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?

Answer 1

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

Answer 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