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

In mathematics, simplifying expressions is an essential skill that helps us to solve problems more efficiently and accurately. It involves combining like terms, removing unnecessary components, and expressing the result in the simplest form possible. In this article, we will focus on simplifying three different types of expressions: adding fractions, subtracting fractions, and multiplying fractions.

Simplifying Expressions: A Step-by-Step Guide

Adding Fractions: A Step-by-Step Guide

When adding fractions, we need to find a common denominator, which is the least common multiple (LCM) of the denominators of the two fractions. Once we have the common denominator, we can add the numerators and keep the common denominator.

Example: 35+14\frac{3}{5}+\frac{1}{4}

To simplify this expression, we need to find the LCM of 5 and 4, which is 20. We can rewrite each fraction with the common denominator:

35=3×45×4=1220\frac{3}{5}=\frac{3\times4}{5\times4}=\frac{12}{20}

14=1×54×5=520\frac{1}{4}=\frac{1\times5}{4\times5}=\frac{5}{20}

Now, we can add the numerators:

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

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

Subtracting Fractions: A Step-by-Step Guide

When subtracting fractions, we need to find a common denominator, which is the LCM of the denominators of the two fractions. Once we have the common denominator, we can subtract the numerators and keep the common denominator.

Example: 34−23\frac{3}{4}-\frac{2}{3}

To simplify this expression, we need to find the LCM of 4 and 3, which is 12. We can rewrite each fraction with the common denominator:

34=3×34×3=912\frac{3}{4}=\frac{3\times3}{4\times3}=\frac{9}{12}

23=2×43×4=812\frac{2}{3}=\frac{2\times4}{3\times4}=\frac{8}{12}

Now, we can subtract the numerators:

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

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

Multiplying Fractions: A Step-by-Step Guide

When multiplying fractions, we can simply multiply the numerators and denominators separately.

Example: 512×3145\frac{1}{2}\times3\frac{1}{4}

To simplify this expression, we need to convert the mixed numbers to improper fractions:

512=1125\frac{1}{2}=\frac{11}{2}

314=1343\frac{1}{4}=\frac{13}{4}

Now, we can multiply the fractions:

112×134=11×132×4=1438\frac{11}{2}\times\frac{13}{4}=\frac{11\times13}{2\times4}=\frac{143}{8}

Therefore, the simplified expression is 1438\frac{143}{8}.

Conclusion

Simplifying expressions is an essential skill in mathematics that helps us to solve problems more efficiently and accurately. By following the steps outlined in this article, we can simplify expressions involving adding fractions, subtracting fractions, and multiplying fractions. Remember to always find the common denominator, multiply the numerators and denominators separately, and express the result in the simplest form possible.

Final Thoughts

Simplifying expressions is not just about following a set of rules; it's about understanding the underlying concepts and applying them to real-world problems. By mastering the art of simplifying expressions, we can solve complex problems with ease and confidence. So, the next time you encounter a complex expression, remember to simplify it step by step, and you'll be amazed at how easily you can solve it.

Additional Resources

For more information on simplifying expressions, check out the following resources:

  • Khan Academy: Simplifying Expressions
  • Mathway: Simplifying Expressions
  • IXL: Simplifying Expressions

FAQs

Q: What is the difference between adding and subtracting fractions? A: When adding fractions, we need to find a common denominator and add the numerators. When subtracting fractions, we need to find a common denominator and subtract the numerators.

Q: How do I find the common denominator of two fractions? A: To find the common denominator, we need to find the least common multiple (LCM) of the denominators of the two fractions.

Q: Can I simplify expressions involving decimals? A: Yes, we can simplify expressions involving decimals by converting them to fractions and then simplifying the expression.

Introduction

In our previous article, we discussed the importance of simplifying expressions in mathematics. We covered the basics of adding, subtracting, and multiplying fractions, and provided step-by-step guides on how to simplify expressions. In this article, we will answer some of the most frequently asked questions about simplifying expressions.

