Finding the Difference between Two Fractions

Oct 13, 2025 by miodeya.com 32 views

Introduction

Fractions are a fundamental concept in mathematics, and understanding how to simplify them is crucial for solving various mathematical problems. In this article, we will explore the process of finding the difference between two fractions, which is a common operation in mathematics. We will use real-world examples to illustrate the concept and provide step-by-step solutions to five different problems.

What is a Fraction?

A fraction is a way of expressing a part of a whole as a ratio of two numbers. It consists of a numerator (the top number) and a denominator (the bottom number). For example, the fraction 1/2 represents one half of a whole.

Finding the Difference between Two Fractions

To find the difference between two fractions, we need to follow these steps:

Find a common denominator for the two fractions.
Convert both fractions to have the common denominator.
Subtract the numerators of the two fractions.
Simplify the resulting fraction, if possible.

Problem 1: $\frac{1}{6} - \frac{1}{10}$

Step 1: Find a Common Denominator

To find a common denominator, we need to find the least common multiple (LCM) of 6 and 10. The LCM of 6 and 10 is 30.

Step 2: Convert Both Fractions to Have the Common Denominator

We need to convert both fractions to have a denominator of 30.

$\frac{1}{6} = \frac{1 \times 5}{6 \times 5} = \frac{5}{30}$

$\frac{1}{10} = \frac{1 \times 3}{10 \times 3} = \frac{3}{30}$

Step 3: Subtract the Numerators

Now that both fractions have the same denominator, we can subtract the numerators.

$\frac{5}{30} - \frac{3}{30} = \frac{5 - 3}{30} = \frac{2}{30}$

Step 4: Simplify the Resulting Fraction

We can simplify the resulting fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 2.

$\frac{2}{30} = \frac{2 \div 2}{30 \div 2} = \frac{1}{15}$

Problem 2: $\frac{7}{10} - \frac{5}{12}$

Step 1: Find a Common Denominator

To find a common denominator, we need to find the LCM of 10 and 12. The LCM of 10 and 12 is 60.

Step 2: Convert Both Fractions to Have the Common Denominator

We need to convert both fractions to have a denominator of 60.

$\frac{7}{10} = \frac{7 \times 6}{10 \times 6} = \frac{42}{60}$

$\frac{5}{12} = \frac{5 \times 5}{12 \times 5} = \frac{25}{60}$

Step 3: Subtract the Numerators

Now that both fractions have the same denominator, we can subtract the numerators.

$\frac{42}{60} - \frac{25}{60} = \frac{42 - 25}{60} = \frac{17}{60}$

Step 4: Simplify the Resulting Fraction

We can simplify the resulting fraction by dividing both the numerator and the denominator by their GCD, which is 1.

$\frac{17}{60}$ cannot be simplified further.

Problem 3: $\frac{5}{6} - \frac{1}{8}$

Step 1: Find a Common Denominator

To find a common denominator, we need to find the LCM of 6 and 8. The LCM of 6 and 8 is 24.

Step 2: Convert Both Fractions to Have the Common Denominator

We need to convert both fractions to have a denominator of 24.

$\frac{5}{6} = \frac{5 \times 4}{6 \times 4} = \frac{20}{24}$

$\frac{1}{8} = \frac{1 \times 3}{8 \times 3} = \frac{3}{24}$

Step 3: Subtract the Numerators

Now that both fractions have the same denominator, we can subtract the numerators.

$\frac{20}{24} - \frac{3}{24} = \frac{20 - 3}{24} = \frac{17}{24}$

Step 4: Simplify the Resulting Fraction

We can simplify the resulting fraction by dividing both the numerator and the denominator by their GCD, which is 1.

$\frac{17}{24}$ cannot be simplified further.

Problem 4: $\frac{5}{8} - \frac{1}{3}$

Step 1: Find a Common Denominator

To find a common denominator, we need to find the LCM of 8 and 3. The LCM of 8 and 3 is 24.

Step 2: Convert Both Fractions to Have the Common Denominator

We need to convert both fractions to have a denominator of 24.

$\frac{5}{8} = \frac{5 \times 3}{8 \times 3} = \frac{15}{24}$

$\frac{1}{3} = \frac{1 \times 8}{3 \times 8} = \frac{8}{24}$

Step 3: Subtract the Numerators

Now that both fractions have the same denominator, we can subtract the numerators.

$\frac{15}{24} - \frac{8}{24} = \frac{15 - 8}{24} = \frac{7}{24}$

Step 4: Simplify the Resulting Fraction

We can simplify the resulting fraction by dividing both the numerator and the denominator by their GCD, which is 1.

$\frac{7}{24}$ cannot be simplified further.

