Question

to gain more customers, the owners of harolds 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. what is the price of the train set without any discounts? $

Explanation:

Step1: Define the variable

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

Step2: Set up the equation for the first scenario

With a \( \$10 \) membership card and \( 50\% \) off the train set, the total cost is \( 0.5x + 10 \).

Step3: Set up the equation for the second scenario

With a \( \$20 \) membership card and \( 75\% \) off the train set, the total cost is \( 0.25x + 20 \).

Step4: Equate the two total costs

Since the total price (train set + membership card) did not change, we set the two expressions equal to each other:
\[
0.5x + 10 = 0.25x + 20
\]

Step5: Solve for \( x \)

Subtract \( 0.25x \) from both sides:
\[
0.5x - 0.25x + 10 = 20
\]
\[
0.25x + 10 = 20
\]
Subtract 10 from both sides:
\[
0.25x = 20 - 10
\]
\[
0.25x = 10
\]
Divide both sides by \( 0.25 \):
\[
x = \frac{10}{0.25}
\]
\[
x = 40
\]

Answer:

The price of the train set without any discounts is \(\$40\).