Solve The Equation:$\[ 8 \frac{7}{18} - \left(1 \frac{5}{6} + Y\right) = 3 \frac{4}{15} + 2 \frac{13}{45} \\]

by ADMIN 110 views

Introduction

When it comes to solving equations involving mixed numbers and fractions, it can be a daunting task, especially for those who are not familiar with the concept. However, with a clear understanding of the steps involved and a bit of practice, anyone can master the art of simplifying mixed numbers and fractions to solve complex equations. In this article, we will delve into the world of mixed numbers and fractions, exploring the steps to simplify them and solve the given equation.

Understanding Mixed Numbers and Fractions

Before we dive into the solution, let's take a moment to understand what mixed numbers and fractions are. A mixed number is a combination of a whole number and a fraction, while a fraction is a part of a whole. For example, 3 1/2 is a mixed number, where 3 is the whole number and 1/2 is the fraction. On the other hand, 1/2 is a fraction, where 1 is the numerator and 2 is the denominator.

Simplifying Mixed Numbers and Fractions

To simplify mixed numbers and fractions, we need to follow a few simple steps:

  1. Convert the mixed number to an improper fraction: To convert a mixed number to an improper fraction, we multiply the whole number by the denominator and add the numerator. For example, 3 1/2 can be converted to an improper fraction as follows: 3 x 2 + 1 = 7/2.
  2. Find the least common multiple (LCM): To add or subtract fractions, we need to find the least common multiple (LCM) of the denominators. The LCM is the smallest number that both denominators can divide into evenly.
  3. Add or subtract the fractions: Once we have the LCM, we can add or subtract the fractions by multiplying each fraction by the LCM and then adding or subtracting the numerators.

Solving the Equation

Now that we have a clear understanding of mixed numbers and fractions, let's move on to solving the equation:

8718−(156+y)=3415+21345{ 8 \frac{7}{18} - \left(1 \frac{5}{6} + y\right) = 3 \frac{4}{15} + 2 \frac{13}{45} }

To solve this equation, we need to follow the steps outlined above:

  1. Convert the mixed numbers to improper fractions: We can convert the mixed numbers to improper fractions as follows:

8718=15118{ 8 \frac{7}{18} = \frac{151}{18} } 156=116{ 1 \frac{5}{6} = \frac{11}{6} } 3415=4915{ 3 \frac{4}{15} = \frac{49}{15} } 21345=7345{ 2 \frac{13}{45} = \frac{73}{45} }

  1. Find the LCM: To add or subtract the fractions, we need to find the LCM of the denominators. The LCM of 18, 6, 15, and 45 is 90.

  2. Add or subtract the fractions: Now that we have the LCM, we can add or subtract the fractions by multiplying each fraction by the LCM and then adding or subtracting the numerators.

15118−(116+y)=4915+7345{ \frac{151}{18} - \left(\frac{11}{6} + y\right) = \frac{49}{15} + \frac{73}{45} }

To add or subtract the fractions, we need to find the LCM of 18, 6, 15, and 45, which is 90. We can then multiply each fraction by the LCM and add or subtract the numerators.

15118×55=75590{ \frac{151}{18} \times \frac{5}{5} = \frac{755}{90} } 116×1515=16590{ \frac{11}{6} \times \frac{15}{15} = \frac{165}{90} } 4915×66=29490{ \frac{49}{15} \times \frac{6}{6} = \frac{294}{90} } 7345×22=14690{ \frac{73}{45} \times \frac{2}{2} = \frac{146}{90} }

Now that we have the fractions multiplied by the LCM, we can add or subtract the numerators.

75590−16590−y=29490+14690{ \frac{755}{90} - \frac{165}{90} - y = \frac{294}{90} + \frac{146}{90} }

Simplifying the equation, we get:

59090−y=44090{ \frac{590}{90} - y = \frac{440}{90} }

To solve for y, we can subtract 440/90 from both sides of the equation.

59090−44090=y{ \frac{590}{90} - \frac{440}{90} = y }

Simplifying the equation, we get:

15090=y{ \frac{150}{90} = y }

Simplifying further, we get:

53=y{ \frac{5}{3} = y }

Therefore, the value of y is 5/3.

Conclusion

In this article, we have explored the world of mixed numbers and fractions, learning how to simplify them and solve complex equations. We have taken a step-by-step approach to solving the given equation, converting mixed numbers to improper fractions, finding the least common multiple (LCM), and adding or subtracting the fractions. With practice and patience, anyone can master the art of simplifying mixed numbers and fractions to solve complex equations.

Frequently Asked Questions

  • What is a mixed number? A mixed number is a combination of a whole number and a fraction.
  • What is a fraction? A fraction is a part of a whole.
  • How do I convert a mixed number to an improper fraction? To convert a mixed number to an improper fraction, multiply the whole number by the denominator and add the numerator.
  • How do I find the least common multiple (LCM)? To find the LCM, list the multiples of each number and find the smallest number that both numbers can divide into evenly.
  • How do I add or subtract fractions? To add or subtract fractions, find the least common multiple (LCM) of the denominators, multiply each fraction by the LCM, and then add or subtract the numerators.

