Alice can vacuum in 40 minutes, thus her rate per min is: 1/40

It takes Bob 45 minutes to do the same job, thus his rate per min is: 1/45 min

let x represents the number of minutes it takes for Alice and Bob to do the job together thus their together rate per min is: 1/x

1/40 + 1/45 = 1/x

by solving we find:

x = 21.18 min

click here to see the step by step solution of the equation:

it takes 21.18 min approximately for Alice and Bob to do the job together.