Simplify Each Of The Following Expressions. Be Sure To Simplify Each Of Your Answers As Much As Possible. Write Any Answers Greater Than One As Mixed Numbers.a. $ \frac{3}{5} + \frac{1}{4} $b. $ \frac{3}{4} - \frac{2}{3} $c. $ 5

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Introduction

Fractions are a fundamental concept in mathematics, and simplifying them is an essential skill to master. In this article, we will explore the process of simplifying fractions, focusing on addition and subtraction. We will use real-world examples to illustrate the concepts and provide step-by-step solutions to each problem.

Simplifying Fractions: A Review

Before we dive into the examples, let's review the basics of simplifying fractions. A fraction is a way of expressing a part of a whole. It consists of a numerator (the top number) and a denominator (the bottom number). To simplify a fraction, we need to find the greatest common divisor (GCD) of the numerator and the denominator and divide both numbers by the GCD.

Example a: Adding Fractions

Problem

Simplify the expression: 35+14\frac{3}{5} + \frac{1}{4}

Solution

To add fractions, we need to have the same denominator. In this case, the least common multiple (LCM) of 5 and 4 is 20. We can rewrite each fraction with a denominator of 20:

35=3×45×4=1220\frac{3}{5} = \frac{3 \times 4}{5 \times 4} = \frac{12}{20}

14=1×54×5=520\frac{1}{4} = \frac{1 \times 5}{4 \times 5} = \frac{5}{20}

Now we can add the fractions:

1220+520=1720\frac{12}{20} + \frac{5}{20} = \frac{17}{20}

Therefore, the simplified expression is 1720\frac{17}{20}.

Example b: Subtracting Fractions

Problem

Simplify the expression: 34−23\frac{3}{4} - \frac{2}{3}

Solution

To subtract fractions, we need to have the same denominator. In this case, the LCM of 4 and 3 is 12. We can rewrite each fraction with a denominator of 12:

34=3×34×3=912\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}

23=2×43×4=812\frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12}

Now we can subtract the fractions:

912−812=112\frac{9}{12} - \frac{8}{12} = \frac{1}{12}

Therefore, the simplified expression is 112\frac{1}{12}.

Example c: Multiplying Fractions

Problem

Simplify the expression: 5×345 \times \frac{3}{4}

Solution

To multiply a fraction by a whole number, we can multiply the numerator by the whole number and keep the denominator the same:

5×34=5×34=1545 \times \frac{3}{4} = \frac{5 \times 3}{4} = \frac{15}{4}

Therefore, the simplified expression is 154\frac{15}{4}, which can be written as a mixed number: 3343\frac{3}{4}.

Conclusion

Simplifying fractions is an essential skill in mathematics, and it requires a deep understanding of the concepts. By following the steps outlined in this article, you can simplify fractions with ease. Remember to always find the greatest common divisor (GCD) of the numerator and the denominator and divide both numbers by the GCD. With practice, you will become proficient in simplifying fractions and be able to tackle more complex problems.

Tips and Tricks

  • Always find the least common multiple (LCM) of the denominators when adding or subtracting fractions.
  • Use the greatest common divisor (GCD) to simplify fractions.
  • When multiplying a fraction by a whole number, multiply the numerator by the whole number and keep the denominator the same.
  • Practice, practice, practice! The more you practice simplifying fractions, the more comfortable you will become with the concepts.

Common Mistakes to Avoid

  • Not finding the least common multiple (LCM) of the denominators when adding or subtracting fractions.
  • Not using the greatest common divisor (GCD) to simplify fractions.
  • Not multiplying the numerator by the whole number when multiplying a fraction by a whole number.
  • Not writing the answer as a mixed number when the answer is greater than one.

Real-World Applications

Simplifying fractions has many real-world applications, including:

  • Cooking: When a recipe calls for a fraction of an ingredient, you need to simplify the fraction to know how much to use.
  • Building: When building a structure, you need to simplify fractions to know how much material to use.
  • Science: When conducting experiments, you need to simplify fractions to know how much of a substance to use.

Introduction

Simplifying fractions is a fundamental concept in mathematics, and it's essential to understand the concepts and techniques involved. In this article, we'll answer some of the most frequently asked questions about simplifying fractions, providing step-by-step solutions and explanations.

Q: What is a fraction?

A: A fraction is a way of expressing a part of a whole. It consists of a numerator (the top number) and a denominator (the bottom number). For example, 34\frac{3}{4} is a fraction where 3 is the numerator and 4 is the denominator.

