Matt gets a `$1,000` commission on a big sale. This commission alone raises his average commission by `$150.` If Matt's new average commission is `$400`, how many sales has Matt made?

Answer

To solve the problem step-by-step, we can follow these steps:

### Step 1: Define Variables
Let:
- \( X \) = original average commission
- \( Y \) = number of sales Matt has made

### Step 2: Calculate Total Commission
The total commission before the big sale is:
\[ \text{Total Commission} = X \times Y \]

### Step 3: Calculate New Total Commission
After the big sale, Matt receives an additional $1,000, so the new total commission becomes:
\[ \text{New Total Commission} = X \times Y + 1000 \]

### Step 4: Calculate New Average Commission
The new average commission after the sale is given by:
\[ \text{New Average Commission} = \frac{X \times Y + 1000}{Y + 1} \]
According to the problem, this new average commission is $400. Therefore, we can set up the equation:
\[ \frac{X \times Y + 1000}{Y + 1} = 400 \]

### Step 5: Rearranging the Equation
Cross-multiply to eliminate the fraction:
\[ X \times Y + 1000 = 400 \times (Y + 1) \]
Expanding the right side gives:
\[ X \times Y + 1000 = 400Y + 400 \]

### Step 6: Simplifying the Equation
Rearranging the equation leads to:
\[ X \times Y - 400Y + 1000 - 400 = 0 \]
This simplifies to:
\[ X \times Y - 400Y + 600 = 0 \]
Or:
\[ X \times Y - 400Y = -600 \]

### Step 7: Use the Information About Average Increase
We know that the new average commission is $150 more than the old average:
\[ 400 = X + 150 \]
From this, we can find \( X \):
\[ X = 400 - 150 = 250 \]

### Step 8: Substitute \( X \) Back into the Equation
Now substitute \( X = 250 \) into the equation:
\[ 250Y - 400Y = -600 \]
This simplifies to:
\[ -150Y = -600 \]
Dividing both sides by -150 gives:
\[ Y = 4 \]

### Step 9: Calculate Total Sales
Since \( Y \) is the number of sales made before the big sale, the total number of sales after the big sale is:
\[ Y + 1 = 4 + 1 = 5 \]

### Final Answer
Thus, the total number of sales Matt has made is:
\[ \boxed{5} \]