Simplify The Expression:$\[ \left(\frac{1}{10} + \frac{7}{10}\right) \times \frac{3}{4} \\]

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Introduction

In this article, we will simplify the given expression using basic algebraic operations. The expression is a combination of fractions, and our goal is to simplify it to its simplest form. We will use the order of operations (PEMDAS) to guide us through the process.

Understanding the Expression

The given expression is:

(110+710)×34\left(\frac{1}{10} + \frac{7}{10}\right) \times \frac{3}{4}

This expression consists of two fractions added together inside parentheses, and then multiplied by another fraction.

Step 1: Simplify the Expression Inside the Parentheses

To simplify the expression inside the parentheses, we need to add the two fractions together. Since the denominators are the same (10), we can simply add the numerators:

110+710=1+710=810\frac{1}{10} + \frac{7}{10} = \frac{1+7}{10} = \frac{8}{10}

Now, we can rewrite the original expression as:

810×34\frac{8}{10} \times \frac{3}{4}

Step 2: Simplify the Expression by Multiplying the Fractions

To simplify the expression, we need to multiply the two fractions together. To do this, we multiply the numerators and the denominators separately:

810×34=8×310×4=2440\frac{8}{10} \times \frac{3}{4} = \frac{8 \times 3}{10 \times 4} = \frac{24}{40}

Step 3: Simplify the Fraction to its Simplest Form

To simplify the fraction to its simplest form, we need to find the greatest common divisor (GCD) of the numerator and the denominator. The GCD of 24 and 40 is 8. We can divide both the numerator and the denominator by 8 to simplify the fraction:

2440=24÷840÷8=35\frac{24}{40} = \frac{24 \div 8}{40 \div 8} = \frac{3}{5}

Conclusion

In this article, we simplified the given expression using basic algebraic operations. We started by simplifying the expression inside the parentheses, then multiplied the fractions together, and finally simplified the fraction to its simplest form. The simplified expression is:

35\frac{3}{5}

Final Answer

The final answer is 35\boxed{\frac{3}{5}}.

Additional Tips and Tricks

  • When simplifying expressions, always follow the order of operations (PEMDAS).
  • When adding or subtracting fractions, make sure the denominators are the same.
  • When multiplying fractions, multiply the numerators and the denominators separately.
  • When simplifying fractions, find the greatest common divisor (GCD) of the numerator and the denominator.

Common Mistakes to Avoid

  • Not following the order of operations (PEMDAS).
  • Not simplifying fractions to their simplest form.
  • Not finding the greatest common divisor (GCD) of the numerator and the denominator.

Real-World Applications

Simplifying expressions is an essential skill in mathematics, and it has many real-world applications. For example, in finance, simplifying expressions can help you calculate interest rates and investment returns. In science, simplifying expressions can help you model complex systems and make predictions. In engineering, simplifying expressions can help you design and optimize systems.

Conclusion

Introduction

In our previous article, we simplified the given expression using basic algebraic operations. In this article, we will answer some frequently asked questions (FAQs) related to simplifying expressions.

Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that tells us which operations to perform first when simplifying expressions. PEMDAS stands for:

  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.

Q: How do I simplify fractions to their simplest form?

A: To simplify fractions to their simplest form, you need to find the greatest common divisor (GCD) of the numerator and the denominator. The GCD is the largest number that divides both the numerator and the denominator without leaving a remainder. Once you find the GCD, you can divide both the numerator and the denominator by the GCD to simplify the fraction.

Q: What is the greatest common divisor (GCD)?

A: The greatest common divisor (GCD) is the largest number that divides both the numerator and the denominator of a fraction without leaving a remainder. For example, the GCD of 24 and 40 is 8, because 8 is the largest number that divides both 24 and 40 without leaving a remainder.

Q: How do I add or subtract fractions?

A: To add or subtract fractions, you need to have the same denominator. If the denominators are different, you need to find the least common multiple (LCM) of the denominators and convert both fractions to have the LCM as the denominator. Once you have the same denominator, you can add or subtract the numerators.

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

A: The least common multiple (LCM) is the smallest number that is a multiple of both 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 multiply fractions?

A: To multiply fractions, you need to multiply the numerators and the denominators separately. For example, to multiply 1/2 and 3/4, you would multiply the numerators (1 and 3) to get 3, and multiply the denominators (2 and 4) to get 8, resulting in 3/8.

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

A: Some common mistakes to avoid when simplifying expressions include:

  • Not following the order of operations (PEMDAS)
  • Not simplifying fractions to their simplest form
  • Not finding the greatest common divisor (GCD) of the numerator and the denominator
  • Not having the same denominator when adding or subtracting fractions

Q: How do I apply simplifying expressions in real-world scenarios?

A: Simplifying expressions is an essential skill in mathematics that has many real-world applications. For example, in finance, simplifying expressions can help you calculate interest rates and investment returns. In science, simplifying expressions can help you model complex systems and make predictions. In engineering, simplifying expressions can help you design and optimize systems.

Conclusion

In conclusion, simplifying expressions is a fundamental skill in mathematics that has many real-world applications. By following the order of operations (PEMDAS) and simplifying fractions to their simplest form, you can simplify complex expressions and make calculations easier. Remember to avoid common mistakes and find the greatest common divisor (GCD) of the numerator and the denominator to simplify fractions to their simplest form.