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?

Esther had 200 cards at first

Step-by-step 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

yep

step-by-step explanation: