How many total pastries did Mama bake originally, and how many did member of the family get?

Question

One evening, Mama baked a large batch of pastries for her family, which
consisted of Papa and their three children: Ed, Lisa, and Judy. During the
night, Papa got up and ate 1/3 of the pastries. He then (separate from the
1/3) gave one pastry to the family dog, Spike.

The oldest son, Ed was the next to get up. He planned to eat 1/4 of the
pastries, but when he separated them into 4 piles, he saw there were 3
pastries left over. He decided to feed those 3 pastries to Spike. Ed then
ate the original planned 1/4 of the pastries.

Lisa awoke next and gobbled down 1/3 of the remaining pastries. Before
going back to bed, Lisa gave 2 pastries to Spike.

The last family member to awaken and visit the kitchen was young Judy.
She ate 8 pastries and then gave 1/8 of what was left to Spike.

The next morning Mama noticed there were only 14 pastries left! She ate
2, gave 2 to Spike, and then split the remaining pastries evenly for her
whole family, (5 people in all: Mama, Papa, Ed, Lisa, and Judy) to have for
breakfast.

Answer 1

I got 84 for the number of pastries (I'll not point out Ashley's error), but I checked my answer (see my comment).

Answer 2

Mrs. Obama would not be happy with this family. But I digress.

 

Start with the part of the word problem which is most specific, in this case we'll mostly be working backwards.

 

The Next morning, only 14 pastries left. M=2, S=2. Now there are 10 left (14-4=10). Now split them evenly between the family. M=2, P=2, E=2, L=2, J=2.

 

Judy at 8 pastries. J=8.

She gave 1/8 of what was left to Spike, and after that happened, the problem says the next morning there were 14 left. So, algebraically, if there were X remaining, X - (1/8)X = 14. --> (7/8)X=14. Multiply both sides by (8/7) ---> X=14(8/7) ---> X=16. So, after Judy ate, there were 16 pastries. (1/8) of that was given to spike. (1/8)16=2. S=2 Also, continuing to work backwards through the problem, we find that before Judy ate hers, there were 8+16=24.

 

Before going to bed, Lisa gave Spike 2: S=2. (Before Spike ate 2, there were 24+2=26) Lisa at 1/3 of the remaining pastries, after which there were 26. So, X-(1/3)X=26 ---> (2/3)X=26 ---> X=26(3/2) ---> X=39. There were 39 pastries when Lisa got up, and she ate (1/3) of them: (1/3)(39)=13. L=13.

 

Ed gave 3 pastries to Spike: S=3. Ed ate 1/4 of the remaining pastries. Since there were 39 when Lisa woke up, 39-3=36. (1/4)36=9. E=9. Also, before Ed woke up, there were the 39+3+9=51 pastries.

 

That night, Papa gave 1 to Spike, separate from the 1/3 he ate: S=1. Before that, there were 51+1=52. He ate 1/3 of the original amount, so X-(1/3)X=52 ---> (2/3)X=52 ---> X=52(3/2) ---> 78. There were 78 pastries originally, and he ate 1/3: (1/3)78=26. P=26.

 

Now, go back through and add up how many each person got.

P=26+2=28

S=1+3+2+2+2=10

E=9+2=11

L=13+2=15

J=8+2=10

M=2+2=4

TOTAL: 78.