To gain more customers, the owners of Harrold's Hobby Shop advertised their new train set for 50% off with the purchase of a $10 membership card. The next month, after not receiving much business, they advertised the train set for 75% off with the purchase of a $20 membership card. Bonnie, always the thrifty shopper, realized that the total price of the train set and membership card did not change.

Question

What is the price of the train set without any discounts?

Answer 1

Let the price of the train set without any discounts be \( x \).

In the first month, the train set is advertised for 50% off with the purchase of a $10 membership card. The price after the discount is:

\[ \text{Discounted price} = x - 0.5x = 0.5x \]

The total cost for the train set and membership card in this month is:

\[ \text{Total cost} = 0.5x + 10 \]

In the second month, the train set is advertised for 75% off with the purchase of a $20 membership card. The price after the discount is:

\[ \text{Discounted price} = x - 0.75x = 0.25x \]

The total cost for the train set and membership card in this month is:

\[ \text{Total cost} = 0.25x + 20 \]

According to the problem, Bonnie realized that the total prices did not change, which gives us:

\[ 0.5x + 10 = 0.25x + 20 \]

Now, we'll solve this equation for \( x \).

First, isolate \( x \):

\[ 0.5x - 0.25x = 20 - 10 \]

\[ 0.25x = 10 \]

Now, solve for \( x \):

\[ x = \frac{10}{0.25} = 40 \]

Thus, the price of the train set without any discounts is \(\boxed{40}\).