Find the following quotient. Reduce to lowest terms. \[ \frac{20}{14} \div \frac{15}{23} \]

To find the quotient \(\frac{20}{14} \div \frac{15}{23}\) and reduce it to its lowest terms, follow these steps: 1. **Convert Division to Multiplication:** \[ \frac{20}{14} \div \frac{15}{23} = \frac{20}{14} \times \frac{23}{15} \] 2. **Factor and Simplify Before Multiplying:** - **Factor the Numbers:** \[ 20 = 4 \times 5,\quad 14 = 2 \times 7,\quad 15 = 3 \times 5,\quad 23 \text{ is prime} \] - **Set Up the Multiplication:** \[ \frac{4 \times 5}{2 \times 7} \times \frac{23}{3 \times 5} \] - **Cancel Common Factors:** - The \(5\) in the numerator and denominator cancels out: \[ \frac{4}{2 \times 7} \times \frac{23}{3} \] - Simplify \(\frac{4}{2}\) to \(2\): \[ \frac{2}{7} \times \frac{23}{3} \] 3. **Multiply the Simplified Fractions:** \[ \frac{2 \times 23}{7 \times 3} = \frac{46}{21} \] 4. **Verify Lowest Terms:** - The greatest common divisor (GCD) of 46 and 21 is 1, so \(\frac{46}{21}\) is already in its lowest terms. **Final Answer:** \[ \boxed{\dfrac{46}{21}} \]