Calculate the Note Value Balancing Daniel's Wins and Losses

Daniel bets on three games. In the first game, he lost three notes. In the second game, he lost 7 notes. In the third game, he won 2 notes and another £400. In total, Daniel left with the same amount of money he started with.

What is the value of each note?

To solve this problem, we'll follow these steps:

• Step 1: Identify the given information and express losses and gains as an equation.
• Step 2: Simplify the equation to solve for the value of a note $x$.

Now, let's work through each step:

Step 1: Define the total outcome equation given the losses and gains.
Daniel starts with an unknown amount equivalent to his final amount.

In the first game, he loses 3 notes, resulting in a loss of $-3x$.
In the second game, he loses 7 notes, resulting in a loss of $-7x$.
In the third game, he wins 2 notes, resulting in a gain of $+2x$, and he also wins an additional £400.

We equate the total changes to start with zero (final balance being the start):

Step 2: Simplify and solve for $x$.

Combine like terms:

Thus, the equation is:

Isolate $x$ by subtracting 400 from both sides:

Divide by $-8$ to solve for $x$:

Therefore, each note is worth $\text{£50}$.

The value of each note is, therefore, $\pounds 50$.