Esther has 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 has 90 cards more than Richard. How many cards did Esther have at first.
e = 5r
e - e/4 + 1/6 (r + e/4) = 90 + 5/6 (r+e/4)
5r - 5r/4 + 1/6 (r + 5r/4) = 90 + 5/6 (r+5r/4)
r=40
so, e=200
check:
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
165 = 75+90
Thank you Steve
Let's solve this problem step by step:
Step 1: Let's assume the number of cards Richard had at first as "x".
Esther has five times as many cards as Richard, so Esther had 5x cards at first.
Step 2: Esther gave 1/4 of her cards to Richard. This means she gave (1/4)*5x = 5/4x cards to Richard.
After giving away 5/4x cards, Esther has 5x - (5/4x) = (20x - 5x)/4 = 15x/4 cards remaining.
Step 3: Richard gave 1/6 of his cards to Esther in return. This means he gave (1/6)*x cards to Esther.
After giving away (1/6)*x cards, Richard has x - (1/6)*x = (6x - x)/6 = 5x/6 cards remaining.
Step 4: In the end, Esther has 90 more cards than Richard, so we can write the equation:
(15x/4) - (5x/6) = 90
Step 5: Multiply both sides of the equation by 12 to eliminate the fractions:
12*(15x/4) - 12*(5x/6) = 12*90
45x - 10x = 1080
35x = 1080
Step 6: Divide both sides of the equation by 35 to isolate x:
x = 1080/35
Step 7: Calculate the value of x:
x ≈ 30.86
Since we can't have a fraction of a card, we can round down x to the nearest whole number:
x = 30
Step 8: Calculate the number of cards Esther had at first:
Esther had 5x cards = 5*30 = 150 cards.
Therefore, Esther had 150 cards at first.
To solve this problem, let's break it down step by step.
Let's assume Richard has "x" cards.
According to the given information, Esther has five times as many cards as Richard, so Esther has 5x cards.
Esther gave 1/4 of her cards to Richard, so she gave (1/4) * 5x = (5/4)x cards to Richard.
After giving these cards, Esther has 5x - (5/4)x = (20/4)x - (5/4)x = (15/4)x cards left.
Richard received (1/4) of Esther's cards, which is (1/4) * 5x = (5/4)x cards. So Richard now has x + (5/4)x = (9/4)x cards.
Then, Richard gave 1/6 of his cards to Esther, which is (1/6) * (9/4)x = (9/24)x = (3/8)x cards
After giving these cards, Richard has (9/4)x - (3/8)x = (18/8)x - (3/8)x = (15/8)x cards left.
Given that Esther has 90 cards more than Richard, we can set up the equation:
(15/4)x = (15/8)x + 90
To solve for x, let's clear the fractions by multiplying each side of the equation by 8:
8 * (15/4)x = 8 * (15/8)x + 8 * 90
30x = 15x + 720
30x - 15x = 720
15x = 720
x = 720 / 15
x = 48
So, Richard initially had 48 cards.
Since Esther has five times as many cards as Richard, she initially had 5 * 48 = 240 cards.
Therefore, Esther initially had 240 cards.