9. Solve For \[$ X \$\] In The Equation:$\[ 1 \frac{1}{9} \cdot 6 = X \\]

Introduction

In this section, we will delve into solving a simple algebraic equation involving a mixed number and a whole number. The equation is ${ 1 \frac{1}{9} \cdot 6 = x }$. To solve for ${ x }$, we need to follow the order of operations and apply the rules of arithmetic operations. This problem requires a basic understanding of fractions, mixed numbers, and multiplication.

Understanding the Equation

The given equation is ${ 1 \frac{1}{9} \cdot 6 = x }$. Here, ${ 1 \frac{1}{9} }$ is a mixed number, which can be written as ${ \frac{10}{9} }$. The equation can be rewritten as ${ \frac{10}{9} \cdot 6 = x }$.

Converting the Mixed Number to an Improper Fraction

To simplify the equation, we can convert the mixed number ${ 1 \frac{1}{9} }$ to an improper fraction. An improper fraction is a fraction where the numerator is greater than or equal to the denominator. In this case, the mixed number ${ 1 \frac{1}{9} }$ can be written as ${ \frac{10}{9} }$.

Multiplying the Fraction by 6

Now that we have the equation in terms of an improper fraction, we can multiply the fraction by 6. To multiply a fraction by a whole number, we multiply the numerator by the whole number and keep the denominator the same. So, ${ \frac{10}{9} \cdot 6 = \frac{10 \cdot 6}{9} }$.

Simplifying the Expression

To simplify the expression, we can multiply the numerator and denominator separately. ${ 10 \cdot 6 = 60 }$ and ${ 9 }$ remains the same. So, the expression becomes ${ \frac{60}{9} }$.

Reducing the Fraction to Its Lowest Terms

To reduce the fraction to its lowest terms, we need to find the greatest common divisor (GCD) of the numerator and denominator. The GCD of 60 and 9 is 3. We can divide both the numerator and denominator by 3 to get the reduced fraction. ${ \frac{60}{9} = \frac{60 \div 3}{9 \div 3} = \frac{20}{3} }$.

Conclusion

In this section, we solved the equation ${ 1 \frac{1}{9} \cdot 6 = x }$ by converting the mixed number to an improper fraction, multiplying the fraction by 6, simplifying the expression, and reducing the fraction to its lowest terms. The solution to the equation is ${ x = \frac{20}{3} }$.

Real-World Applications

This type of problem may seem trivial, but it is essential to understand the concept of multiplying fractions and mixed numbers. In real-world applications, you may encounter situations where you need to multiply fractions and mixed numbers, such as in cooking, carpentry, or finance.

Tips and Tricks

• When multiplying a fraction by a whole number, multiply the numerator by the whole number and keep the denominator the same.
• To simplify an expression, multiply the numerator and denominator separately.
• To reduce a fraction to its lowest terms, find the greatest common divisor (GCD) of the numerator and denominator and divide both by the GCD.

Practice Problems

1. Solve the equation ${ 2 \frac{1}{4} \cdot 5 = x }$.
2. Solve the equation ${ 3 \frac{1}{2} \cdot 4 = x }$.
3. Solve the equation ${ 1 \frac{1}{3} \cdot 6 = x }$.

Solutions to Practice Problems

1. ${ 2 \frac{1}{4} \cdot 5 = x }$
   • Convert the mixed number to an improper fraction: ${ 2 \frac{1}{4} = \frac{9}{4} }$
   • Multiply the fraction by 5: ${ \frac{9}{4} \cdot 5 = \frac{9 \cdot 5}{4} = \frac{45}{4} }$
   • Reduce the fraction to its lowest terms: ${ \frac{45}{4} = \frac{45 \div 1}{4 \div 1} = \frac{45}{4} }$
   • The solution to the equation is ${ x = \frac{45}{4} }$.
2. ${ 3 \frac{1}{2} \cdot 4 = x }$
   • Convert the mixed number to an improper fraction: ${ 3 \frac{1}{2} = \frac{7}{2} }$
   • Multiply the fraction by 4: ${ \frac{7}{2} \cdot 4 = \frac{7 \cdot 4}{2} = \frac{28}{2} }$
   • Reduce the fraction to its lowest terms: ${ \frac{28}{2} = \frac{28 \div 2}{2 \div 2} = \frac{14}{1} = 14 }$
   • The solution to the equation is ${ x = 14 }$.
3. ${ 1 \frac{1}{3} \cdot 6 = x }$
   • Convert the mixed number to an improper fraction: ${ 1 \frac{1}{3} = \frac{4}{3} }$
   • Multiply the fraction by 6: ${ \frac{4}{3} \cdot 6 = \frac{4 \cdot 6}{3} = \frac{24}{3} }$
   • Reduce the fraction to its lowest terms: ${ \frac{24}{3} = \frac{24 \div 3}{3 \div 3} = \frac{8}{1} = 8 }$
   • The solution to the equation is ${ x = 8 }$.
     

