Problem 2
Question
For exercises 1-66, simplify. $$ \frac{240}{540} $$
Step-by-Step Solution
Verified Answer
\( \frac{4}{9} \)
1Step 1: 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.
2Step 2: Divide the numerator and denominator by the GCD
Next, divide both the numerator (240) and the denominator (540) by the GCD (60): \(\frac{240 \div 60}{540 \div 60} = \frac{4}{9}\).
3Step 3: Simplified fraction
After dividing, the simplified form of \(\frac{240}{540}\) is \(\frac{4}{9}\).
Key Concepts
greatest common divisornumerator and denominatorsimplified fraction
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.
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.
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.
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.
Other exercises in this chapter
Problem 2
For exercises \(1-4\), evaluate. $$ \frac{9}{50}+\frac{3}{50} $$
View solution Problem 2
Describe how to multiply two fractions.
View solution Problem 3
Explain why the relationship of the number of bags of leaves per hour that are raked, \(x\), and the hours it takes to rake a yard, \(y\), is an inverse variati
View solution Problem 3
For exercises 1-10, (a) solve. (b) check. $$ \frac{4}{9} p-\frac{1}{8}=\frac{25}{72} $$
View solution