Question 6·Hard·Linear Equations in One Variable
Townsend Realty Group Investments
| Property address | Purchase price ($) | Monthly rental price ($) |
|---|---|---|
| Clearwater Lane | 128,000 | 950 |
| Driftwood Drive | 176,000 | 1,310 |
| Edgemont Street | 70,000 | 515 |
| Glenview Street | 140,000 | 1,040 |
| Hamilton Circle | 450,000 | 3,365 |
Townsend Realty purchased the Glenview Street property and received a 40% discount off the original price along with an additional 20% off the discounted price for purchasing the property in cash. Which of the following best approximates the original price, in dollars, of the Glenview Street property?
For multi-step percent discount problems, first convert each discount into the percent you pay (for example, 40% off means you pay 60%, or 0.6) and then multiply these decimals to get one combined multiplier. Let be the original price, set (combined multiplier) equal to the final (given) price, and solve the simple linear equation. Avoid the common mistake of adding discount percentages; for sequential discounts on the SAT, you almost always need to multiply the remaining percents instead.
Hints
What do the discounts mean in terms of percent paid?
Instead of thinking about how much is taken off, think about how much of the price is left after each discount. If a price is reduced by 40%, what percent of the original price do you actually pay? What about a 20% discount?
Combine the discounts correctly
The two discounts are applied one after the other, so you should multiply the remaining percents (as decimals), not add the percentages. First multiply by the percent you pay after a 40% discount, then multiply by the percent you pay after a 20% discount.
Write an equation for the original price
Let be the original price. After both discounts, the price is some decimal times , and the table says the purchase price is $140,000. Write an equation of the form (decimal) and solve for , then choose the option closest to your result.
Desmos Guide
Compute the original price from the final price
Since the final price is of the original, you can enter the expression 140000/0.48 into Desmos. The value Desmos outputs is the original price; compare that value to the answer choices and select the closest one.
Step-by-step Explanation
Translate each discount into a multiplier
A discount of 40% off means the buyer pays the remaining 60% of the original price.
- Paying 60% is the same as multiplying by (or ).
Then there is an additional 20% off the discounted price, so the buyer pays 80% of that discounted price.
- Paying 80% is the same as multiplying by (or ).
The combined effect of the two discounts is the product of these multipliers:
So the final price is () of the original price.
Relate the final (purchase) price to the original price
Let be the original price of the Glenview Street property.
After both discounts, the price is .
From the table, the purchase price for Glenview Street is , so we set up the equation:
Now we just need to solve this equation for .
Solve the equation and match to the closest answer choice
Solve by dividing both sides by :
Compute the fraction:
Rounding to the nearest hundred gives approximately , which corresponds to answer choice B) $291,700.