SOLUTION: A Prince picked a basketful of golden apples in the enchanted orchard. On his way home, he was stopped by a troll. The troll demanded payment of one-half of the apples plus two mor

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Question 213456: A Prince picked a basketful of golden apples in the enchanted orchard. On his way home, he was stopped by a troll. The troll demanded payment of one-half of the apples plus two more. the prince gave him the apples and continued on home. The prince was then stopped by a second troll who demanded one- half of the apples plus two more, the prince did as he was told and continued on home. Stopped by a third troll he demanded the prince give him one- half of the apples plus two more. The prince did as asked and sadly continued on home only having two golden apples left. How many apples did he pick to begin with?
Found 2 solutions by josmiceli, rapaljer:
Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
Let a = number of apples he started with
After 1st troll:
He has a%2F2+-+2
After 2nd troll:
He has %28a%2F2+-+2%29%2F2+-+2
After 3rd troll:
He has %28%28a%2F2+-+2%29%2F2+-+2%29%2F2+-+2
and the problem says:
%28%28a%2F2+-+2%29%2F2+-+2%29%2F2+-+2+=+2
Add 2 to both sides
%28%28a%2F2+-+2%29%2F2+-+2%29%2F2+=+4
Multiply both sides by 2
%28a%2F2+-+2%29%2F2+-+2+=+8
Add 2 to both sides
%28a%2F2+-+2%29%2F2+=+10
Multiply both sides by 2
a%2F2+-+2+=+20
Add 2 to both sides
a%2F2+=+22
a+=+44
The Prince picked 44 apples
check answer:
1st troll took 44%2F2+%2B+2+=+24
which left the Prince 44+-+24+=+20
2nd troll took 20%2F2+%2B+2+=+12
which left the prince 20+-+12+=+8
3rd troll took 8%2F2+%2B+2+=+6
Which left the Prince 8+-+6+=+2
OK

Answer by rapaljer(4671) About Me  (Show Source):
You can put this solution on YOUR website!
Let x = total number of apples

The first troll demands half of his apples plus 2, so after leaving the first troll, the prince has
x-%281%2F2%29%2Ax+-2+apples
%281%2F2%29%2Ax-2

The 2nd troll demands half of what is left and 2 more, so now the prince has
%281%2F2%29%28%281%2F2%29%2Ax-2%29-2
%281%2F4%29x-1-2
%281%2F4%29x-3

The 3rd troll demands half of what is left and 2 more, so now the prince has
%281%2F2%29%2A%28%281%2F4%29x-3%29-2

And this equals exactly 2 apples!
%281%2F2%29%2A%28%281%2F4%29x-3%29-2=2

Now, work this backwards by adding +2 to each side:
%281%2F2%29%2A%28%281%2F4%29x-3%29=4

Multiply both sides by 2:
(1/4)x-3=8

Add +3 to each side:
(1/4)x=11

Multiply both sides by 4:
x=44 apples.

Check: If the prince starts with 44 apples, the first troll gets 22 +2 =24 apples. The prince is left with 20. The second troll demands 10+2, which leaves the prince with 8. The third troll demands 4 apples plus 2, leaving the prince with 2 apples.

Very nice problem!! Poor princess!!

R^2

Dr. Robert J. Rapalje, Retired
Seminole Community College
Altamonte Springs Campus
Florida