Solving a Money Puzzle: Using Algebra to Find Out How Much Each Person Has

Let's dive into a fun money puzzle involving Marie, Sonia, and Toni. By using algebra, we'll figure out how much money each of them has.

Introduction to the Problem

We know that Marie, Sonia, and Toni have a total of $580. Additionally, Sonia has $120 more than Marie, and Toni has $190 more than Sonia. The puzzle is to determine how much money Toni has. This problem will be solved using algebra, specifically by translating common language into mathematical language and solving the resulting equations.

Puzzle Breakdown

To set up our problem, let's assign variables to the unknowns:

M: The amount of money Marie has S: The amount of money Sonia has T: The amount of money Toni has

The problem states that the total amount of money is $580, so we can write the first equation as:

M S T 580

Given that Toni has $190 more than Sonia, we can write:

T S 190

Similarly, since Sonia has $120 more than Marie, we can express:

S M 120

Algebraic Transformation

Let's start by expressing all variables in terms of one another:

Since T is expressed in terms of S, and S is in terms of M, we can substitute:

First, replace S with M 120 in the equation for T:

T (M 120) 190

T M 310

Now we have:

T M 310 S M 120 M S T 580

Substitute the expressions for S and T into the total money equation:

M (M 120) (M 310) 580

3M 430 580

Solving for M

Now, let's solve for M:

3M 430 580

3M 580 - 430

3M 150

M 150 / 3

M 50

Finding S and T

Now that we have M, we can find S and T:

First, find S:

S M 120

S 50 120

S 170

Next, find T:

T M 310

T 50 310

T 360

Verification

Finally, let's verify that the solution works:

M S T 50 170 360

M S T 580

The solution checks out, so Toni has $360.

Conclusion

We've successfully used algebra to solve the money puzzle. The final answer is that Toni has $360. This method teaches us how to break down and solve complex problems systematically by translating language into mathematical expressions.