Problem 48 - Introductory Algebra for College Students

Question

Divide as indicated. $$\frac{y^{2}-2 y}{15} \div \frac{y-2}{5}$$

Step-by-Step Solution

Answer

The result of the division \( \frac{y^{2}-2 y}{15} \div \frac{y-2}{5} \) is \( \frac{y}{3} \).

Step 1: Convert into Multiplication

Convert the division operation into multiplication, by finding the reciprocal of \( \frac{y-2}{5} \). The reciprocal of a fraction is obtained by reversing (or 'flipping') the positions of the numerator and the denominator. This yields: \( \frac{y^{2}-2 y}{15} \times \frac{5}{y-2} \).

Step 2: Simplify Multiplication

Next, cancel out the common factors between the numerators and denominators. Here, on splitting \( y^{2}-2 y \) as \( y(y-2) \), we find that \( y-2 \) common in the numerator of the first fraction and the denominator of the second fraction. The number 15 in the denominator of the first fraction and 5 in the numerator of the second fraction also have a common factor of 5. Therefore, cancel out the common factors to get: \( \frac{y \times 1}{3 \times 1} \), which simplifies to \( \frac{y}{3} \).

Step 3: Final Solution

The simplified form of the given expression after performing the division operation is \( \frac{y}{3} \). This is the final answer. Proceed to write this in final solution.