Esth had five times as many cards as Rich. Esth gave 1/4 of her cards to Rich. Rich. gave 1/6 of his cards to Esth in return. In the end, Esth had 90 cards more than Rich. How many cards did Esth have at first?
is the swap simultaneous, or did Rich give her his 1/6 after he received her 1/4?
Rich gave her his 1/6 after he received her 1/4. Please help
To solve this problem, let's break it down step by step.
1. Let's assume that Rich had x cards.
- So, Esth had 5 times as many cards as Rich, which means Esth had 5x cards.
2. Esth gave 1/4 of her cards to Rich.
- Esth gave (1/4) * 5x = 5x/4 cards to Rich.
- After giving away her cards, Esth had 5x - 5x/4 = 20x/4 - 5x/4 = 15x/4 cards.
3. Rich gave 1/6 of his cards to Esth in return.
- Rich gave (1/6) * x = x/6 cards back to Esth.
- After receiving cards from Rich, Esth had 15x/4 + x/6 = (90x + 2x)/12 = (92x)/12 = 46x/6 cards.
4. In the end, Esth had 90 cards more than Rich.
- So, we can set up an equation: 46x/6 - x = x + 90.
- Solving this equation will give us the value of x, which represents the number of cards Rich had at first.
Let's continue solving the equation:
46x/6 - x = x + 90
(46x - 6x)/6 = x + 90
40x/6 = x + 90
40x = 6x + 540
40x - 6x = 540
34x = 540
x = 540/34
x ≈ 15.88
Therefore, Rich had approximately 15.88 cards at first.
Now, we can find the number of cards Esth had at first.
Esth had 5 times as many cards as Rich, which is 5 * 15.88 ≈ 79.40 cards.
Therefore, Esth had approximately 79.40 cards at first.