Calculate the Note Value Balancing Daniel's Wins and Losses

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):

$-3x - 7x + 2x + 400 = 0$

Step 2: Simplify and solve for $x$.

Combine like terms:

$-3x - 7x + 2x = -8x$

Thus, the equation is:

$-8x + 400 = 0$

Isolate $x$ by subtracting 400 from both sides:

$-8x = -400$

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

$x = \frac{-400}{-8}$ $x = 50$

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

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