Which Number Makes The Equation True?$-8 \frac{1}{2} + ? = -5 \frac{1}{2}$

Introduction

In mathematics, equations are a fundamental concept that helps us understand and solve problems. An equation is a statement that expresses the equality of two mathematical expressions. In this article, we will explore how to solve an equation that involves fractions and mixed numbers. We will use the equation $-8 \frac{1}{2} + ? = -5 \frac{1}{2}$ as an example to demonstrate the steps involved in solving it.

Understanding the Equation

The given equation is $-8 \frac{1}{2} + ? = -5 \frac{1}{2}$. To solve this equation, we need to isolate the variable, which in this case is the unknown number represented by the question mark. The equation involves a mixed number, which is a combination of a whole number and a fraction. In this case, the mixed number is $-8 \frac{1}{2}$.

Converting Mixed Numbers to Improper Fractions

To simplify the equation, we can convert the mixed number $-8 \frac{1}{2}$ to an improper fraction. An improper fraction is a fraction where the numerator is greater than or equal to the denominator. To convert a mixed number to an improper fraction, we multiply the whole number by the denominator and then add the numerator.

-8 \frac{1}{2} = \frac{(-8 \times 2) + 1}{2} = \frac{-16 + 1}{2} = \frac{-15}{2}

Rewriting the Equation

Now that we have converted the mixed number to an improper fraction, we can rewrite the equation as follows:

$$\frac{-15}{2} + ? = -5 \frac{1}{2}$$

Converting the Right-Hand Side to an Improper Fraction

To make it easier to solve the equation, we can convert the right-hand side to an improper fraction. We can do this by converting the mixed number $-5 \frac{1}{2}$ to an improper fraction.

-5 \frac{1}{2} = \frac{(-5 \times 2) + 1}{2} = \frac{-10 + 1}{2} = \frac{-9}{2}

Rewriting the Equation Again

Now that we have converted the right-hand side to an improper fraction, we can rewrite the equation as follows:

$$\frac{-15}{2} + ? = \frac{-9}{2}$$

Isolating the Variable

To isolate the variable, we need to get rid of the fraction on the left-hand side. We can do this by multiplying both sides of the equation by 2.

2 \times \left(\frac{-15}{2} + ?\right) = 2 \times \frac{-9}{2}

Simplifying the Equation

After multiplying both sides of the equation by 2, we get:

$$-15 + 2? = -9$$

Solving for the Variable

To solve for the variable, we need to isolate it on one side of the equation. We can do this by adding 15 to both sides of the equation.

2? = -9 + 15
2? = 6

Dividing Both Sides by 2

To find the value of the variable, we need to divide both sides of the equation by 2.

? = \frac{6}{2}
? = 3

Conclusion

In this article, we have solved the equation $-8 \frac{1}{2} + ? = -5 \frac{1}{2}$ by converting the mixed numbers to improper fractions, rewriting the equation, isolating the variable, and solving for it. We have demonstrated the steps involved in solving an equation that involves fractions and mixed numbers. By following these steps, you can solve similar equations and become proficient in solving mathematical problems.

Final Answer

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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 where the numerator is greater than or equal to the denominator.

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

A: To convert a mixed number to an improper fraction, you multiply the whole number by the denominator and then add the numerator. For example, to convert $-8 \frac{1}{2}$ to an improper fraction, you would multiply $-8$ by $2$ and then add $1$, resulting in $\frac{-15}{2}$.

Q: How do I convert a mixed number to a decimal?

A: To convert a mixed number to a decimal, you can divide the numerator by the denominator. For example, to convert $-8 \frac{1}{2}$ to a decimal, you would divide $-8$ by $2$ and then add $0.5$, resulting in $-4.5$.

Q: How do I add and subtract mixed numbers?

A: To add and subtract mixed numbers, you need to first convert them to improper fractions. Then, you can add or subtract the numerators and keep the same denominator. For example, to add $-8 \frac{1}{2}$ and $-5 \frac{1}{2}$, you would first convert them to improper fractions, resulting in $\frac{-15}{2}$ and $\frac{-9}{2}$. Then, you can add the numerators, resulting in $\frac{-24}{2}$, which simplifies to $-12$.

Q: How do I multiply and divide mixed numbers?

A: To multiply and divide mixed numbers, you need to first convert them to improper fractions. Then, you can multiply or divide the numerators and keep the same denominator. For example, to multiply $-8 \frac{1}{2}$ and $-5 \frac{1}{2}$, you would first convert them to improper fractions, resulting in $\frac{-15}{2}$ and $\frac{-9}{2}$. Then, you can multiply the numerators, resulting in $\frac{-135}{4}$.

Q: What are some common mistakes to avoid when working with mixed numbers?

A: Some common mistakes to avoid when working with mixed numbers include:

• Not converting mixed numbers to improper fractions when necessary
• Not simplifying mixed numbers when possible
• Not using the correct operations (addition, subtraction, multiplication, or division) when working with mixed numbers
• Not checking for errors when converting mixed numbers to improper fractions or decimals

Q: How can I practice working with mixed numbers?

A: You can practice working with mixed numbers by:

• Using online resources, such as math websites or apps, to practice converting mixed numbers to improper fractions and decimals
• Working with real-world examples, such as measuring ingredients for a recipe or calculating the cost of items
• Creating your own practice problems and solving them on your own
• Asking a teacher or tutor for help if you need it

Q: What are some real-world applications of mixed numbers?

A: Mixed numbers have many real-world applications, including:

• Measuring ingredients for a recipe
• Calculating the cost of items
• Working with time and schedules
• Measuring distances and lengths
• Calculating areas and volumes

Q: How can I use mixed numbers in my everyday life?

A: You can use mixed numbers in your everyday life by:

• Using them to measure ingredients for a recipe
• Calculating the cost of items
• Working with time and schedules
• Measuring distances and lengths
• Calculating areas and volumes

Q: What are some common misconceptions about mixed numbers?

A: Some common misconceptions about mixed numbers include:

• Thinking that mixed numbers are only used in math class
• Thinking that mixed numbers are only used for simple calculations
• Thinking that mixed numbers are only used for negative numbers
• Thinking that mixed numbers are only used for fractions

Q: How can I overcome my fear of mixed numbers?

A: You can overcome your fear of mixed numbers by:

• Practicing working with mixed numbers regularly
• Using online resources, such as math websites or apps, to practice converting mixed numbers to improper fractions and decimals
• Working with real-world examples, such as measuring ingredients for a recipe or calculating the cost of items
• Asking a teacher or tutor for help if you need it

Q: What are some tips for mastering mixed numbers?

A: Some tips for mastering mixed numbers include:

• Practicing working with mixed numbers regularly
• Using online resources, such as math websites or apps, to practice converting mixed numbers to improper fractions and decimals
• Working with real-world examples, such as measuring ingredients for a recipe or calculating the cost of items
• Asking a teacher or tutor for help if you need it
• Reviewing and practicing mixed numbers regularly to build your confidence and skills.