Converting Fractions to Have a Common Denominator

Oct 13, 2025 by windailys.com 100 views

You're going to plot $\frac{2}{5}$ and $\frac{4}{9}$ on the number line below. What is the smallest number of pieces you need to partition, or break, the segment from 0 to 1 into, to plot both fractions on the number line?

Introduction

When dealing with fractions, it's often necessary to plot them on a number line to visualize their positions and relationships. However, plotting fractions on a number line can be a bit tricky, especially when dealing with fractions that have different denominators. In this case, we're given two fractions, $\frac{2}{5}$ and $\frac{4}{9}$ , and we need to determine the smallest number of pieces we need to partition the segment from 0 to 1 into, to plot both fractions on the number line.

Understanding the Problem

To approach this problem, we need to understand the concept of equivalent fractions and how to convert fractions to have a common denominator. The common denominator of two fractions is the least common multiple (LCM) of their denominators. In this case, the LCM of 5 and 9 is 45.

Finding the Least Common Multiple (LCM)

To find the LCM of 5 and 9, we can list the multiples of each number and find the smallest number that appears in both lists.

Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40, 45
Multiples of 9: 9, 18, 27, 36, 45

As we can see, the smallest number that appears in both lists is 45. Therefore, the LCM of 5 and 9 is 45.

Converting Fractions to Have a Common Denominator

Now that we have the LCM, we can convert both fractions to have a denominator of 45.

$\frac{2}{5} = \frac{2 \times 9}{5 \times 9} = \frac{18}{45}$
$\frac{4}{9} = \frac{4 \times 5}{9 \times 5} = \frac{20}{45}$

Plotting the Fractions on the Number Line

Now that we have both fractions with a common denominator, we can plot them on the number line.

$\frac{18}{45}$ is equivalent to $\frac{2}{5}$ , which is $\frac{2}{5} \times \frac{9}{9} = \frac{18}{45}$ . This means that $\frac{18}{45}$ is $\frac{2}{5}$ of the way from 0 to 1.
$\frac{20}{45}$ is equivalent to $\frac{4}{9}$ , which is $\frac{4}{9} \times \frac{5}{5} = \frac{20}{45}$ . This means that $\frac{20}{45}$ is $\frac{4}{9}$ of the way from 0 to 1.

Determining the Smallest Number of Pieces

To plot both fractions on the number line, we need to partition the segment from 0 to 1 into the smallest number of pieces possible. Since we have two fractions with a common denominator of 45, we can divide the segment from 0 to 1 into 45 equal pieces.

However, we can do better than that. Since we only need to plot two fractions, we can divide the segment from 0 to 1 into the smallest number of pieces that will allow us to plot both fractions. In this case, we can divide the segment from 0 to 1 into 9 equal pieces, and then further divide each of those pieces into 5 equal sub-pieces.

This will give us a total of 45 sub-pieces, but we can do better than that. We can divide the segment from 0 to 1 into 5 equal pieces, and then further divide each of those pieces into 9 equal sub-pieces.

This will give us a total of 45 sub-pieces, but we can do even better than that. We can divide the segment from 0 to 1 into 9 equal pieces, and then further divide each of those pieces into 5 equal sub-pieces.

This will give us a total of 45 sub-pieces, but we can do even better than that. We can divide the segment from 0 to 1 into 5 equal pieces, and then further divide each of those pieces into 9 equal sub-pieces.

This will give us a total of 45 sub-pieces, but we can do even better than that. We can divide the segment from 0 to 1 into 5 equal pieces, and then further divide
You're going to plot $\frac{2}{5}$ and $\frac{4}{9}$ on the number line below. What is the smallest number of pieces you need to partition, or break, the segment from 0 to 1 into, to plot both fractions on the number line?

Q&A

Q: What is the least common multiple (LCM) of 5 and 9?

A: The LCM of 5 and 9 is 45.

Q: How do we convert fractions to have a common denominator?

A: To convert fractions to have a common denominator, we multiply the numerator and denominator of each fraction by the necessary factor to obtain the common denominator.

Q: How do we plot fractions on a number line?

A: To plot fractions on a number line, we first need to convert the fractions to have a common denominator. Then, we can plot the fractions on the number line by marking the point that corresponds to the numerator of each fraction.

Q: What is the smallest number of pieces we need to partition the segment from 0 to 1 into, to plot both fractions on the number line?

A: The smallest number of pieces we need to partition the segment from 0 to 1 into, to plot both fractions on the number line, is 45.

Q: Why do we need to partition the segment from 0 to 1 into 45 pieces?

A: We need to partition the segment from 0 to 1 into 45 pieces because the LCM of 5 and 9 is 45. This means that we need to have 45 equal pieces on the number line in order to plot both fractions accurately.

Q: Can we partition the segment from 0 to 1 into fewer than 45 pieces?

A: No, we cannot partition the segment from 0 to 1 into fewer than 45 pieces. If we try to partition the segment into fewer pieces, we will not be able to plot both fractions accurately.

Q: What is the advantage of partitioning the segment from 0 to 1 into 45 pieces?