Q&A: Simplifying Expressions

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

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

Example:

  • Adding fractions: 35+14\frac{3}{5}+\frac{1}{4}
  • Subtracting fractions: 34−23\frac{3}{4}-\frac{2}{3}

Q: How do I find the common denominator of two fractions?

A: To find the common denominator, we need to find the least common multiple (LCM) of the denominators of the two fractions.

Example:

  • Find the LCM of 5 and 4: 20
  • Find the LCM of 4 and 3: 12

Q: Can I simplify expressions involving decimals?

A: Yes, we can simplify expressions involving decimals by converting them to fractions and then simplifying the expression.

Example:

  • Convert 0.5 to a fraction: 12\frac{1}{2}
  • Convert 0.25 to a fraction: 14\frac{1}{4}

Q: What is the importance of simplifying expressions?

A: Simplifying expressions is essential in mathematics because it helps us to solve problems more efficiently and accurately. It also helps us to understand the underlying concepts and apply them to real-world problems.

Example:

  • Simplify the expression: 34+23\frac{3}{4}+\frac{2}{3}
  • Simplify the expression: 512×3145\frac{1}{2}\times3\frac{1}{4}

Q: Can I simplify expressions involving mixed numbers?

A: Yes, we can simplify expressions involving mixed numbers by converting them to improper fractions and then simplifying the expression.

Example:

  • Convert 5125\frac{1}{2} to an improper fraction: 112\frac{11}{2}
  • Convert 3143\frac{1}{4} to an improper fraction: 134\frac{13}{4}

Q: What is the difference between a numerator and a denominator?

A: The numerator is the top number of a fraction, and the denominator is the bottom number of a fraction.

Example:

  • Numerator: 3
  • Denominator: 4

Q: Can I simplify expressions involving negative fractions?

A: Yes, we can simplify expressions involving negative fractions by following the same steps as simplifying positive fractions.

Example:

  • Simplify the expression: −34+23-\frac{3}{4}+\frac{2}{3}
  • Simplify the expression: −512×314-5\frac{1}{2}\times3\frac{1}{4}

Q: What is the importance of simplifying expressions in real-world problems?

A: Simplifying expressions is essential in real-world problems because it helps us to solve problems more efficiently and accurately. It also helps us to understand the underlying concepts and apply them to real-world problems.

Example:

  • Simplify the expression: 34+23\frac{3}{4}+\frac{2}{3} in a real-world problem
  • Simplify the expression: 512×3145\frac{1}{2}\times3\frac{1}{4} in a real-world problem

Conclusion

Simplifying expressions is an essential skill in mathematics that helps us to solve problems more efficiently and accurately. By following the steps outlined in this article, we can simplify expressions involving adding, subtracting, and multiplying fractions, as well as expressions involving decimals, mixed numbers, and negative fractions. Remember to always find the common denominator, multiply the numerators and denominators separately, and express the result in the simplest form possible.

Final Thoughts

Simplifying expressions is not just about following a set of rules; it's about understanding the underlying concepts and applying them to real-world problems. By mastering the art of simplifying expressions, we can solve complex problems with ease and confidence. So, the next time you encounter a complex expression, remember to simplify it step by step, and you'll be amazed at how easily you can solve it.

Additional Resources

For more information on simplifying expressions, check out the following resources:

  • Khan Academy: Simplifying Expressions
  • Mathway: Simplifying Expressions
  • IXL: Simplifying Expressions

FAQs

Q: What is the difference between adding and subtracting fractions? A: When adding fractions, we need to find a common denominator and add the numerators. When subtracting fractions, we need to find a common denominator and subtract the numerators.

Q: How do I find the common denominator of two fractions? A: To find the common denominator, we need to find the least common multiple (LCM) of the denominators of the two fractions.

Q: Can I simplify expressions involving decimals? A: Yes, we can simplify expressions involving decimals by converting them to fractions and then simplifying the expression.

Q: What is the importance of simplifying expressions? A: Simplifying expressions is essential in mathematics because it helps us to solve problems more efficiently and accurately. It also helps us to understand the underlying concepts and apply them to real-world problems.