Determine whether \( \frac{196}{266}=\frac{210}{285} \) (a) by reducing each side to lowest terms. (b) by using the equality test for fractions. (This is the cross-product method.) (c) Which method was fastor? Why? (b) Determine whether the statement \( \frac{196}{268}=\frac{210}{285} \) is a proportion by using the equality lest for fractions. (This is the cross-product method.) Select the correct choice below, and fint in the answer boxes if necessary. (Simplity your answers.) A. Yes, \( 196 \times 285=\square \) and \( 266 \times 210=\square \) B. No, \( 196 \times 285=\square \) and \( 266 \times 210=\square \)

To determine whether \( \frac{196}{266} = \frac{210}{285} \), we will use both methods as requested. ### (a) Reducing each side to lowest terms 1. **Reducing \( \frac{196}{266} \)**: - The GCD (greatest common divisor) of 196 and 266 can be found. - The prime factorization of 196 is \( 2^2 \times 7^2 \) and for 266 is \( 2 \times 133 \) (where 133 is \( 7 \times 19 \)). - The GCD is 2. - Thus, \( \frac{196}{266} = \frac{196 \div 2}{266 \div 2} = \frac{98}{133} \). 2. **Reducing \( \frac{210}{285} \)**: - The GCD of 210 and 285 can be found. - The prime factorization of 210 is \( 2 \times 3 \times 5 \times 7 \) and for 285 is \( 3 \times 5 \times 19 \). - The GCD is 15. - Thus, \( \frac{210}{285} = \frac{210 \div 15}{285 \div 15} = \frac{14}{19} \). Now we compare the reduced forms: - \( \frac{98}{133} \) and \( \frac{14}{19} \) are not equal, so \( \frac{196}{266} \neq \frac{210}{285} \). ### (b) Using the equality test for fractions (cross-product method) We will cross-multiply to check if \( \frac{196}{266} = \frac{210}{285} \): - Calculate \( 196 \times 285 \) and \( 266 \times 210 \). 1. **Calculating \( 196 \times 285 \)**: \[ 196 \times 285 = 55860 \] 2. **Calculating \( 266 \times 210 \)**: \[ 266 \times 210 = 55860 \] Since both products are equal: \[ 196 \times 285 = 266 \times 210 \] Thus, \( \frac{196}{266} = \frac{210}{285} \). ### (c) Which method was faster? Why? The cross-product method was faster because it involved straightforward multiplication and comparison, while reducing fractions required finding the GCD and performing division, which can be more time-consuming. ### (d) Determine whether \( \frac{196}{268} = \frac{210}{285} \) using the equality test for fractions. We will cross-multiply again: 1. **Calculating \( 196 \times 285 \)** (already calculated): \[ 196 \times 285 = 55860 \] 2. **Calculating \( 268 \times 210 \)**: \[ 268 \times 210 = 56280 \] Now we compare: - \( 196 \times 285 = 55860 \) - \( 268 \times 210 = 56280 \) Since \( 55860 \neq 56280 \), we conclude that \( \frac{196}{268} \neq \frac{210}{285} \). ### Final Answer for (b): B. No, \( 196 \times 285 = 55860 \) and \( 268 \times 210 = 56280 \).