Question 138·Hard·Linear Equations in One Variable
Redwood Antiques Auction Fees
At Redwood Antiques, the buyer of an item must pay two additional charges on top of the winning bid:
- a buyer’s premium equal to 12% of the winning bid, and
- a local sales tax of 6.5% applied to the sum of the winning bid and the buyer’s premium.
If the total amount paid for an antique vase was $1,135.58, which of the following amounts is closest to the winning bid for the vase?
For percentage mark-up problems that give you the final total and ask for the original amount, always turn each percentage increase into a multiplier: adding 12% becomes multiplying by , adding 6.5% becomes multiplying by , and so on. Multiply these factors together to get a single overall multiplier on the original price, set that equal to the given total, then divide to solve for the original amount. Finally, if the question asks for the closest amount, compute or approximate the result and pick the nearest answer choice rather than trying to overcomplicate the arithmetic.
Hints
Represent the unknown
Let be the winning bid for the vase. How can you write the 12% buyer’s premium in terms of ?
Build the subtotal before tax
Once you have the buyer’s premium , add it to to get the subtotal before tax. Express that subtotal as a single multiple of .
Incorporate the tax correctly
The 6.5% tax is applied to the subtotal (bid + premium), not just the bid. Write the tax as 6.5% of the subtotal, and then add it to the subtotal to get an expression for the total amount paid.
Set up and solve the equation
Set your expression for the total equal to 1135.58, combine the constants into a single multiplier on , then divide to find and pick the closest answer choice.
Desmos Guide
Compute the original bid with one expression
In Desmos, type the expression 1135.58 / (1.12 * 1.065) and press Enter. The value Desmos outputs is the winning bid; compare that number to the answer choices and select the one that is closest.
Step-by-step Explanation
Define the variable and write the premium
Let the winning bid be dollars.
The buyer’s premium is 12% of the winning bid, so it is .
So, the subtotal before tax (bid + premium) is
Express the sales tax and total paid
The local sales tax is 6.5% of the subtotal .
So the tax amount is .
Total paid :
We know this total equals , so:
Combine the multipliers and solve for b symbolically
First multiply the constants:
So the equation becomes
Solve for by dividing both sides by :
Now we just need to evaluate this quotient and compare it to the choices.
Compute the bid and choose the closest option
Evaluate the quotient (by hand, calculator, or estimation):
So the winning bid is about 952 dollars, which is closest to $952 (choice B).