Word problem on combined variation Suppose that the amount of time it takes to build a highway varies directly, with the length of the highway and inversely, with the number of workers. Suppose also that it takes 150 workers 14 weeks to build 12 miles of highway. How many miles of highway could 125 workers build in 21 weeks? miles

1. Let the relationship between the time \( T \) (in weeks), the length \( L \) (in miles), and the number of workers \( W \) be given by \[ T = k \frac{L}{W}, \] where \( k \) is the constant of proportionality. 2. We know that 150 workers take 14 weeks to build 12 miles of highway. Substitute these values into the equation: \[ 14 = k \frac{12}{150}. \] 3. Solve for \( k \) by multiplying both sides of the equation by 150: \[ 14 \times 150 = k \times 12. \] Thus, \[ 2100 = 12k. \] 4. Divide both sides by 12 to find \( k \): \[ k = \frac{2100}{12} = 175. \] 5. Now, we use the found value of \( k \) to determine how many miles of highway (\( L \)) 125 workers could build in 21 weeks: \[ 21 = 175 \frac{L}{125}. \] 6. Solve for \( L \) by multiplying both sides by 125: \[ 21 \times 125 = 175L. \] This gives: \[ 2625 = 175L. \] 7. Divide both sides by 175 to solve for \( L \): \[ L = \frac{2625}{175}. \] 8. Simplify the fraction: \[ L = \frac{2625 \div 35}{175 \div 35} = \frac{75}{5} = 15. \] Thus, \( 125 \) workers can build \( 15 \) miles of highway in \( 21 \) weeks.