Let, Ann decorates a cupcake in x minutes,
So, in one minute Ann decorates 1/x cupcakes.
Joe would decorate a cupcake in x/2 minutes.
and, in one minute Joe decorates 1/(x/2) = 2/x cupcakes.
Together, the number of cupcakes they would decorate in one minute is: (1/x + 2/x) = 3/x.
Together, they decorate 3/x cupcakes in 1 minute.
Therefore, they would decorate 1 cupcake in 1/(3/x) = x/3 minutes. and they would decorate a dozen =12 cupcakes in x/3 * 12 = 4x minutes.
By the condition of the problem, 4x = 8
`rArr` x = 2 minutes
So, Ann alone decorates a cupcake in 2 minutes and Joe in 2/2 = 1 minute.
Therefore, Ann alone would decorate a dozen cupcakes in 12*2 = 24 minutes and Joe alone in 12*1 = 12 minutes.