Comparing Fractions with Different Denominators

Oct 13, 2025 by leolavanderia.com.br 89 views

Introduction

Comparing fractions is an essential skill in mathematics, and it's often used in various real-world applications. In this article, we will compare two fractions, $\frac{5}{7}$ and $\frac{9}{14}$ , to determine which one is greater and by how much.

Understanding the Problem

To compare fractions, we need to understand the concept of equivalent fractions. Equivalent fractions are fractions that have the same value, but with different numerators and denominators. For example, $\frac{1}{2}$ and $\frac{2}{4}$ are equivalent fractions.

Comparing Fractions with Different Denominators

When comparing fractions with different denominators, we need to find a common denominator. The common denominator is the least common multiple (LCM) of the two denominators. In this case, the denominators are 7 and 14. The LCM of 7 and 14 is 14.

Converting Fractions to Have a Common Denominator

To convert $\frac{5}{7}$ to have a denominator of 14, we multiply the numerator and denominator by 2:

\frac{5}{7} \times \frac{2}{2} = \frac{10}{14}

Now, we can compare $\frac{10}{14}$ and $\frac{9}{14}$ .

Comparing the Fractions

Since the denominators are the same, we can compare the numerators. The numerator of $\frac{10}{14}$ is greater than the numerator of $\frac{9}{14}$ . Therefore, $\frac{10}{14}$ is greater than $\frac{9}{14}$ .

Finding the Difference

To find the difference between the two fractions, we subtract the smaller fraction from the larger fraction:

\frac{10}{14} - \frac{9}{14} = \frac{1}{14}

Therefore, the difference between $\frac{5}{7}$ and $\frac{9}{14}$ is $\frac{1}{14}$ .

Conclusion

In conclusion, $\frac{5}{7}$ is greater than $\frac{9}{14}$ by $\frac{1}{14}$ . This is the correct answer.

Answer Key

The correct answer is A. $\frac{1}{14}$ .

Why is this Important?

Comparing fractions is an essential skill in mathematics, and it's often used in various real-world applications. For example, in cooking, you may need to compare the ratio of ingredients in a recipe. In finance, you may need to compare the interest rates of different investments. In science, you may need to compare the concentrations of different substances.

Real-World Applications

Comparing fractions has many real-world applications. Here are a few examples:

Cooking: When cooking, you may need to compare the ratio of ingredients in a recipe. For example, if a recipe calls for 2 cups of flour and 1 cup of sugar, you can compare the ratio of flour to sugar.
Finance: When investing, you may need to compare the interest rates of different investments. For example, if you have two investment options with different interest rates, you can compare the interest rates to determine which one is better.
Science: When conducting experiments, you may need to compare the concentrations of different substances. For example, if you are testing the effect of different concentrations of a substance on a plant, you can compare the concentrations to determine which one has the greatest effect.

Tips and Tricks

Here are a few tips and tricks for comparing fractions:

Find a common denominator: When comparing fractions with different denominators, find a common denominator to make the comparison easier.
Convert fractions to have a common denominator: Convert fractions to have a common denominator to make the comparison easier.
Compare numerators: When the denominators are the same, compare the numerators to determine which fraction is greater.
Subtract fractions: To find the difference between two fractions, subtract the smaller fraction from the larger fraction.

Conclusion

Q: What is the first step in comparing fractions with different denominators?

A: The first step in comparing fractions with different denominators is to find a common denominator. The common denominator is the least common multiple (LCM) of the two denominators.

Q: How do I find the least common multiple (LCM) of two numbers?

A: To find the LCM of two numbers, you can list the multiples of each number and find the smallest multiple that is common to both. Alternatively, you can use the following formula:

LCM(a, b) = (a × b) / GCD(a, b)

where GCD(a, b) is the greatest common divisor of a and b.

Q: What is the greatest common divisor (GCD) of two numbers?

A: The GCD of two numbers is the largest number that divides both numbers without leaving a remainder.

Q: How do I convert a fraction to have a common denominator?

A: To convert a fraction to have a common denominator, you can multiply the numerator and denominator by the same number. For example, to convert $\frac{1}{2}$ to have a denominator of 4, you can multiply the numerator and denominator by 2:

\frac{1}{2} \times \frac{2}{2} = \frac{2}{4}

Q: What is the difference between two fractions?

A: The difference between two fractions is found by subtracting the smaller fraction from the larger fraction. For example, to find the difference between $\frac{3}{4}$ and $\frac{1}{4}$ , you can subtract the smaller fraction from the larger fraction:

\frac{3}{4} - \frac{1}{4} = \frac{2}{4}

Q: Can I compare fractions with different numerators and denominators?

A: Yes, you can compare fractions with different numerators and denominators. To do this, you need to find a common denominator and convert both fractions to have that denominator.

Q: What is the order of operations for comparing fractions?

A: The order of operations for comparing fractions is as follows:

Find a common denominator.
Convert both fractions to have the common denominator.
Compare the numerators.
Subtract the smaller fraction from the larger fraction to find the difference.

Q: Can I use a calculator to compare fractions?

A: Yes, you can use a calculator to compare fractions. However, it's often easier to use a calculator to find the decimal equivalent of a fraction and then compare the decimals.

Q: What are some real-world applications of comparing fractions?

A: Comparing fractions has many real-world applications, including:

Cooking: When cooking, you may need to compare the ratio of ingredients in a recipe.
Finance: When investing, you may need to compare the interest rates of different investments.
Science: When conducting experiments, you may need to compare the concentrations of different substances.

Q: How can I practice comparing fractions?

A: You can practice comparing fractions by using online resources, such as fraction comparison games and worksheets. You can also practice by creating your own fraction comparison problems and solving them.

Conclusion

In conclusion, comparing fractions is an essential skill in mathematics, and it's often used in various real-world applications. By following the steps outlined in this article, you can compare fractions with different denominators and find the difference between them. Remember to find a common denominator, convert fractions to have a common denominator, compare numerators, and subtract fractions to find the difference.