Simplify The Expression: 3 5 + 2 3 ⋅ 1 5 + 1 3 \frac{3}{5} + \frac{2}{3} \cdot \frac{1}{5} + \frac{1}{3} 5 3 ​ + 3 2 ​ ⋅ 5 1 ​ + 3 1 ​

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Introduction

When dealing with fractions, it's not uncommon to encounter expressions that involve multiple fractions. In this article, we'll explore how to simplify the expression 35+23⋅15+13\frac{3}{5} + \frac{2}{3} \cdot \frac{1}{5} + \frac{1}{3} using a step-by-step approach.

Understanding the Expression

The given expression involves three fractions: 35\frac{3}{5}, 23⋅15\frac{2}{3} \cdot \frac{1}{5}, and 13\frac{1}{3}. To simplify this expression, we need to follow the order of operations (PEMDAS):

  1. Parentheses: Evaluate expressions inside parentheses first.
  2. Exponents: Evaluate any exponential expressions next.
  3. Multiplication and Division: Evaluate multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Step 1: Multiply the Fractions

The first step is to multiply the fractions 23\frac{2}{3} and 15\frac{1}{5}. To do this, we multiply the numerators (2 and 1) and multiply the denominators (3 and 5):

23⋅15=2⋅13⋅5=215\frac{2}{3} \cdot \frac{1}{5} = \frac{2 \cdot 1}{3 \cdot 5} = \frac{2}{15}

Step 2: Add the Fractions

Now that we have the product of the fractions, we can add the fractions together. However, the fractions have different denominators, so we need to find a common denominator. The least common multiple (LCM) of 5 and 15 is 15. We can rewrite the fractions with the common denominator:

35=3⋅35⋅3=915\frac{3}{5} = \frac{3 \cdot 3}{5 \cdot 3} = \frac{9}{15}

215=215\frac{2}{15} = \frac{2}{15}

13=1⋅53⋅5=515\frac{1}{3} = \frac{1 \cdot 5}{3 \cdot 5} = \frac{5}{15}

Now that the fractions have the same denominator, we can add them together:

915+215+515=1615\frac{9}{15} + \frac{2}{15} + \frac{5}{15} = \frac{16}{15}

Conclusion

In this article, we simplified the expression 35+23⋅15+13\frac{3}{5} + \frac{2}{3} \cdot \frac{1}{5} + \frac{1}{3} using a step-by-step approach. We first multiplied the fractions 23\frac{2}{3} and 15\frac{1}{5} to get 215\frac{2}{15}. Then, we added the fractions together, finding a common denominator of 15. The final simplified expression is 1615\frac{16}{15}.

Tips and Tricks

When dealing with fractions, it's essential to follow the order of operations and find a common denominator when adding fractions with different denominators. Additionally, make sure to multiply the numerators and denominators correctly when multiplying fractions.

Common Mistakes

When simplifying expressions involving fractions, it's easy to make mistakes. Some common mistakes include:

  • Not following the order of operations
  • Not finding a common denominator when adding fractions
  • Multiplying the numerators and denominators incorrectly

Real-World Applications

Simplifying expressions involving fractions has many real-world applications. For example, in cooking, you may need to combine different ingredients with different measurements. In finance, you may need to calculate interest rates or investment returns. In science, you may need to calculate probabilities or statistics.

Final Thoughts

Simplifying expressions involving fractions may seem daunting at first, but with practice and patience, it becomes second nature. By following the order of operations and finding a common denominator, you can simplify even the most complex expressions. Remember to always double-check your work and make sure to multiply the numerators and denominators correctly.

Additional Resources

If you're struggling with simplifying expressions involving fractions, here are some additional resources to help you:

  • Khan Academy: Fractions and Decimals
  • Mathway: Simplifying Fractions
  • Wolfram Alpha: Simplifying Fractions

Frequently Asked Questions

Q: What is the order of operations? A: The order of operations is PEMDAS: Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction.

Q: How do I find a common denominator? A: To find a common denominator, you can list the multiples of each denominator and find the least common multiple (LCM).

Q: What is the difference between adding and multiplying fractions? A: When adding fractions, you need to find a common denominator. When multiplying fractions, you multiply the numerators and denominators separately.

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Introduction

In our previous article, we explored how to simplify the expression 35+23⋅15+13\frac{3}{5} + \frac{2}{3} \cdot \frac{1}{5} + \frac{1}{3} using a step-by-step approach. However, we know that math can be a challenging subject, and sometimes it's helpful to have a Q&A guide to clarify any doubts. In this article, we'll answer some frequently asked questions about simplifying expressions involving fractions.

