Simplify. Express Your Answer As A Proper Fraction:${ \frac{1-\frac{2}{5}}{2} = \square }$

Understanding the Problem

To simplify the given fraction, we need to follow the order of operations (PEMDAS) and apply the rules of fractions. The given expression is $\frac{1-\frac{2}{5}}{2}$. Our goal is to simplify this expression and express the answer as a proper fraction.

Applying the Order of Operations

The first step is to evaluate the expression inside the parentheses, which is $1-\frac{2}{5}$. To do this, we need to find a common denominator for the two fractions. The common denominator is 5, so we can rewrite the expression as $\frac{5}{5}-\frac{2}{5}$.

Simplifying the Expression Inside the Parentheses

Now, we can simplify the expression inside the parentheses by subtracting the two fractions. $\frac{5}{5}-\frac{2}{5} = \frac{5-2}{5} = \frac{3}{5}$.

Substituting the Simplified Expression

Now that we have simplified the expression inside the parentheses, we can substitute it back into the original expression. $\frac{1-\frac{2}{5}}{2} = \frac{\frac{3}{5}}{2}$.

Simplifying the Final Expression

To simplify the final expression, we need to divide the fraction by 2. To do this, we can multiply the fraction by the reciprocal of 2, which is $\frac{1}{2}$. $\frac{\frac{3}{5}}{2} = \frac{3}{5} \times \frac{1}{2} = \frac{3}{10}$.

Conclusion

In conclusion, the simplified expression is $\frac{3}{10}$. This is the final answer to the given problem.

Final Answer

The final answer is $\boxed{\frac{3}{10}}$.

Discussion

This problem requires the application of the order of operations and the rules of fractions. The key concept is to simplify the expression inside the parentheses and then substitute it back into the original expression. The final answer is a proper fraction, which is $\frac{3}{10}$.

Related Problems

• Simplifying fractions with variables
• Applying the order of operations to expressions with fractions
• Simplifying expressions with multiple fractions

Example Problems

• Simplify the expression $\frac{1-\frac{3}{4}}{3}$
• Simplify the expression $\frac{2-\frac{1}{2}}{4}$
• Simplify the expression $\frac{3-\frac{2}{3}}{2}$

Solutions

• Simplify the expression $\frac{1-\frac{3}{4}}{3} = \frac{\frac{1}{4}}{3} = \frac{1}{12}$
• Simplify the expression $\frac{2-\frac{1}{2}}{4} = \frac{\frac{3}{2}}{4} = \frac{3}{8}$
• Simplify the expression $\frac{3-\frac{2}{3}}{2} = \frac{\frac{7}{3}}{2} = \frac{7}{6}$
  

Frequently Asked Questions

Q: What is the order of operations in simplifying fractions?

A: The order of operations in simplifying fractions is 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.

Q: How do I simplify an expression with multiple fractions?

A: To simplify an expression with multiple fractions, follow these steps:

1. Find a common denominator for all the fractions.
2. Rewrite each fraction with the common denominator.
3. Add or subtract the fractions as needed.
4. Simplify the resulting fraction.

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

A: A proper fraction is a fraction where the numerator is less than the denominator, such as $\frac{1}{2}$. An improper fraction is a fraction where the numerator is greater than or equal to the denominator, such as $\frac{3}{2}$.

Q: How do I simplify a fraction with a variable in the numerator or denominator?

A: To simplify a fraction with a variable in the numerator or denominator, follow these steps:

1. Factor out any common factors from the numerator and denominator.
2. Simplify the resulting fraction.
3. If the variable is in the denominator, multiply the numerator and denominator by the reciprocal of the variable.

Q: What is the reciprocal of a fraction?

A: The reciprocal of a fraction is obtained by swapping the numerator and denominator. For example, the reciprocal of $\frac{1}{2}$ is $\frac{2}{1}$.

Q: How do I simplify a fraction with a negative exponent?

A: To simplify a fraction with a negative exponent, follow these steps:

1. Rewrite the fraction with a positive exponent.
2. Simplify the resulting fraction.

Q: What is the difference between a fraction and a decimal?

A: A fraction is a ratio of two integers, such as $\frac{1}{2}$. A decimal is a way of representing a fraction as a series of digits, such as 0.5.

Q: How do I convert a fraction to a decimal?

A: To convert a fraction to a decimal, follow these steps:

1. Divide the numerator by the denominator.
2. Simplify the resulting decimal.

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 proper fraction, such as $2\frac{1}{2}$. An improper fraction is a fraction where the numerator is greater than or equal to the denominator, such as $\frac{3}{2}$.

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

A: To convert a mixed number to an improper fraction, follow these steps:

1. Multiply the whole number by the denominator.
2. Add the numerator to the result.
3. Write the result as an improper fraction.

Q: What is the difference between a fraction and a percentage?

A: A fraction is a ratio of two integers, such as $\frac{1}{2}$. A percentage is a way of representing a fraction as a percentage of 100, such as 50%.

Q: How do I convert a fraction to a percentage?

