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Discounts and Sale Prices

29 questions

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Question 21Question →

A local hardware store is purchasing a bulk wholesale order of industrial shelving. The distributor offers a 20%20\% volume discount on the order. Because the hardware store pays the invoice in full within 10 days, they receive an additional 5%5\% discount on the reduced price. If the hardware store pays $3,420\$3,420 for the shelving (before shipping and taxes), what was the original price of the bulk order?

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Answer: $4,500\$4,500

Answer

$4,500\$4,500
The correct answer is found by representing the unknown original price as a variable, applying the 20% discount (multiplying by 0.80), and then applying the 5% discount to that new amount (multiplying by 0.95). Setting the resulting expression (0.76 times the original price) equal to the final price of 3,420anddividingyieldsanoriginalpriceof3,420 and dividing yields an original price of 4,500.

Step-by-Step Solution

1
Set up an equation representing the first discount.
Let the original price be PP. A 20%20\% discount means the price becomes 100%−20%=80%100\% - 20\% = 80\% of the original price, or 0.80P0.80P.
The volume discount is applied to the starting original price.
2
Apply the second successive discount to the result of the first.
An additional 5%5\% discount on the reduced price means the hardware store pays 100%−5%=95%100\% - 5\% = 95\% of that reduced price. This gives 0.95×0.80P=0.76P0.95 \times 0.80P = 0.76P.
Successive discounts are calculated based on the new, already-discounted price, not the original price.
3
Solve for the original price PP.
Set the expression equal to the final paid amount: 0.76P=34200.76P = 3420. Divide both sides by 0.760.76 to find P=$4,500P = \$4,500.
This isolates PP to determine the starting value before any discounts were applied.

Key Concept

Calculating the original price after successive discounts by working backward using algebraic representation.
Question 22Question →

Read the scenario below and fill in the missing financial values.

Fill in the blanks below

A musician is purchasing a new drum kit for $1,200\$1,200 and a set of cymbals for $400\$400. The music store is running a promotion that offers a 15%15\% discount on the total purchase if it exceeds $1,000\$1,000. After the discount is applied, a 6%6\% state sales tax is added to the sale price.

Based on this information, the total amount saved from the discount is $\$,andthefinalamountpaidaftertaxis, and the final amount paid after tax is \$$.
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Answer

The discount amount is $240\$240, and the final total paid is $1,441.60\$1,441.60.
The solution requires finding the total original price of $1,600\$1,600. The 15%15\% discount on this amount is $240\$240, leaving a sale price of $1,360\$1,360. Finally, calculating a 6%6\% sales tax on $1,360\$1,360 adds $81.60\$81.60, bringing the final total to $1,441.60\$1,441.60.

Step-by-Step Solution

1
Calculate the total original price of the items.
$1,200+$400=$1,600\$1,200 + \$400 = \$1,600
To find the base amount before any discounts are applied.
2
Calculate the discount amount.
15%15\% of $1,600=0.15×1600=$240\$1,600 = 0.15 \times 1600 = \$240
Since the purchase exceeds $1,000\$1,000, the 15%15\% discount applies to the total original price. This provides the first missing value.
3
Calculate the sale price before tax.
$1,600−$240=$1,360\$1,600 - \$240 = \$1,360
Subtracting the discount from the original price gives the taxable amount.
4
Calculate the sales tax.
6%6\% of $1,360=0.06×1360=$81.60\$1,360 = 0.06 \times 1360 = \$81.60
Sales tax is calculated based on the discounted sale price.
5
Calculate the final total paid.
$1,360+$81.60=$1,441.60\$1,360 + \$81.60 = \$1,441.60
Adding the sales tax to the sale price yields the final cost. This provides the second missing value.

Key Concept

Calculating a percentage discount on a total sum and then applying sales tax to the discounted price.

