Solving a compound ratio problem

Question

There is food for 800 men in a fortress for 30 days. After 10 days 400 more men join in. How long the food will last while food per man is 2/3 from before.

I listed ratios as following

Days    Men            Food
30      800            1   (all food)
x       1200 (+400)    2/3 (since 1/3 of food has been consumed already by 800 men)

30 : x ::   1200 : 800   inversely proportional
            1    : 2/3   directly proportional
 x * 1200 * 1 = 30 * 800 * (2/3)
 x = 13.33

I get the answer while the book tells me it is 20.

I can figure out what I am doing wrong.

Answer 1

Suppose the daily ration starts at R per man. So there must be 24000R initially. After 10 days there is 16000R left. The number of men increases to 1200 or 40R/3 per man. The ration is reduced to 2R/3, so the food will last for 20 days.