Question

Esther had five times as many cards as Richard. Esther gave 1/4 of her cards to Richard. Then, Richard gave 1/6 of his cards to Esther in return. In the end, Esther had 90 more than Richard. How many cards did Esther have at first?

Answer 1

Esther had 200 cards at first

Explanation:

Variables:

Esther: 5r

Richard: r

Equalities:

e = 5r

e - e/4 + 1/6 (r + e/4) = 5/6 (r + e/4) + 90

Solve:

Replace e with 5r and solve for r:

e - e/4 + 1/6 (r + e/4) = 5/6 (r + e/4) + 90

5r - 5r/4 + 1/6 (r + 5r/4) = 5/6 (r + 5r/4) + 90

5r - 5r/4 + 1/6 (2.25r) = 5/6 (2.25r) + 90

5r - 1.25r + 0.375r = 1.875r + 90

5r - 1.25r + 0.375r - 1.875r = 90

2.25r = 90

r = 40

Esther: 5r = 5(40) = 200

Richard: r = 40

Answer 2

Esther has 200, Richard has 40

Esther gives Richard 50 cards. Now they have

Esther: 150, Richard: 90

Richard gives Esther 15, so they now have

Esther: 165, Richard 75

Explanation: