An electrical store is having a sale. If a customer spends a total of $3,500, then the payable price for the television is 80% of that portion that is left after paying for the refrigerator. The payable price of the refrigerator is 85% of that portion that is left after the television has been paid for. Find the payable price of the television and refrigerator.

Let's denote the payable price of the television by $ T $ and the payable price of the refrigerator by $ R $.

According to the problem, we have the following two conditions:

1. The payable price for the television is 80% of the amount left after paying for the refrigerator:
   \[
   T = 0.8 \times (3500 - R)
   \]

2. The payable price of the refrigerator is 85% of the amount left after paying for the television:
   \[
   R = 0.85 \times (3500 - T)
   \]

We now have two equations:
\[
T = 0.8 \times (3500 - R)
\]
\[
R = 0.85 \times (3500 - T)
\]

### Step 1: Solve for $ T $ and $ R $

First, let's substitute the expression for $ T $ from the first equation into the second equation:

\[
R = 0.85 \times \left(3500 - 0.8 \times (3500 - R)\right)
\]

Now, let's simplify the equation:

\[
R = 0.85 \times \left(3500 - 0.8 \times 3500 + 0.8R\right)
\]
\[
R = 0.85 \times \left(3500(1 - 0.8) + 0.8R\right)
\]
\[
R = 0.85 \times \left(3500 \times 0.2 + 0.8R\right)
\]
\[
R = 0.85 \times \left(700 + 0.8R\right)
\]
\[
R = 0.85 \times 700 + 0.85 \times 0.8R
\]
\[
R = 595 + 0.68R
\]

### Step 2: Isolate $ R $

Now, let's isolate $ R $ on one side of the equation:

\[
R - 0.68R = 595
\]
\[
0.32R = 595
\]
\[
R = \frac{595}{0.32}
\]
\[
R \approx 1859.375
\]

### Step 3: Find $ T $

Substitute the value of $ R $ back into the first equation to find $ T $:

\[
T = 0.8 \times (3500 - 1859.375)
\]
\[
T = 0.8 \times 1640.625
\]
\[
T \approx 1312.5
\]

### Final Answer:

- The payable price of the television is approximately **$1,312.50**.
- The payable price of the refrigerator is approximately **$1,859.38**.

Would you like further details or have any questions about this solution?

Here are 5 related questions to expand your understanding:

1. How would the solution change if the total amount spent was different, say $4,000?
2. What if the television's discount rate was different, such as 75%?
3. How can we solve this problem using a system of linear equations?
4. What would be the effect on the solution if the refrigerator's discount rate were increased to 90%?
5. Can you derive a general formula for $ T $ and $ R $ based on different percentages and a different total amount?

**Tip:** Always double-check your algebraic manipulations to ensure accuracy in solving equations.

Learn how to calculate the payable prices for a television and refrigerator when specific percentage discounts apply based on a total spending amount of $3,500. Follow a step-by-step solution involving algebraic equations and percentage calculations.

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