A: To convert a fraction to a percentage, follow these steps:

1. Divide the numerator by the denominator.
2. Multiply the result by 100.
3. Simplify the resulting percentage.

Q: What is the difference between a fraction and a ratio?

A: A fraction is a ratio of two integers, such as $\frac{1}{2}$. A ratio is a comparison of two quantities, such as 1:2.

Q: How do I simplify a ratio?

A: To simplify a ratio, follow these steps:

1. Find the greatest common divisor (GCD) of the two quantities.
2. Divide both quantities by the GCD.
3. Simplify the resulting ratio.

Q: What is the difference between a fraction and a proportion?

A: A fraction is a ratio of two integers, such as $\frac{1}{2}$. A proportion is a statement that two ratios are equal, such as $\frac{1}{2} = \frac{3}{6}$.

Q: How do I solve a proportion?

A: To solve a proportion, follow these steps:

1. Cross-multiply the two ratios.
2. Simplify the resulting equation.
3. Solve for the unknown quantity.

Q: What is the difference between a fraction and a fraction with a variable?

A: A fraction is a ratio of two integers, such as $\frac{1}{2}$. A fraction with a variable is a fraction where one or both of the quantities are variables, such as $\frac{x}{2}$.

Q: How do I simplify a fraction with a variable?

A: To simplify a fraction with a variable, follow these steps:

1. Factor out any common factors from the numerator and denominator.
2. Simplify the resulting fraction.
3. If the variable is in the denominator, multiply the numerator and denominator by the reciprocal of the variable.

Q: What is the difference between a fraction and a decimal with a variable?

A: A fraction is a ratio of two integers, such as $\frac{1}{2}$. A decimal with a variable is a decimal where one or both of the quantities are variables, such as 0.5x.

Q: How do I convert a fraction with a variable to a decimal with a variable?

A: To convert a fraction with a variable to a decimal with a variable, follow these steps:

1. Divide the numerator by the denominator.
2. Simplify the resulting decimal.
3. If the variable is in the denominator, multiply the numerator and denominator by the reciprocal of the variable.

Q: What is the difference between a fraction and a percentage with a variable?

A: A fraction is a ratio of two integers, such as $\frac{1}{2}$. A percentage with a variable is a percentage where one or both of the quantities are variables, such as 50x%.

Q: How do I convert a fraction with a variable to a percentage with a variable?

A: To convert a fraction with a variable to a percentage with a variable, follow these steps:

1. Divide the numerator by the denominator.
2. Multiply the result by 100.
3. Simplify the resulting percentage.
4. If the variable is in the denominator, multiply the numerator and denominator by the reciprocal of the variable.

Q: What is the difference between a fraction and a ratio with a variable?

A: A fraction is a ratio of two integers, such as $\frac{1}{2}$. A ratio with a variable is a comparison of two quantities where one or both of the quantities are variables, such as x:y.

Q: How do I simplify a ratio with a variable?

A: To simplify a ratio with a variable, follow these steps:

1. Find the greatest common divisor (GCD) of the two quantities.
2. Divide both quantities by the GCD.
3. Simplify the resulting ratio.

Q: What is the difference between a fraction and a proportion with a variable?

A: A fraction is a ratio of two integers, such as $\frac{1}{2}$. A proportion with a variable is a statement that two ratios are equal where one or both of the quantities are variables, such as $\frac{x}{2} = \frac{3}{6}$.

Q: How do I solve a proportion with a variable?

A: To solve a proportion with a variable, follow these steps:

1. Cross-multiply the two ratios.
2. Simplify the resulting equation.
3. Solve for the unknown quantity.

Q: What is the difference between a fraction and a fraction with a negative exponent?

A: A fraction is a ratio of two integers, such as $\frac{1}{2}$. A fraction with a negative exponent is a fraction where the exponent is negative, such as $\frac{1}{2^{-1}}$.

Q: How do I simplify a fraction with a negative exponent?

A: To simplify a fraction with a negative exponent, follow these steps:

1. Rewrite the fraction with a positive exponent.
2. Simplify the resulting fraction.

Q: What is the difference between a fraction and a fraction with a zero denominator?

A: A fraction is a ratio of two integers, such as $\frac{1}{2}$. A fraction with a zero denominator is a fraction where the denominator is zero, such as $\frac{1}{0}$.

Q: How do I simplify a fraction with a zero denominator?

A: To simplify a fraction with a zero denominator, follow these steps:

1. The fraction is undefined, as division by zero is not allowed.

Q: What is the difference between a fraction and a fraction with a negative numerator?

A: A fraction is a ratio of two integers, such as $\frac{1}{2}$. A fraction with a negative numerator is a fraction where the numerator is negative, such as $-\frac{1}{2}$.

Q: How do I simplify a fraction with a negative numerator?

A: To simplify a fraction with a negative numerator, follow these steps:

1. Rewrite the fraction with a positive numerator.
2. Simplify the resulting fraction.

Q: What is the difference between a fraction and a fraction with a negative denominator?

A: A fraction is a ratio of two integers, such as $\frac{1}{2}$. A fraction with a negative denominator is a fraction where the denominator