Mary, Joe, Cathy, Anne, and Tony each have some $1 bills. Mary has one of them, Joe has two, Cathy has three, Anne has four, and Tony has five. Each person may also have a $5 bill. None of them has any other kind of money.
At least one person has more money than Joe and less money than Tony. Mary does not have less money than both Cathy and Anne. Tony does not have more money than both Mary and Joe. The person with the most money has six dollars more than the person with the least amount of money. How much does each person have?
The wording is ambiguous
Mary does not have less money than both Cathy and Anne
Does that mean that M >= C+A or that M>=C or M>=A ?
same for
Tony does not have more money than both Mary and Joe
To solve this problem, let's assign variables to represent the number of $1 bills each person has. Let's say Mary has x $1 bills, Joe has y $1 bills, Cathy has z $1 bills, Anne has w $1 bills, and Tony has v $1 bills.
We know that:
- Mary has one of the $1 bills, so x = 1.
- Joe has two $1 bills, so y = 2.
- Cathy has three $1 bills, so z = 3.
- Anne has four $1 bills, so w = 4.
- Tony has five $1 bills, so v = 5.
Now, let's consider the additional conditions given:
1. At least one person has more money than Joe and less money than Tony.
This means there is at least one person whose number of $1 bills is greater than 2 (Joe's amount) but less than 5 (Tony's amount). Given the numbers we already have, this condition is already satisfied, so we don't need to make any changes.
2. Mary does not have less money than both Cathy and Anne.
Mary has 1 $1 bill, so we need to determine if she has more money than both Cathy and Anne. To do that, we need to compare their values of $1 bills.
- Mary's $1 bills: x = 1
- Cathy's $1 bills: z = 3
- Anne's $1 bills: w = 4
Since Mary has fewer $1 bills than both Cathy and Anne, this condition is not satisfied. We need to give Mary a $5 bill to make her have more money than Cathy and Anne. So, x = 1 + 5.
3. Tony does not have more money than both Mary and Joe.
Tony's $1 bills: v = 5
Mary's $1 bills: x = 1 + 5 = 6
Joe's $1 bills: y = 2
Since Tony has more $1 bills (5) than both Mary (6) and Joe (2), this condition is not satisfied. We need to take away one of Tony's $1 bills and give it to Joe. So, v = 5 - 1 and y = 2 + 1.
Now, we have the following values:
- Mary's $1 bills: x = 6
- Joe's $1 bills: y = 3
- Cathy's $1 bills: z = 3
- Anne's $1 bills: w = 4
- Tony's $1 bills: v = 4
Finally, we know that the person with the most money has six dollars more than the person with the least amount of money. Since we've assigned $1 bills to everyone, we can conclude that the person with the most $1 bills also has six more $1 bills than the person with the least $1 bills.
In this case, Mary has the least amount of money with 6 $1 bills, and Tony has the most with 4 $1 bills. Since we know the person with the most money has six dollars more, we can infer that Tony has 4 + 6 = 10 $1 bills.
Therefore, the final solution is:
- Mary has 6 $1 bills.
- Joe has 3 $1 bills.
- Cathy has 3 $1 bills.
- Anne has 4 $1 bills.
- Tony has 10 $1 bills.