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

To find the quotient of \(\frac{20}{14} \div \frac{15}{23}\), we can rewrite the division as multiplication by the reciprocal: \[ \frac{20}{14} \div \frac{15}{23} = \frac{20}{14} \times \frac{23}{15} \] Next, we multiply the numerators and the denominators: \[ \frac{20 \times 23}{14 \times 15} \] Calculating the products: - For the numerator: \(20 \times 23 = 460\) - For the denominator: \(14 \times 15 = 210\) So, we have: \[ \frac{460}{210} \] Now, we should reduce this fraction to its lowest terms. First, we find the greatest common divisor (GCD) of 460 and 210. The prime factorizations are: - \(460 = 2 \times 2 \times 5 \times 23 = 2^2 \times 5^1 \times 23^1\) - \(210 = 2 \times 3 \times 5 \times 7 = 2^1 \times 3^1 \times 5^1 \times 7^1\) The common factors are \(2^1\) and \(5^1\), so: \[ \text{GCD}(460, 210) = 2 \times 5 = 10 \] Now we divide both the numerator and the denominator by the GCD: \[ \frac{460 \div 10}{210 \div 10} = \frac{46}{21} \] Thus, the final reduced form of the quotient is: \[ \boxed{\frac{46}{21}} \]