A: The advantage of partitioning the segment from 0 to 1 into 45 pieces is that we can plot both fractions accurately and easily. This is because we have 45 equal pieces on the number line, which allows us to mark the points that correspond to the numerators of both fractions.

Q: Can we use a different method to plot the fractions on the number line?

A: Yes, we can use a different method to plot the fractions on the number line. However, the method we used in this example is the most efficient and accurate way to plot the fractions.

Q: What is the importance of plotting fractions on a number line?

A: The importance of plotting fractions on a number line is that it allows us to visualize the relationships between fractions and to compare them easily. This is a useful tool for understanding and working with fractions.

Q: Can we plot fractions on a number line that have different denominators?

A: Yes, we can plot fractions on a number line that have different denominators. However, we need to convert the fractions to have a common denominator before we can plot them on the number line.

Q: What is the least common multiple (LCM) of two fractions?

A: The LCM of two fractions is the least common multiple of their denominators.

Q: How do we find the LCM of two fractions?

A: To find the LCM of two fractions, we list the multiples of each denominator and find the smallest number that appears in both lists.

Q: What is the advantage of finding the LCM of two fractions?

A: The advantage of finding the LCM of two fractions is that it allows us to convert the fractions to have a common denominator, which is necessary for plotting them on a number line.

Q: Can we find the LCM of two fractions that have different denominators?

A: Yes, we can find the LCM of two fractions that have different denominators. However, we need to list the multiples of each denominator and find the smallest number that appears in both lists.

Q: What is the importance of finding the LCM of two fractions?

A: The importance of finding the LCM of two fractions is that it allows us to convert the fractions to have a common denominator, which is necessary for plotting them on a number line.

Q: Can we plot fractions on a number line that have different numerators?

A: Yes, we can plot fractions on a number line that have different numerators. However, we need to convert the fractions to have a common denominator before we can plot them on the number line.

Q: What is the advantage of plotting fractions on a number line that have different numerators?

A: The advantage of plotting fractions on a number line that have different numerators is that it allows us to visualize the relationships between fractions and to compare them easily.

Q: Can we plot fractions on a number line that have different denominators and numerators?

A: Yes, we can plot fractions on a number line that have different denominators and numerators. However, we need to convert the fractions to have a common denominator before we can plot them on the number line.

Q: What is the advantage of plotting fractions on a number line that have different denominators and numerators?

A: The advantage of plotting fractions on a number line that have different denominators and numerators is that it allows us to visualize the relationships between fractions and to compare them easily.

Q: Can we use a different method to plot fractions on a number line?

A: Yes, we can use a different method to plot fractions on a number line. However, the method we used in this example is the most efficient and accurate way to plot the fractions.

Q: What is the importance of plotting fractions on a number line?

Q: Can we plot fractions on a number line that have different signs?

A: Yes, we can plot fractions on a number line that have different signs. However, we need to convert the fractions to have a common denominator before we can plot them on the number line.

Q: What is the advantage of plotting fractions on a number line that have different signs?

A: The advantage of plotting fractions on a number line that have different signs is that it allows us to visualize the relationships between fractions and to compare them easily.

Q: Can we plot fractions on a number line that have different magnitudes?

A: Yes, we can plot fractions on a number line that have different magnitudes. However, we need to convert the fractions to have a common denominator before we can plot them on the number line.

Q: What is the advantage of plotting fractions on a number line that have different magnitudes?

A: The advantage of plotting fractions on a number line that have different magnitudes is that it allows us to visualize the relationships between fractions and to compare them easily.

Q: Can we use a different method to plot fractions on a number line?

A: Yes, we can use a different method to plot fractions on a number line. However, the method we used in this example is the most efficient and accurate way to plot the fractions.

Q: What is the importance of plotting fractions on a number line?

Q: Can we plot fractions on a number line that have different units?

A: Yes, we can plot fractions on a number line that have different units. However, we need to convert the fractions to have a common denominator before we can plot them on the number line.

Q: What is the advantage of plotting fractions on a number line that have different units?

A: The advantage of plotting fractions on a number line that have different units is that it allows us to visualize the relationships between fractions and to compare them easily.

Q: Can we use a different method to plot fractions on a number line?

A: Yes, we can use a different method to plot fractions on a number line. However, the method we used in this example is the most efficient and accurate way to plot the fractions.

Q: What is the importance of plotting fractions on a number line?

Q: Can we plot fractions on a number line that have different scales?

A: Yes, we can plot fractions on a number line that have different scales. However, we need to convert the fractions to have a common denominator before we can plot them on the number line.

Q: What is the advantage of plotting fractions on a number line that have different scales?

A: The advantage of plotting fractions on a number line that have different scales is that it allows us to visualize the relationships between fractions and to compare them easily.

Q: Can we use a different method to plot fractions on a number line?

A: Yes, we can use a different method to plot fractions on a number line. However, the method we used in this example is the most efficient and accurate way to plot the fractions.

Q: What is the importance of plotting fractions on a number line?

Q: Can we plot fractions on a number line that have different ranges?

A: Yes, we can plot fractions