Q&A: Solving Equations with Mixed Numbers and Fractions

Q: What is the first step in solving the equation ${ 1 \frac{1}{9} \cdot 6 = x }$?

A: The first step is to convert the mixed number ${ 1 \frac{1}{9} }$ to an improper fraction. An improper fraction is a fraction where the numerator is greater than or equal to the denominator. In this case, the mixed number ${ 1 \frac{1}{9} }$ can be written as ${ \frac{10}{9} }$.

Q: How do I multiply a fraction by a whole number?

A: To multiply a fraction by a whole number, multiply the numerator by the whole number and keep the denominator the same. So, ${ \frac{10}{9} \cdot 6 = \frac{10 \cdot 6}{9} }$.

Q: What is the next step after multiplying the fraction by 6?

A: After multiplying the fraction by 6, we need to simplify the expression. To simplify the expression, we can multiply the numerator and denominator separately. ${ 10 \cdot 6 = 60 }$ and ${ 9 }$ remains the same. So, the expression becomes ${ \frac{60}{9} }$.

Q: How do I reduce a fraction to its lowest terms?

A: To reduce a fraction to its lowest terms, we need to find the greatest common divisor (GCD) of the numerator and denominator. The GCD of 60 and 9 is 3. We can divide both the numerator and denominator by 3 to get the reduced fraction. ${ \frac{60}{9} = \frac{60 \div 3}{9 \div 3} = \frac{20}{3} }$.

Q: What is the solution to the equation ${ 1 \frac{1}{9} \cdot 6 = x }$?

A: The solution to the equation is ${ x = \frac{20}{3} }$.

Q: Can you provide some real-world applications of solving equations with mixed numbers and fractions?

A: Yes, this type of problem may seem trivial, but it is essential to understand the concept of multiplying fractions and mixed numbers. In real-world applications, you may encounter situations where you need to multiply fractions and mixed numbers, such as in cooking, carpentry, or finance.

Q: What are some tips and tricks for solving equations with mixed numbers and fractions?

A: Here are some tips and tricks:

• When multiplying a fraction by a whole number, multiply the numerator by the whole number and keep the denominator the same.
• To simplify an expression, multiply the numerator and denominator separately.
• To reduce a fraction to its lowest terms, find the greatest common divisor (GCD) of the numerator and denominator and divide both by the GCD.

Q: Can you provide some practice problems for solving equations with mixed numbers and fractions?

A: Yes, here are some practice problems:

1. Solve the equation ${ 2 \frac{1}{4} \cdot 5 = x }$.
2. Solve the equation ${ 3 \frac{1}{2} \cdot 4 = x }$.
3. Solve the equation ${ 1 \frac{1}{3} \cdot 6 = x }$.

Q: Can you provide the solutions to the practice problems?

A: Yes, here are the solutions to the practice problems:

1. ${ 2 \frac{1}{4} \cdot 5 = x }$
   • Convert the mixed number to an improper fraction: ${ 2 \frac{1}{4} = \frac{9}{4} }$
   • Multiply the fraction by 5: ${ \frac{9}{4} \cdot 5 = \frac{9 \cdot 5}{4} = \frac{45}{4} }$
   • Reduce the fraction to its lowest terms: ${ \frac{45}{4} = \frac{45 \div 1}{4 \div 1} = \frac{45}{4} }$
   • The solution to the equation is ${ x = \frac{45}{4} }$.
2. ${ 3 \frac{1}{2} \cdot 4 = x }$
   • Convert the mixed number to an improper fraction: ${ 3 \frac{1}{2} = \frac{7}{2} }$
   • Multiply the fraction by 4: ${ \frac{7}{2} \cdot 4 = \frac{7 \cdot 4}{2} = \frac{28}{2} }$
   • Reduce the fraction to its lowest terms: ${ \frac{28}{2} = \frac{28 \div 2}{2 \div 2} = \frac{14}{1} = 14 }$
   • The solution to the equation is ${ x = 14 }$.
3. ${ 1 \frac{1}{3} \cdot 6 = x }$
   • Convert the mixed number to an improper fraction: ${ 1 \frac{1}{3} = \frac{4}{3} }$
   • Multiply the fraction by 6: ${ \frac{4}{3} \cdot 6 = \frac{4 \cdot 6}{3} = \frac{24}{3} }$
   • Reduce the fraction to its lowest terms: ${ \frac{24}{3} = \frac{24 \div 3}{3 \div 3} = \frac{8}{1} = 8 }$
   • The solution to the equation is ${ x = 8 }$.