Further Reading

  • Simplifying Fractions
  • Adding and Subtracting Fractions
  • Multiplying and Dividing Fractions
  • Mixed Numbers and Fractions: A Comprehensive Guide

Note: The above article is a comprehensive guide to solving the equation involving mixed numbers and fractions. It provides a step-by-step approach to simplifying mixed numbers and fractions and solving complex equations. The article also includes frequently asked questions and further reading resources for those who want to learn more about the topic.

Introduction

Mixed numbers and fractions can be a challenging topic for many students and professionals. With so many rules and formulas to remember, it's easy to get confused and lose track of what's going on. But don't worry, we're here to help! In this article, we'll answer some of the most frequently asked questions about mixed numbers and fractions, covering topics from the basics to more advanced concepts.

Q&A: Mixed Numbers and Fractions

Q: What is a mixed number?

A: A mixed number is a combination of a whole number and a fraction. For example, 3 1/2 is a mixed number, where 3 is the whole number and 1/2 is the fraction.

Q: What is a fraction?

A: A fraction is a part of a whole. For example, 1/2 is a fraction, where 1 is the numerator and 2 is the denominator.

Q: How do I convert a mixed number to an improper fraction?

A: To convert a mixed number to an improper fraction, multiply the whole number by the denominator and add the numerator. For example, 3 1/2 can be converted to an improper fraction as follows: 3 x 2 + 1 = 7/2.

Q: How do I find the least common multiple (LCM)?

A: To find the LCM, list the multiples of each number and find the smallest number that both numbers can divide into evenly. For example, the LCM of 6 and 8 is 24.

Q: How do I add or subtract fractions?

A: To add or subtract fractions, find the least common multiple (LCM) of the denominators, multiply each fraction by the LCM, and then add or subtract the numerators. For example, to add 1/2 and 1/4, find the LCM of 2 and 4, which is 4. Multiply each fraction by the LCM: 1/2 x 2/2 = 2/4 and 1/4 x 1/1 = 1/4. Then add the numerators: 2/4 + 1/4 = 3/4.

Q: How do I multiply and divide fractions?

A: To multiply fractions, multiply the numerators and denominators separately. For example, to multiply 1/2 and 1/4, multiply the numerators: 1 x 1 = 1, and multiply the denominators: 2 x 4 = 8. Then write the result as a fraction: 1/8.

To divide fractions, invert the second fraction and multiply. For example, to divide 1/2 by 1/4, invert the second fraction: 1/4 becomes 4/1. Then multiply the fractions: 1/2 x 4/1 = 4/2 = 2.

Q: What is the difference between a mixed number and an improper fraction?

A: A mixed number is a combination of a whole number and a fraction, while an improper fraction is a fraction with a numerator that is greater than or equal to the denominator.

Q: How do I simplify a fraction?

A: To simplify a fraction, find the greatest common divisor (GCD) of the numerator and denominator, and divide both numbers by the GCD. For example, to simplify 6/8, find the GCD of 6 and 8, which is 2. Then divide both numbers by 2: 6/2 = 3 and 8/2 = 4. The simplified fraction is 3/4.

Q: What is the order of operations for fractions?

A: The order of operations for fractions is the same as for whole numbers: parentheses, exponents, multiplication and division (from left to right), and addition and subtraction (from left to right).

Q: Can I use a calculator to simplify fractions?

A: Yes, you can use a calculator to simplify fractions. However, it's always a good idea to check your work by hand to make sure the calculator is giving you the correct answer.

Conclusion

We hope this Q&A article has helped you understand mixed numbers and fractions better. Remember, practice makes perfect, so be sure to try out the examples and exercises in this article to reinforce your understanding. If you have any more questions or need further clarification, don't hesitate to ask.

Frequently Asked Questions

  • What is a mixed number? A mixed number is a combination of a whole number and a fraction.
  • What is a fraction? A fraction is a part of a whole.
  • How do I convert a mixed number to an improper fraction? To convert a mixed number to an improper fraction, multiply the whole number by the denominator and add the numerator.
  • How do I find the least common multiple (LCM)? To find the LCM, list the multiples of each number and find the smallest number that both numbers can divide into evenly.
  • How do I add or subtract fractions? To add or subtract fractions, find the least common multiple (LCM) of the denominators, multiply each fraction by the LCM, and then add or subtract the numerators.

Further Reading

  • Simplifying Fractions
  • Adding and Subtracting Fractions
  • Multiplying and Dividing Fractions
  • Mixed Numbers and Fractions: A Comprehensive Guide

Note: The above article is a comprehensive Q&A guide to mixed numbers and fractions. It covers topics from the basics to more advanced concepts, providing clear and concise answers to frequently asked questions. The article also includes frequently asked questions and further reading resources for those who want to learn more about the topic.