Alternative Method

You can calculate the final price in one continuous equation by multiplying the original price by the complement of the discount (1−0.15=0.851 - 0.15 = 0.85) to get the sale price, and then multiplying by the tax multiplier (1+0.06=1.061 + 0.06 = 1.06): $1,600×0.85×1.06=$1,441.60\$1,600 \times 0.85 \times 1.06 = \$1,441.60.
Estimated Time:1m 30s
Question 23Question →

A fleet manager is purchasing a replacement heavy-duty transmission. The original price of the transmission is $3,200\$3,200. The supplier offers a 15%15\% commercial discount, and an additional 5%5\% discount off the reduced price if the invoice is paid within 1010 days. If the manager pays the invoice on the same day, what is the final price of the transmission before any taxes?

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Answer: $2,584\$2,584

Answer

The final price of the transmission is $2,584\$2,584.
The correct answer accurately calculates the successive discounts step by step. First, a 15%15\% discount on $3,200\$3,200 saves $480\$480, bringing the price to $2,720\$2,720. Then, a 5%5\% discount on the new price of $2,720\$2,720 saves $136\$136, resulting in a final price of $2,584\$2,584.

Step-by-Step Solution

1
Calculate the price after the first discount of 15%15\%.
$3,200×0.15=$480\$3,200 \times 0.15 = \$480 in savings. The new price is $3,200−$480=$2,720\$3,200 - \$480 = \$2,720.
The commercial discount is applied to the original price of the item.
2
Calculate the final price after the second discount of 5%5\%.
$2,720×0.05=$136\$2,720 \times 0.05 = \$136 in savings. The final price is $2,720−$136=$2,584\$2,720 - \$136 = \$2,584.
Successive (chain) discounts are always applied to the previously reduced price, not the original price.

Key Concept

Discounts and Sale Prices
Question 24Question →

A commercial photographer is purchasing a new studio lighting kit. The original retail price of the kit is $1,200\$1,200. The camera supply store is currently offering a 25%25\% seasonal discount on all studio equipment. In addition, the photographer receives a 10%10\% professional discount, which is applied to the already reduced price. What is the final price of the lighting kit before sales tax?

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Answer: 810

Answer

The final price of the lighting kit is $810\$810.
The correct calculation requires applying the first discount to find a new subtotal, and then applying the second discount to that new subtotal. Applying 25%25\% off $1,200\$1,200 yields $900\$900. Applying the 10%10\% professional discount to the new base of $900\$900 reduces the price by another $90\$90, resulting in a final price of $810\$810.

Step-by-Step Solution

1
Calculate the first discount amount.
25%25\% of $1,200\$1,200 is $300\$300.
The seasonal discount applies to the original retail price.
2
Find the intermediate reduced price.
$1,200−$300=$900\$1,200 - \$300 = \$900.
Subtracting the first discount gives the new base price before the second discount.
3
Calculate the second discount amount.
10%10\% of $900\$900 is $90\$90.
The professional discount applies to the already reduced price, not the original price.
4
Determine the final price.
$900−$90=$810\$900 - \$90 = \$810.
Subtracting the second discount from the intermediate price gives the final amount paid.

Key Concept

Calculating successive discounts requires applying the second percentage to the new, already discounted price, rather than the original amount.
Question 25Question →

A cafe owner is purchasing a commercial espresso machine. Supplier A lists the machine for $2,000\$2,000 with a single 30%30\% discount. Supplier B lists the identical machine for $2,400\$2,400 with successive discounts of 25%25\% and 20%20\%. How much money will the owner save by purchasing the machine from the less expensive supplier?

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Answer: $40\$40

Answer

The owner will save $40 by choosing Supplier A.
Supplier A's final price is 1,400(whichis1,400 (which is 2,000 minus a 600discount).SupplierB′sfinalpriceis600 discount). Supplier B's final price is 1,440 (calculated by first reducing 2,400by252,400 by 25% to get 1,800, and then reducing 1,800by201,800 by 20% to get 1,440). The difference between the two prices is $40, making Supplier A the less expensive option by that amount.

