${$1 \frac{7}{10} + \frac{7}{4} =$}$6) { \frac{8}{5} + \frac{19}{12} =$}$

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Introduction

In mathematics, fractions are a fundamental concept that helps us represent parts of a whole. When we add or subtract fractions, we need to ensure that the denominators are the same. However, what if we have fractions with different denominators? In this article, we will explore how to solve complex fractions, including adding and subtracting fractions with different denominators.

Understanding Complex Fractions

A complex fraction is a fraction that contains one or more fractions in its numerator or denominator. For example, the expression \frac{1 \frac{7}{10} + \frac{7}{4}} is a complex fraction because it contains a mixed number in the numerator and a fraction in the denominator.

Solving Complex Fractions: A Step-by-Step Guide

To solve complex fractions, we need to follow a step-by-step approach. Here's a general guide:

Step 1: Simplify the Numerator

The first step is to simplify the numerator of the complex fraction. If the numerator is a mixed number, we need to convert it to an improper fraction. For example, in the expression \frac{1 \frac{7}{10} + \frac{7}{4}}, we need to convert the mixed number 17101 \frac{7}{10} to an improper fraction.

Step 2: Find the Least Common Multiple (LCM)

Once we have simplified the numerator, we need to find the least common multiple (LCM) of the denominators. The LCM is the smallest number that both denominators can divide into evenly. For example, in the expression \frac{1 \frac{7}{10} + \frac{7}{4}}, the denominators are 10 and 4. The LCM of 10 and 4 is 20.

Step 3: Convert the Fractions

Now that we have found the LCM, we need to convert both fractions to have the same denominator. We can do this by multiplying the numerator and denominator of each fraction by the necessary factor. For example, in the expression \frac{1 \frac{7}{10} + \frac{7}{4}}, we need to multiply the first fraction by 2 and the second fraction by 5 to get:

\frac{2 \cdot 1 \frac{7}{10} + 5 \cdot \frac{7}{4}}

Step 4: Add or Subtract the Fractions

Now that we have converted both fractions to have the same denominator, we can add or subtract them. For example, in the expression \frac{2 \cdot 1 \frac{7}{10} + 5 \cdot \frac{7}{4}}, we can add the fractions:

\frac{2 \cdot 1 \frac{7}{10} + 5 \cdot \frac{7}{4}} = \frac{2 \cdot \frac{27}{10} + 5 \cdot \frac{7}{4}}

Step 5: Simplify the Result

Finally, we need to simplify the result by combining the fractions. For example, in the expression \frac{2 \cdot \frac{27}{10} + 5 \cdot \frac{7}{4}}, we can simplify the result:

\frac{2 \cdot \frac{27}{10} + 5 \cdot \frac{7}{4}} = \frac{\frac{54}{10} + \frac{35}{4}} = \frac{\frac{216}{40} + \frac{175}{40}} = \frac{\frac{391}{40}}

Example 1: Adding Fractions with Different Denominators

Let's consider the expression \frac{1 \frac{7}{10} + \frac{7}{4}}. We can follow the steps outlined above to solve this expression:

  1. Simplify the numerator: 1710=27101 \frac{7}{10} = \frac{27}{10}
  2. Find the LCM: The LCM of 10 and 4 is 20.
  3. Convert the fractions: \frac{2 \cdot \frac{27}{10} + 5 \cdot \frac{7}{4}}
  4. Add the fractions: \frac{2 \cdot \frac{27}{10} + 5 \cdot \frac{7}{4}} = \frac{\frac{54}{10} + \frac{35}{4}}
  5. Simplify the result: \frac{\frac{54}{10} + \frac{35}{4}} = \frac{\frac{216}{40} + \frac{175}{40}} = \frac{\frac{391}{40}}

Example 2: Subtracting Fractions with Different Denominators

Let's consider the expression 85+1912\frac{8}{5} + \frac{19}{12}. We can follow the steps outlined above to solve this expression:

  1. Simplify the numerator: None required
  2. Find the LCM: The LCM of 5 and 12 is 60.
  3. Convert the fractions: \frac{12 \cdot \frac{8}{5} + 5 \cdot \frac{19}{12}}
  4. Add the fractions: \frac{12 \cdot \frac{8}{5} + 5 \cdot \frac{19}{12}} = \frac{\frac{96}{5} + \frac{95}{12}}
  5. Simplify the result: \frac{\frac{96}{5} + \frac{95}{12}} = \frac{\frac{1152}{60} + \frac{475}{60}} = \frac{\frac{1627}{60}}

Conclusion

Solving complex fractions requires a step-by-step approach. By simplifying the numerator, finding the least common multiple, converting the fractions, adding or subtracting the fractions, and simplifying the result, we can solve complex fractions with different denominators. With practice and patience, you can become proficient in solving complex fractions and tackle even the most challenging math problems.

Final Answer

The final answer to the expression \frac{1 \frac{7}{10} + \frac{7}{4}} is \frac{\frac{391}{40}}.

Q: What is a complex fraction?

A: A complex fraction is a fraction that contains one or more fractions in its numerator or denominator.

Q: How do I simplify a complex fraction?

A: To simplify a complex fraction, you need to follow these steps:

  1. Simplify the numerator
  2. Find the least common multiple (LCM) of the denominators
  3. Convert the fractions to have the same denominator
  4. Add or subtract the fractions
  5. Simplify the result

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

A: The least common multiple (LCM) is the smallest number that both denominators can divide into evenly.

Q: How do I find the LCM?

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

Q: Can I use a calculator to find the LCM?

A: Yes, you can use a calculator to find the LCM. Most calculators have a built-in function to find the LCM.

Q: What if I have a fraction with a variable in the numerator or denominator?

A: If you have a fraction with a variable in the numerator or denominator, you need to follow the same steps as before, but you will also need to use algebraic techniques to simplify the expression.

Q: Can I use a formula to simplify complex fractions?

A: Yes, there are formulas that can be used to simplify complex fractions, such as the formula for adding fractions with different denominators:

ab+cd=ad+bcbd\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}

Q: What if I have a complex fraction with multiple levels of nesting?

A: If you have a complex fraction with multiple levels of nesting, you need to follow the same steps as before, but you will also need to use parentheses to group the expressions correctly.

Q: Can I use a graphing calculator to solve complex fractions?

A: Yes, you can use a graphing calculator to solve complex fractions. Graphing calculators can be used to visualize the expressions and find the solutions.

Q: What are some common mistakes to avoid when solving complex fractions?

A: Some common mistakes to avoid when solving complex fractions include:

  • Not simplifying the numerator
  • Not finding the LCM
  • Not converting the fractions to have the same denominator
  • Not adding or subtracting the fractions correctly
  • Not simplifying the result

Q: How can I practice solving complex fractions?

A: You can practice solving complex fractions by working through examples and exercises in a textbook or online resource. You can also try solving complex fractions on your own and checking your answers with a calculator or a teacher.

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

A: Complex fractions have many real-world applications, including:

  • Finance: Complex fractions are used to calculate interest rates and investment returns.
  • Science: Complex fractions are used to calculate rates of change and solve problems in physics and engineering.
  • Engineering: Complex fractions are used to design and optimize systems, such as electrical circuits and mechanical systems.

Conclusion

Solving complex fractions requires a step-by-step approach and a good understanding of algebraic techniques. By following the steps outlined in this article and practicing regularly, you can become proficient in solving complex fractions and tackle even the most challenging math problems.