Q&A: Simplifying Expressions Involving Fractions

Q: What is the order of operations when simplifying expressions involving fractions?

A: The order of operations is PEMDAS: Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction.

Q: How do I find a common denominator when adding fractions?

A: To find a common denominator, you can list the multiples of each denominator and find the least common multiple (LCM). Alternatively, you can use the following formula:

  • If the denominators are 2 and 3, the common denominator is 6.
  • If the denominators are 3 and 4, the common denominator is 12.
  • If the denominators are 5 and 7, the common denominator is 35.

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

A: When adding fractions, you need to find a common denominator. When multiplying fractions, you multiply the numerators and denominators separately.

Q: Can I simplify an expression involving fractions by canceling out common factors?

A: Yes, you can simplify an expression involving fractions by canceling out common factors. For example:

68=34\frac{6}{8} = \frac{3}{4}

Q: How do I simplify an expression involving fractions with negative numbers?

A: To simplify an expression involving fractions with negative numbers, you can follow the same steps as before. For example:

−34+23=−912+812=−112-\frac{3}{4} + \frac{2}{3} = -\frac{9}{12} + \frac{8}{12} = -\frac{1}{12}

Q: Can I simplify an expression involving fractions with decimals?

A: Yes, you can simplify an expression involving fractions with decimals. For example:

34+0.5=34+24=54\frac{3}{4} + 0.5 = \frac{3}{4} + \frac{2}{4} = \frac{5}{4}

Q: How do I simplify an expression involving fractions with mixed numbers?

A: To simplify an expression involving fractions with mixed numbers, you can convert the mixed number to an improper fraction. For example:

213+23=73+23=93=32\frac{1}{3} + \frac{2}{3} = \frac{7}{3} + \frac{2}{3} = \frac{9}{3} = 3

Q: Can I simplify an expression involving fractions with variables?

A: Yes, you can simplify an expression involving fractions with variables. For example:

xy+2y=x+2y\frac{x}{y} + \frac{2}{y} = \frac{x+2}{y}

Conclusion

In this article, we've answered some frequently asked questions about simplifying expressions involving fractions. We've covered topics such as the order of operations, finding a common denominator, adding and multiplying fractions, and simplifying expressions with negative numbers, decimals, and variables. By following these steps and practicing regularly, you'll become more confident in your ability to simplify expressions involving fractions.

Tips and Tricks

  • When simplifying expressions involving fractions, make sure to follow the order of operations.
  • Use a common denominator when adding fractions.
  • Multiply the numerators and denominators separately when multiplying fractions.
  • Cancel out common factors to simplify expressions.
  • Convert mixed numbers to improper fractions when necessary.

Common Mistakes

  • Not following the order of operations.
  • Not finding a common denominator when adding fractions.
  • Multiplying the numerators and denominators incorrectly.
  • Not canceling out common factors.

Real-World Applications

Simplifying expressions involving fractions has many real-world applications. For example, in cooking, you may need to combine different ingredients with different measurements. In finance, you may need to calculate interest rates or investment returns. In science, you may need to calculate probabilities or statistics.

Final Thoughts

Simplifying expressions involving fractions may seem daunting at first, but with practice and patience, it becomes second nature. By following the order of operations and finding a common denominator, you can simplify even the most complex expressions. Remember to always double-check your work and make sure to multiply the numerators and denominators correctly.

Additional Resources

If you're struggling with simplifying expressions involving fractions, here are some additional resources to help you:

  • Khan Academy: Fractions and Decimals
  • Mathway: Simplifying Fractions
  • Wolfram Alpha: Simplifying Fractions

Frequently Asked Questions

Q: What is the order of operations? A: The order of operations is PEMDAS: Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction.

Q: How do I find a common denominator? A: To find a common denominator, you can list the multiples of each denominator and find the least common multiple (LCM).

Q: What is the difference between adding and multiplying fractions? A: When adding fractions, you need to find a common denominator. When multiplying fractions, you multiply the numerators and denominators separately.

Q: Can I simplify an expression involving fractions by canceling out common factors? A: Yes, you can simplify an expression involving fractions by canceling out common factors.

Q: How do I simplify an expression involving fractions with negative numbers? A: To simplify an expression involving fractions with negative numbers, you can follow the same steps as before.

Q: Can I simplify an expression involving fractions with decimals? A: Yes, you can simplify an expression involving fractions with decimals.

Q: How do I simplify an expression involving fractions with mixed numbers? A: To simplify an expression involving fractions with mixed numbers, you can convert the mixed number to an improper fraction.

Q: Can I simplify an expression involving fractions with variables? A: Yes, you can simplify an expression involving fractions with variables.