Do unto Caesar
At the beginning of the night three poker players; Alan, Bernie and Craig had money in the ratios 7 : 6 : 5. At the end of the night the ratio was 6 : 5 : 4. One of them won $1 200. What were the assets of the players at the beginning of the evening?
Problem
At Caesar's Palace in Vegas, were three poker players ? Alan, Bernie and Craig. At the beginning of the night the amount of money each had was in the ratios 7 : 6 : 5. At the end of the night the ratio was 6 : 5 : 4.
One of the players won $1 200.
What were the assets of the players at the beginning of the evening?
Getting Started
In Poker the total money in the game doesn't change, just the distribution between the players. What fraction of that total does each player have at the start and at the finish? They can't all be winners.
Student Solutions
At Caesar's Palace in Vegas, were three poker players - Alan, Bernie and Craig. At the beginning of the night the amount of money each had was in the ratios 7 : 6 : 5. At the end of the night the ratio was 6 : 5 : 4. One of the players won 1 200. Suppose at the outset they had a, b and c.
This solution was sent to you by Prav Idaikkadar and Sheli Levenson (YR9) of the North London Collegiate School Maths Puzzle Club. Have a nice day
a : b: c = 7 : 6 : 5 = 7/18 : 6/18 : 5/18
6 : 5 : 4 = 6/15 : 5/15 : 4/15The lowest common multiple of 15 and 18 is 90.
This is 35/90 : 30/90 : 25/90
36/90 : 30/90 : 24/90
We can see that Craig lost 1/90 of his money and Alan gained it. We are told that the sum of money gained is 1,200. Therefore 1/90 of the money = 1,200.
The total amount of money is 90 * 1,200 = 108,000.
In the beginning of the game Alan had 35/90 of the total money
35/90 * 108, 000 = 42, 000.
Bernie had 30/90 of the total money and this is
30/90 * 108, 000 = 36, 000.
Craig had 25/90 of the total money and this is
25/90 * 108, 000 = 30, 000.
These were the assets of the players at the beginning of the evening
Well done Prav and Sheli. Other very good solutions were sent in by Elizabeth Wells, also from the N.C.L.S. puzzle club and James Page from Hethersett High School, Norwich.
Teachers' Resources
This problem encourages students to think carefully about what the ratio actually means. The total sum of money doesn't change, so students need to convert the distribution given as a ratio into a fraction form to help them consider what each player has before and after the game.