Plum dumplings

Three journeymen on their apprenticeship arrived completely exhausted at a tavern and ordered – it was at the end of summer – plum dumplings for supper. While supper was being prepared, they chattered and dozed at the table, until they fell soundly asleep. Then the tavern-keeper brought the bowl with the supper and called them to eat. Of the three companions, only one woke up and, not caring about his companions, ate his share, folded his arms under his head, and fell asleep again. After a while the second one woke up, and thinking that it was supper, he ate one third of the actual remainder, lay down, and fell asleep. Finally the third one woke up and, taking himself also for the first, he ate also one third of the remainder, and stretching his stiff limbs, fell asleep again. After some time, the first journeyman woke up from his drowsiness and, seeing that there were still 24 dumplings on the bowl, woke up the others. And it turned out that they had all already had their supper. How many dumplings were there, how many did each one himself eat, and how was the remainder divided?

Final Answer:

a =  27
b =  18
c =  12
n =  81

Step-by-step explanation:

a+b+c+24= n
a = n / 3
b = (n - a)/3
c = (n - a -b )/3

a+b+c-n = -24
3a-n = 0
a+3b-n = 0
a+b+3c-n = 0

Pivot: Row 1 ↔ Row 2
3a-n = 0
a+b+c-n = -24
a+3b-n = 0
a+b+3c-n = 0

Row 2 - 1/3 · Row 1 → Row 2
3a-n = 0
b+c-0.6667n = -24
a+3b-n = 0
a+b+3c-n = 0

Row 3 - 1/3 · Row 1 → Row 3
3a-n = 0
b+c-0.6667n = -24
3b-0.6667n = 0
a+b+3c-n = 0

Row 4 - 1/3 · Row 1 → Row 4
3a-n = 0
b+c-0.6667n = -24
3b-0.6667n = 0
b+3c-0.6667n = 0

Pivot: Row 2 ↔ Row 3
3a-n = 0
3b-0.6667n = 0
b+c-0.6667n = -24
b+3c-0.6667n = 0

Row 3 - 1/3 · Row 2 → Row 3
3a-n = 0
3b-0.6667n = 0
c-0.4444n = -24
b+3c-0.6667n = 0

Row 4 - 1/3 · Row 2 → Row 4
3a-n = 0
3b-0.6667n = 0
c-0.4444n = -24
3c-0.4444n = 0

Pivot: Row 3 ↔ Row 4
3a-n = 0
3b-0.6667n = 0
3c-0.4444n = 0
c-0.4444n = -24

Row 4 - 1/3 · Row 3 → Row 4
3a-n = 0
3b-0.6667n = 0
3c-0.4444n = 0
-0.2963n = -24


n = -24/-0.2962963 = 81
c = 0+0.44444444444444n/3 = 0+0.44444444 · 81/3 = 12
b = 0+0.66666666666667n/3 = 0+0.66666667 · 81/3 = 18
a = 0+n/3 = 0+81/3 = 27

a = 27
b = 18
c = 12
n = 81

Our linear equations calculator calculates it.

Try another example