Simplify (7/y)/(1/4-2/y)

Solution:

Step 1:

Multiply the numerator by the reciprocal of the denominator. $\frac{7}{y} \times \left(\frac{1}{\frac{1}{4} - \frac{2}{y}}\right)$

Step 2:

Simplify the denominator.

Step 2.1:

Convert $\frac{1}{4}$ to a fraction with a common denominator by multiplying by $\frac{y}{y}$. $\frac{7}{y} \times \left(\frac{1}{\frac{1}{4} \times \frac{y}{y} - \frac{2}{y}}\right)$

Step 2.2:

Convert $-\frac{2}{y}$ to a fraction with a common denominator by multiplying by $\frac{4}{4}$. $\frac{7}{y} \times \left(\frac{1}{\frac{1}{4} \times \frac{y}{y} - \frac{2}{y} \times \frac{4}{4}}\right)$

Step 2.3:

Establish a common denominator of $4y$ for each term by multiplying by the appropriate form of 1.

Step 2.3.1:

Multiply $\frac{1}{4}$ by $\frac{y}{y}$. $\frac{7}{y} \times \left(\frac{1}{\frac{y}{4y} - \frac{2}{y} \times \frac{4}{4}}\right)$

Step 2.3.2:

Multiply $\frac{2}{y}$ by $\frac{4}{4}$. $\frac{7}{y} \times \left(\frac{1}{\frac{y}{4y} - \frac{8}{4y}}\right)$

Step 2.3.3:

Rearrange the factors to show the common denominator $4y$. $\frac{7}{y} \times \left(\frac{1}{\frac{y}{4y} - \frac{8}{4y}}\right)$

Step 2.4:

Combine the terms over the common denominator. $\frac{7}{y} \times \left(\frac{1}{\frac{y - 8}{4y}}\right)$

Step 3:

Multiply the numerator by the reciprocal of the simplified denominator. $\frac{7}{y} \times \left(\frac{4y}{y - 8}\right)$

Step 4:

Multiply $\frac{4y}{y - 8}$ by 1 to maintain equality. $\frac{7}{y} \times \frac{4y}{y - 8}$

Step 5:

Eliminate the common factor of $y$.

Step 5.1:

Extract $y$ from $4y$. $\frac{7}{y} \times \frac{y \times 4}{y - 8}$

Step 5.2:

Cancel out the common $y$ term. $\frac{7}{\cancel{y}} \times \frac{\cancel{y} \times 4}{y - 8}$

Step 5.3:

Rewrite the simplified expression. $7 \times \frac{4}{y - 8}$

Step 6:

Combine the constant $7$ with the fraction. $\frac{7 \times 4}{y - 8}$

Step 7:

Calculate the product of $7$ and $4$. $\frac{28}{y - 8}$

Knowledge Notes:

The problem involves simplifying a complex fraction, which is a fraction where the numerator, denominator, or both are also fractions. The steps to simplify such a fraction typically involve:

1. Multiplying by the Reciprocal: When dividing by a fraction, you multiply by its reciprocal (i.e., you flip the numerator and denominator of the fraction you are dividing by).

2. Common Denominator: To combine fractions, you need a common denominator. This often involves finding the least common multiple of the denominators and adjusting each fraction accordingly.

3. Simplifying Expressions: After establishing a common denominator, you combine the fractions and simplify the expression by canceling out common factors.

4. Multiplication of Fractions: When multiplying fractions, you multiply the numerators together and the denominators together.

5. Cancellation: If a factor appears in both the numerator and the denominator, it can be canceled out because any number divided by itself is 1.

6. Multiplication of Integers: Finally, if there are integers outside of the fractions, they can be multiplied with the numerators of the fractions to simplify the expression further.

In this problem, we use these principles to simplify the given complex fraction step by step, ensuring that we maintain the equality throughout the process by using equivalent forms of 1 (like $\frac{y}{y}$ or $\frac{4}{4}$) and by canceling common factors.