Ethan and Frank had 78 picture cards. After Ethan gave 15 picture cards to Frank. Frank, in return, gave half of his picture cards to Ethan. In the end, Ethan had twice as many picture cards as Frank. How many picture cards did Ethan have at first?
E = Ethan's picture cards
F = Frank's picture cards
Ethan and Frank had 78 picture cards means:
E + F = 78
After Ethan gave 15 picture cards to Frank, Ethan have E - 15 picture card.
Frank have F + 15 picture cards.
Frank, in return, gave half of his picture cards to Ethan means:
Frank gave ( F + 15 ) / 2 picture cards
Ethan had:
E - 15 + ( F + 15 ) / 2 picture cards
Frank had:
( F + 15 ) / 2 picture cards
In the end, Ethan had twice as many picture cards as Frank means:
E - 15 + ( F + 15 ) / 2 = 2 ( F +15 ) / 2
E - 15 + ( F + 15 ) / 2 = F + 15
Multiply both sides by 2
2 E - 30 + F + 15 = 2 F + 30
2 E - 15 + F = 2 F + 30
Add 15 to both sides
2 E + F = 2 F + 45
Subtract F to both sides
2 E = F + 45
E = ( F + 45 ) / 2
Put this value in equation:
E + F = 78
( F + 45 ) / 2 + F = 78
Multiply both sides by 2
F + 45 + 2 F = 156
3 F + 45 = 156
Subtract 45 to both sides
3 F = 111
F = 111 / 3
F = 37
E + F = 78
E + 37 = 78
E = 78 - 37
E = 41
Frank have at first 37 picture cards
Ethan have at first 41 picture cards
Check of result.
E + F = 78
41 + 37 = 78
After Ethan gave 15 picture cards to Frank, Ethan have:
41 - 15 = 26 picture card.
Frank have:
37 + 15 = 52 picture cards
Frank, in return, gave half of his picture cards.
Frank, in return, gave:
52 / 2 = 26 picture cards
Now Frank had 52 - 26 = 26 picture cards
Now Ethan had 26 + 26 = 52 picture cards
Number of current Ethan's cards / Number of current Frank's cards:
52 / 26 = 2
Let Ethan have x picture cards.
Frank have = (78 - x) picture cards
Ethan = x - 15
Frank = 78 - x + 15 = 93 - x
Frank gave 1/2 of (93 - x) to Ethan
Ethan = x - 15 + (93 - x)/2
=> x - 15 + (93 - x)/2 = 2[(93 - x) - ((93 - x)/2)]
=> (2x - 30 + 93 - x)/2 = 2((2(93) - 2x - 93 + x))
=> (x + 63)/2 = 186 - x - 93
=> x + 63 = 372 - 2x - 186
=> x + 63 = 186 - 2x
=> 3x = 186 - 23
=> x = 123/3
x = 41
Ethan had 41 picture cards at first