How do I solve this word problem?

[caffeineandbooks]

All the others in the grouping seem straightforward, but for some reason I'm struggling with this Singapore challenging word problem for DS.  The book gives an answer but no working to show how they approached it.

 

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 cards more than Richard.  How many cards did Esther have at first?

[Noreen Claire]

I solved it, but I haven't had my first cuppa tea yet this morning so it's not pretty.

Maybe it will still help?

[wendyroo]

I got the same.

I started with a bar diagram divided into 6 pieces. Esther had 5 blocks and Richard had 1.

Then Esther gives Richard 1/4 of her cards, so I divided each block into 4ths (so the whole bar was divided into 24ths with Esther initially having 20/24 and Richard having 4/24). When she gives him the cards, Esther now has 15 blocks and Richard has 9 blocks.

Then Richard gives Esther 1/6 of his cards. One sixth of Richard's 9 blocks is 1.5 blocks (I could have divided the whole bar into 48th, but I didn't want to!). When he gives her the cards, Esther now has 16.5 blocks and Richard has 7.5 blocks.

We are told that Esther's new total is 90 more than Richard's new total. So, 16.5 blocks - 7.5 blocks = 90. Or 9 blocks = 90 cards. 1 block = 10 cards.

Each block represents 1/24th of the cards, and initially Esther had 20/24ths of the cards. So Esther had 10 * 20 = 200 cards. 

[caffeineandbooks]

Thank you, ladies!  Noreen, I set it out similarly to you and tracked with you to almost the last step where I tried to do 33C = 15C + 90.  Wendyroo, I think your way uses some friendlier numbers.  Thank you both so much for taking the time!  I'm glad to be able to see where I went wrong and offer my kiddo two different approaches to try.  Thank you 🙂