Problem 5: $\frac{3}{4} - \frac{2}{5}$

Step 1: Find a Common Denominator

To find a common denominator, we need to find the LCM of 4 and 5. The LCM of 4 and 5 is 20.

Step 2: Convert Both Fractions to Have the Common Denominator

We need to convert both fractions to have a denominator of 20.

$\frac{3}{4} = \frac{3 \times 5}{4 \times 5} = \frac{15}{20}$

$\frac{2}{5} = \frac{2 \times 4}{5 \times 4} = \frac{8}{20}$

Step 3: Subtract the Numerators

Now that both fractions have the same denominator, we can subtract the numerators.

$\frac{15}{20} - \frac{8}{20} = \frac{15 - 8}{20} = \frac{7}{20}$

Step 4: Simplify the Resulting Fraction

We can simplify the resulting fraction by dividing both the numerator and the denominator by their GCD, which is 1.

$\frac{7}{20}$ cannot be simplified further.

Conclusion

$Finding the difference between two fractions involves finding a common denominator, converting both fractions to have the common denominator, subtracting the numerators, and simplifying the resulting fraction. By following these steps, we can solve various mathematical problems involving fractions. In this article, we have used real-world examples to illustrate the concept and provide step-by-step solutions to five different problems.<br/> **Find the Difference: Simplifying Fractions - Q&A**$

Introduction

In our previous article, we explored the process of finding the difference between two fractions. We provided step-by-step solutions to five different problems and discussed the importance of finding a common denominator, converting both fractions to have the common denominator, subtracting the numerators, and simplifying the resulting fraction. In this article, we will answer some frequently asked questions (FAQs) related to finding the difference between two fractions.

Q&A

Q: What is the difference between a fraction and a decimal?

A: A fraction is a way of expressing a part of a whole as a ratio of two numbers, while a decimal is a way of expressing a fraction as a number with a point (.) separating the whole number part from the fractional part.

Q: How do I find the difference between two fractions with different denominators?

A: To find the difference between two fractions with different denominators, you need to find a common denominator, convert both fractions to have the common denominator, subtract the numerators, and simplify the resulting fraction.

Q: What is the least common multiple (LCM) and how do I find it?

A: The LCM is the smallest multiple that two or more numbers have in common. To find the LCM, you can list the multiples of each number and find the smallest multiple that appears in both lists.

Q: How do I simplify a fraction?

A: To simplify a fraction, you need to divide both the numerator and the denominator by their greatest common divisor (GCD).

Q: What is the greatest common divisor (GCD) and how do I find it?

A: The GCD is the largest number that divides two or more numbers without leaving a remainder. To find the GCD, you can use the Euclidean algorithm or list the factors of each number and find the largest factor that appears in both lists.

Q: Can I subtract a fraction from a whole number?

A: Yes, you can subtract a fraction from a whole number by converting the whole number to a fraction with the same denominator as the fraction you are subtracting.

Q: How do I add and subtract fractions with different signs?

A: When adding or subtracting fractions with different signs, you need to follow the rules of arithmetic operations. For example, when adding a positive fraction and a negative fraction, you need to subtract the absolute values of the fractions.

Q: Can I simplify a fraction with a negative sign?

A: Yes, you can simplify a fraction with a negative sign by simplifying the absolute value of the fraction and then applying the negative sign.

Examples

Example 1: Subtracting a fraction from a whole number

$\frac{3}{4} - 2 = \frac{3}{4} - \frac{8}{4} = \frac{3 - 8}{4} = \frac{-5}{4}$

Example 2: Adding and subtracting fractions with different signs

$\frac{3}{4} + \left(-\frac{2}{3}\right) = \frac{3}{4} - \frac{2}{3} = \frac{9}{12} - \frac{8}{12} = \frac{1}{12}$

Example 3: Simplifying a fraction with a negative sign

$\frac{-5}{6} = \frac{-5}{6} = \frac{-5 \div 1}{6 \div 1} = \frac{-5}{6}$

Conclusion

Finding the difference between two fractions involves finding a common denominator, converting both fractions to have the common denominator, subtracting the numerators, and simplifying the resulting fraction. By following these steps, we can solve various mathematical problems involving fractions. In this article, we have answered some frequently asked questions related to finding the difference between two fractions and provided examples to illustrate the concepts.

Additional Resources

Khan Academy: Fractions
Mathway: Fractions
Wolfram Alpha: Fractions

Final Thoughts

Finding the difference between two fractions is an essential skill in mathematics, and it requires a deep understanding of fractions and arithmetic operations. By following the steps outlined in this article and practicing with examples, you can become proficient in finding the difference between two fractions and solve various mathematical problems with confidence.