Q: How do I simplify a fraction?

A: To simplify a fraction, you need to find the greatest common divisor (GCD) of the numerator and the denominator and divide both numbers by the GCD. For example, to simplify 68\frac{6}{8}, you need to find the GCD of 6 and 8, which is 2. Then, you divide both numbers by 2 to get 34\frac{3}{4}.

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

A: The least common multiple (LCM) is the smallest number that is a multiple of two or more numbers. For example, the LCM of 4 and 6 is 12, because 12 is the smallest number that is a multiple of both 4 and 6.

Q: How do I add fractions with different denominators?

A: To add fractions with different denominators, you need to find the least common multiple (LCM) of the denominators and rewrite each fraction with the LCM as the denominator. Then, you can add the fractions. For example, to add 14\frac{1}{4} and 16\frac{1}{6}, you need to find the LCM of 4 and 6, which is 12. Then, you rewrite each fraction with 12 as the denominator: 312\frac{3}{12} and 212\frac{2}{12}. Finally, you can add the fractions: 312+212=512\frac{3}{12} + \frac{2}{12} = \frac{5}{12}.

Q: How do I subtract fractions with different denominators?

A: To subtract fractions with different denominators, you need to find the least common multiple (LCM) of the denominators and rewrite each fraction with the LCM as the denominator. Then, you can subtract the fractions. For example, to subtract 34\frac{3}{4} and 26\frac{2}{6}, you need to find the LCM of 4 and 6, which is 12. Then, you rewrite each fraction with 12 as the denominator: 912\frac{9}{12} and 412\frac{4}{12}. Finally, you can subtract the fractions: 912−412=512\frac{9}{12} - \frac{4}{12} = \frac{5}{12}.

Q: How do I multiply a fraction by a whole number?

A: To multiply a fraction by a whole number, you can multiply the numerator by the whole number and keep the denominator the same. For example, to multiply 34\frac{3}{4} by 5, you can multiply the numerator by 5 and keep the denominator the same: 154\frac{15}{4}.

Q: How do I divide a fraction by a whole number?

A: To divide a fraction by a whole number, you can multiply the fraction by the reciprocal of the whole number. For example, to divide 34\frac{3}{4} by 5, you can multiply the fraction by the reciprocal of 5, which is 15\frac{1}{5}: 34×15=320\frac{3}{4} \times \frac{1}{5} = \frac{3}{20}.

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

A: A fraction is a way of expressing a part of a whole, while a decimal is a way of expressing a number as a sum of powers of 10. For example, the fraction 34\frac{3}{4} is equal to the decimal 0.75.

Q: How do I convert a fraction to a decimal?

A: To convert a fraction to a decimal, you can divide the numerator by the denominator. For example, to convert 34\frac{3}{4} to a decimal, you can divide 3 by 4: 0.75.

Conclusion

Simplifying fractions is a fundamental concept in mathematics, and it's essential to understand the concepts and techniques involved. By following the steps outlined in this article, you can simplify fractions with ease and tackle complex problems with confidence. Remember to always find the greatest common divisor (GCD) of the numerator and the denominator and divide both numbers by the GCD. With practice, you will become proficient in simplifying fractions and be able to tackle more complex problems.

Tips and Tricks

  • Always find the least common multiple (LCM) of the denominators when adding or subtracting fractions.
  • Use the greatest common divisor (GCD) to simplify fractions.
  • When multiplying a fraction by a whole number, multiply the numerator by the whole number and keep the denominator the same.
  • Practice, practice, practice! The more you practice simplifying fractions, the more comfortable you will become with the concepts.

Common Mistakes to Avoid

  • Not finding the least common multiple (LCM) of the denominators when adding or subtracting fractions.
  • Not using the greatest common divisor (GCD) to simplify fractions.
  • Not multiplying the numerator by the whole number when multiplying a fraction by a whole number.
  • Not writing the answer as a mixed number when the answer is greater than one.

Real-World Applications

Simplifying fractions has many real-world applications, including:

  • Cooking: When a recipe calls for a fraction of an ingredient, you need to simplify the fraction to know how much to use.
  • Building: When building a structure, you need to simplify fractions to know how much material to use.
  • Science: When conducting experiments, you need to simplify fractions to know how much of a substance to use.

By mastering the art of simplifying fractions, you will be able to tackle complex problems and make informed decisions in your personal and professional life.