4. Simplify The Following:(a) $\frac{-7}{6}+\frac{3}{5}+\frac{1}{3}$(b) $\frac{1}{8}+\frac{7}{12}+\left(\frac{-2}{9}\right$\](c) $\frac{2}{2}+\frac{-11}{10}+\left(\frac{-7}{60}\right$\](d) $\frac{1}{3} + 2 -

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Introduction

Fractions are an essential part of mathematics, and simplifying complex fractions is a crucial skill to master. In this article, we will explore the process of simplifying complex fractions, focusing on four different examples. We will break down each example step by step, using a clear and concise approach to ensure that you understand the concept.

Example (a): Simplifying −76+35+13\frac{-7}{6}+\frac{3}{5}+\frac{1}{3}

To simplify this complex fraction, we need to find a common denominator. The least common multiple (LCM) of 6, 5, and 3 is 30.

Step 1: Convert each fraction to have a denominator of 30

  • −76=−7×56×5=−3530\frac{-7}{6} = \frac{-7 \times 5}{6 \times 5} = \frac{-35}{30}
  • 35=3×65×6=1830\frac{3}{5} = \frac{3 \times 6}{5 \times 6} = \frac{18}{30}
  • 13=1×103×10=1030\frac{1}{3} = \frac{1 \times 10}{3 \times 10} = \frac{10}{30}

Step 2: Add the fractions

−3530+1830+1030=−35+18+1030=−730\frac{-35}{30} + \frac{18}{30} + \frac{10}{30} = \frac{-35 + 18 + 10}{30} = \frac{-7}{30}

Step 3: Simplify the fraction

The fraction −730\frac{-7}{30} is already in its simplest form.

Example (b): Simplifying 18+712+(−29)\frac{1}{8}+\frac{7}{12}+\left(\frac{-2}{9}\right)

To simplify this complex fraction, we need to find a common denominator. The LCM of 8, 12, and 9 is 72.

Step 1: Convert each fraction to have a denominator of 72

  • 18=1×98×9=972\frac{1}{8} = \frac{1 \times 9}{8 \times 9} = \frac{9}{72}
  • 712=7×612×6=4272\frac{7}{12} = \frac{7 \times 6}{12 \times 6} = \frac{42}{72}
  • −29=−2×89×8=−1672\frac{-2}{9} = \frac{-2 \times 8}{9 \times 8} = \frac{-16}{72}

Step 2: Add the fractions

972+4272+−1672=9+42−1672=3572\frac{9}{72} + \frac{42}{72} + \frac{-16}{72} = \frac{9 + 42 - 16}{72} = \frac{35}{72}

Step 3: Simplify the fraction

The fraction 3572\frac{35}{72} is already in its simplest form.

Example (c): Simplifying 22+−1110+(−760)\frac{2}{2}+\frac{-11}{10}+\left(\frac{-7}{60}\right)

To simplify this complex fraction, we need to find a common denominator. The LCM of 2, 10, and 60 is 60.

Step 1: Convert each fraction to have a denominator of 60

  • 22=2×302×30=6060\frac{2}{2} = \frac{2 \times 30}{2 \times 30} = \frac{60}{60}
  • −1110=−11×610×6=−6660\frac{-11}{10} = \frac{-11 \times 6}{10 \times 6} = \frac{-66}{60}
  • −760=−7×160×1=−760\frac{-7}{60} = \frac{-7 \times 1}{60 \times 1} = \frac{-7}{60}

Step 2: Add the fractions

6060+−6660+−760=60−66−760=−1360\frac{60}{60} + \frac{-66}{60} + \frac{-7}{60} = \frac{60 - 66 - 7}{60} = \frac{-13}{60}

Step 3: Simplify the fraction

The fraction −1360\frac{-13}{60} is already in its simplest form.

Example (d): Simplifying 13+2\frac{1}{3} + 2

To simplify this complex fraction, we need to convert the integer 2 to a fraction with a denominator of 3.

Step 1: Convert the integer 2 to a fraction with a denominator of 3

2=2×33=632 = \frac{2 \times 3}{3} = \frac{6}{3}

Step 2: Add the fractions

13+63=1+63=73\frac{1}{3} + \frac{6}{3} = \frac{1 + 6}{3} = \frac{7}{3}

Step 3: Simplify the fraction

The fraction 73\frac{7}{3} is already in its simplest form.

Conclusion

Simplifying complex fractions requires finding a common denominator and adding the fractions. In this article, we have explored four different examples of simplifying complex fractions, using a clear and concise approach to ensure that you understand the concept. By following these steps, you can simplify complex fractions with ease.

Key Takeaways

  • To simplify a complex fraction, find a common denominator.
  • Convert each fraction to have the common denominator.
  • Add the fractions.
  • Simplify the resulting fraction.

Final Thoughts

Introduction

Simplifying complex fractions can be a challenging task, but with the right approach, it can be made easier. In this article, we will answer some of the most frequently asked questions about simplifying complex fractions, providing you with a clear and concise guide to help you master this essential skill.

Q: What is a complex fraction?

A complex fraction is a fraction that contains one or more fractions in its numerator or denominator. For example, 1234\frac{\frac{1}{2}}{\frac{3}{4}} is a complex fraction.

Q: How do I simplify a complex fraction?

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

  1. Find a common denominator for the fractions in the numerator and denominator.
  2. Convert each fraction to have the common denominator.
  3. Add or subtract the fractions in the numerator and denominator.
  4. Simplify the resulting fraction.

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

The least common multiple (LCM) is the smallest multiple that two or more numbers have in common. For example, the LCM of 6 and 8 is 24.

Q: How do I find the LCM of two or more numbers?

To find the LCM of two or more numbers, you can use the following steps:

  1. List the multiples of each number.
  2. Identify the smallest multiple that is common to all the numbers.
  3. The LCM is the smallest multiple that is common to all the numbers.

Q: What is the difference between adding and subtracting fractions?

When adding fractions, you need to find a common denominator and add the numerators. When subtracting fractions, you need to find a common denominator and subtract the numerators.

Q: How do I add or subtract fractions with different denominators?

To add or subtract fractions with different denominators, you need to follow these steps:

  1. Find a common denominator for the fractions.
  2. Convert each fraction to have the common denominator.
  3. Add or subtract the fractions.
  4. Simplify the resulting fraction.

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

Some common mistakes to avoid when simplifying complex fractions include:

  • Not finding a common denominator.
  • Not converting each fraction to have the common denominator.
  • Not adding or subtracting the fractions correctly.
  • Not simplifying the resulting fraction.

Q: How can I practice simplifying complex fractions?

You can practice simplifying complex fractions by working through examples and exercises. You can also use online resources and practice tests to help you master this essential skill.

Conclusion

Simplifying complex fractions can be a challenging task, but with the right approach, it can be made easier. By following the steps outlined in this article, you can simplify complex fractions with ease and confidence. Remember to always find a common denominator, convert each fraction to have the common denominator, add or subtract the fractions, and simplify the resulting fraction. With practice, you can master this essential skill and become proficient in simplifying complex fractions.

Key Takeaways

  • To simplify a complex fraction, find a common denominator.
  • Convert each fraction to have the common denominator.
  • Add or subtract the fractions.
  • Simplify the resulting fraction.
  • Practice simplifying complex fractions to master this essential skill.

Final Thoughts

Simplifying complex fractions is an essential skill in mathematics, and with practice, you can master it. Remember to always find a common denominator, convert each fraction to have the common denominator, add or subtract the fractions, and simplify the resulting fraction. With these steps, you can simplify complex fractions with ease and confidence.