Question: Section 4: Multiplication & Division of Fractions (with Variables)
1. Multiply and simplify completely: $\frac{4x}{3y^2} \cdot \frac{9y}{8x^3}$
2. Divide and simplify completely: $\frac{6a^2}{5b} \div \frac{12a}{15b^3}$
3. Multiply and simplify completely: $\frac{x+3}{4} \cdot \frac{8}{x^2-9}$
4. Word Problem: The length of a rectangle is represented by $\frac{6x^2}{y^3}$ meters, and its width is represented by $\frac{y^2}{2x}$ meters. Find the area of the rectangle in simplest form.
1. **Multiply and simplify:**
Problem: Multiply $\frac{4x}{3y^2}$ and $\frac{9y}{8x^3}$ and simplify completely.
Formula: When multiplying fractions, multiply numerators and denominators:
$$\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}$$
Step 1: Multiply numerators and denominators:
$$\frac{4x \cdot 9y}{3y^2 \cdot 8x^3} = \frac{36xy}{24x^3y^2}$$
Step 2: Simplify the fraction by dividing numerator and denominator by common factors:
$$\frac{\cancel{12} \cdot 3xy}{\cancel{12} \cdot 2x^3y^2} = \frac{3xy}{2x^3y^2}$$
Step 3: Simplify variables by subtracting exponents:
$$= \frac{3 \cancel{x} y}{2 x^{3-1} y^{2-1}} = \frac{3y}{2x^2 y}$$
Step 4: Cancel $y$:
$$= \frac{3 \cancel{y}}{2x^2 \cancel{y}} = \frac{3}{2x^2}$$
**Answer:** $\frac{3}{2x^2}$
2. **Divide and simplify:**
Problem: Divide $\frac{6a^2}{5b}$ by $\frac{12a}{15b^3}$ and simplify completely.
Formula: Dividing fractions is multiplying by the reciprocal:
$$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}$$
Step 1: Rewrite division as multiplication:
$$\frac{6a^2}{5b} \cdot \frac{15b^3}{12a}$$
Step 2: Multiply numerators and denominators:
$$\frac{6a^2 \cdot 15b^3}{5b \cdot 12a} = \frac{90 a^2 b^3}{60 a b}$$
Step 3: Simplify by dividing numerator and denominator by 30:
$$\frac{\cancel{30} \cdot 3 a^2 b^3}{\cancel{30} \cdot 2 a b} = \frac{3 a^2 b^3}{2 a b}$$
Step 4: Simplify variables by subtracting exponents:
$$= \frac{3 a^{2-1} b^{3-1}}{2} = \frac{3 a b^2}{2}$$
**Answer:** $\frac{3 a b^2}{2}$
3. **Multiply and simplify:**
Problem: Multiply $\frac{x+3}{4}$ and $\frac{8}{x^2-9}$ and simplify completely.
Note: $x^2 - 9$ is a difference of squares:
$$x^2 - 9 = (x-3)(x+3)$$
Step 1: Multiply numerators and denominators:
$$\frac{(x+3) \cdot 8}{4 \cdot (x-3)(x+3)} = \frac{8(x+3)}{4(x-3)(x+3)}$$
Step 2: Cancel common factor $(x+3)$:
$$\frac{8 \cancel{(x+3)}}{4 (x-3) \cancel{(x+3)}} = \frac{8}{4(x-3)}$$
Step 3: Simplify $\frac{8}{4} = 2$:
$$= \frac{2}{x-3}$$
**Answer:** $\frac{2}{x-3}$
4. **Word Problem: Find area of rectangle**
Given:
Length $= \frac{6x^2}{y^3}$ meters
Width $= \frac{y^2}{2x}$ meters
Formula for area of rectangle:
$$\text{Area} = \text{Length} \times \text{Width}$$
Step 1: Multiply length and width:
$$\frac{6x^2}{y^3} \times \frac{y^2}{2x} = \frac{6x^2 y^2}{y^3 \cdot 2x} = \frac{6x^2 y^2}{2x y^3}$$
Step 2: Simplify numerator and denominator by dividing by 2:
$$\frac{\cancel{2} \cdot 3 x^2 y^2}{\cancel{2} x y^3} = \frac{3 x^2 y^2}{x y^3}$$
Step 3: Simplify variables by subtracting exponents:
$$= 3 x^{2-1} y^{2-3} = 3 x y^{-1} = \frac{3x}{y}$$
**Answer:** $\frac{3x}{y}$ meters squared