Adding and Subtracting Fractions with Cross Multiplication

Adding and subtracting fractions can be a tricky concept for many students, especially when the fractions have different denominators. The traditional method of finding a common denominator can be time-consuming and confusing. However, there’s a simpler and more intuitive approach called cross multiplication that can make these operations much easier.

Understanding Cross Multiplication

Cross multiplication is a technique used to simplify the process of adding and subtracting fractions with different denominators. It involves multiplying the numerator of one fraction by the denominator of the other and vice versa. The resulting products are then added or subtracted, and the final result is placed over the product of the original denominators.

Steps for Adding Fractions Using Cross Multiplication

1. Write down the fractions you want to add or subtract, side by side. For example, let’s add 1/2 and 1/3.
2. Cross multiply the numerators and denominators. In this case, we multiply 1 (numerator of the first fraction) by 3 (denominator of the second fraction), and 1 (numerator of the second fraction) by 2 (denominator of the first fraction). This gives us 3 and 2.
3. Add or subtract the results from step 2. Since we’re adding, we get 3 + 2 = 5.
4. Multiply the original denominators. 2 x 3 = 6.
5. Write the final result as the sum from step 3 over the product from step 4. Therefore, 1/2 + 1/3 = 5/6.

Steps for Subtracting Fractions Using Cross Multiplication

1. Write down the fractions you want to subtract, side by side. For example, let’s subtract 2/5 from 3/4.
2. Cross multiply the numerators and denominators. We multiply 3 (numerator of the first fraction) by 5 (denominator of the second fraction), and 2 (numerator of the second fraction) by 4 (denominator of the first fraction). This gives us 15 and 8.
3. Subtract the results from step 2. 15 – 8 = 7.
4. Multiply the original denominators. 4 x 5 = 20.
5. Write the final result as the difference from step 3 over the product from step 4. Therefore, 3/4 – 2/5 = 7/20.

Example:

Let’s say we want to add 2/3 and 1/4.

1. Fractions: 2/3 + 1/4
2. Cross multiply: 2 x 4 = 8 and 1 x 3 = 3
3. Add: 8 + 3 = 11
4. Multiply denominators: 3 x 4 = 12
5. Final result: 2/3 + 1/4 = 11/12

Advantages of Cross Multiplication

• Simpler and faster than finding a common denominator.
• More intuitive for many students.
• Reduces the risk of errors associated with finding a common denominator.

Conclusion

Cross multiplication is a valuable tool for adding and subtracting fractions with different denominators. It’s a simple, efficient, and intuitive method that can make these operations much easier to understand and perform. By mastering this technique, students can confidently tackle fraction problems and build a strong foundation in mathematics.