joe can decorate cupcakes in half the time of ann. together, they can dfecorate the dozen cupcakes in 8 minutes. How would it take each chef to decorate a dozen cupcakes alone.

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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.**

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