Problem 47 - Introductory Algebra for College Students

Question

Divide as indicated. $$\frac{y^{2}-y}{15} \div \frac{y-1}{5}$$

Step-by-Step Solution

Answer

The result of \(\frac{y^{2}-y}{15} \div \frac{y-1}{5}\) is \(\frac{y^{2}}{3(y-1)}-\frac{y}{3(y-1)}\).

Step 1: Reciprocal of Divisor

Instead of dividing, we are going to multiply the first fraction by the reciprocal of the second fraction. The reciprocal of a fraction is obtained by switching its numerator and its denominator. So we get: \[\frac{y^{2}-y}{15} \times \frac{5}{y-1}\]

Step 2: Multiplication

The multiplication of two fractions is done by multiplying the numerators together and the denominators together. Apply this rule: \[\frac{y^{2}-y}{15} \times \frac{5}{y-1} = \frac{(y^{2}-y)×5}{15×(y-1)}\]

Step 3: Simplify

Now simplify the fraction as much as possible. Notice that 5 in the numerator and 15 in the denominator can be simplified as 5/15=1/3. So, the fraction simplifies to:\[\frac{(y^{2}-y)}{3(y-1)}\]

Step 4: Distributive Property

Apply the distributive property to simplify the fraction further:\[\frac{(y^{2}-y)}{3(y-1)} = \frac{y^{2}}{3(y-1)}-\frac{y}{3(y-1)}\]