \(y\) when \(x=2\).

The formula \(R=\frac{V}{I}\) represents the relationship of the resistance \(R\), voltage \(V\), and current \(I\) in an electric circuit. Assume that \(V\) is constant. Is the relationship of \(R\) and \(I\) a direct variation or an inverse variation?
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