answer:Let's break down the problem step by step. First, let's find out the undiscounted amount of the sale. The customer received a 5% discount on the 13,000 sale. To find the undiscounted amount, we need to calculate the original price before the discount. Let's call the original price before the discount P. [ P - 5% times P = 13,000 ] [ P times (1 - 0.05) = 13,000 ] [ P times 0.95 = 13,000 ] [ P = frac{13,000}{0.95} ] [ P = 13,684.21 ] Now, let's calculate the amount that would have been charged with an additional 8% sales tax on the undiscounted amount. [ Taxed Amount = P + 8% times P ] [ Taxed Amount = P times (1 + 0.08) ] [ Taxed Amount = 13,684.21 times 1.08 ] [ Taxed Amount = 14,779.34 ] Now, let's calculate the commission Matt earned on this big sale with the 15% increase, which is 11.5% of the sale. [ Commission = 11.5% times 13,000 ] [ Commission = 0.115 times 13,000 ] [ Commission = 1,495 ] This 1,495 commission alone raises his average commission by 150. So, without this commission, his average commission would have been 400 - 150 = 250. Let's call the number of sales Matt has made N. The total commission from all sales, including the big sale, is 400 * N. The total commission from all sales, excluding the big sale, is 250 * (N - 1). [ 400N = 250(N - 1) + 1,495 ] [ 400N = 250N - 250 + 1,495 ] [ 400N - 250N = 1,495 + 250 ] [ 150N = 1,745 ] [ N = frac{1,745}{150} ] [ N = 11.6333 ] Since the number of sales cannot be a fraction, we round up to the nearest whole number. Therefore, Matt has made boxed{12} sales.

answer:To find the cost of the ball, we can start by setting up an equation based on the given information. Let's denote the cost of the ball as 'x'. Since the bat costs 1.00 more than the ball, the cost of the bat is x + 1.00. We know that the total cost of the bat and the ball together is 1.10. Therefore, we can write the equation: x + (x + 1.00) = 1.10. Combining like terms in the equation, we get 2x + 1.00 = 1.10. To isolate the variable 'x', we need to get rid of the constant term on the left side of the equation. We can do this by subtracting 1.00 from both sides, yielding 2x = 0.10. Dividing both sides by 2, we find that x, the cost of the ball, is equal to 0.05.

answer:To verify that our answer is correct, we can plug the value of x back into the original equation and check if it holds true. We found that the cost of the ball, x, is 0.05. Since the bat costs 1.00 more than the ball, the cost of the bat is 0.05 + 1.00 = 1.05. Now, let's add the cost of the bat and the ball together: 0.05 (ball) + 1.05 (bat) = 1.10. This matches the given total cost of 1.10, so our answer is consistent with the problem's conditions. This check confirms that our calculation is correct, and the ball indeed costs 0.05.

answer:It's interesting that you were presented with two variations of the problem. The way the problem is worded can indeed influence our initial intuition, especially when we're younger and still developing our critical thinking skills. In the first scenario, the bat costs 1.00 more than the ball, which might lead to the incorrect assumption that the ball's price is simply a small addition to the 1.00, rather than considering the total cost and working from there. The second scenario, where the bat is 1.00 less than the ball, is a great example of how a simple change in wording can lead to a different, but equally incorrect, assumption. In this case, the mistake is assuming that the bat's price is a straightforward subtraction of 1.00 from a larger amount, rather than considering the relationship between the two prices. It's great that you can look back on those experiences and recognize how your thinking has evolved. It's a testament to the importance of practicing critical thinking and problem-solving skills, and how they can help us develop a more nuanced understanding of complex problems.