Step-by-Step Solution

1
Calculate the final price for Supplier A.
$2,000×0.30=$600\$2,000 \times 0.30 = \$600 discount. $2,000−$600=$1,400\$2,000 - \$600 = \$1,400.
A single discount is taken directly from the initial price.
2
Calculate the price after the first discount for Supplier B.
$2,400×0.25=$600\$2,400 \times 0.25 = \$600 discount. $2,400−$600=$1,800\$2,400 - \$600 = \$1,800.
The first of the successive discounts is applied to the original listing price.
3
Calculate the price after the second discount for Supplier B.
$1,800×0.20=$360\$1,800 \times 0.20 = \$360 discount. $1,800−$360=$1,440\$1,800 - \$360 = \$1,440.
The second discount must be applied to the new subtotal, not the original price.
4
Find the difference between the two final prices.
$1,440−$1,400=$40\$1,440 - \$1,400 = \$40.
Subtracting the lower price from the higher price gives the total savings.

Key Concept

Calculating successive discounts and comparing final sale prices.
Question 26Question →

An auto repair shop owner orders a bulk supply of replacement brake pads with an original total cost of $800\$800. The automotive parts distributor applies a 15%15\% bulk discount to the order. In addition, the shop receives an extra 5%5\% discount on the reduced price for paying the invoice in full upon delivery.

Determine the shop's final purchase price for the brake pads and the total amount of money saved compared to the original cost.

Fill in the blanks below

The final purchase price is $\$andthetotalamountsavedis and the total amount saved is \$$.
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Answer

The final purchase price is 646andthetotalamountsavedis646 and the total amount saved is 154.
The correct final price is found by taking 15% off 800toget800 to get 680, then taking 5% off 680toarriveat680 to arrive at 646. The total savings is the sum of both discounts (120+120 + 34 = $154).

Step-by-Step Solution

1
Calculate the amount of the 15% bulk discount.
$800×0.15=$120\$800 \times 0.15 = \$120
To find the savings provided by the initial distributor discount.
2
Calculate the intermediate price after the first discount.
$800−$120=$680\$800 - \$120 = \$680
The subsequent 5% discount must be applied to this reduced price, not the original price.
3
Calculate the amount of the 5% payment discount.
$680×0.05=$34\$680 \times 0.05 = \$34
To determine the additional savings from paying the invoice in full upon delivery.
4
Calculate the final purchase price.
$680−$34=$646\$680 - \$34 = \$646
This is the final amount the shop must pay after all successive discounts have been applied.
5
Calculate the total amount saved.
$120+$34=$154\$120 + \$34 = \$154
To find the complete difference between the original cost and the final price.

Key Concept

Successive discounts must be calculated sequentially on the new reduced price at each step, rather than adding the percentages together.
Question 27Question →

A restaurant manager orders a bulk supply of premium cooking oil. The invoice shows a final cost of $510\$510 after a 15%15\% wholesale discount was applied to the standard price. What was the original standard price of the bulk supply before the discount?

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Answer: $600.00\$600.00

Answer

The original standard price of the bulk supply was $600.00.
The final cost of 510represents85510 represents 85% of the original price, because a 15% discount was subtracted (100% - 15% = 85%). To find the original whole amount, divide the part by the percentage as a decimal: 510 \div 0.85 = \600600.

Step-by-Step Solution

1
Determine the percentage of the original price that the final cost represents.
Since a 15%15\% discount was applied, the manager paid 100%−15%=85%100\% - 15\% = 85\% of the original price.
The final discounted price represents the remainder of the original cost after the discount is subtracted.
2
Set up an equation and solve for the original price.
Let xx be the original price. The equation is 0.85x=5100.85x = 510. Solving for xx gives x=510÷0.85=600x = 510 \div 0.85 = 600.
Dividing the part (the discounted price) by its equivalent percentage (expressed as a decimal) gives the whole original amount.

Key Concept

To find the original price before a discount, divide the final discounted price by the complement of the discount percentage (100% minus the discount rate).
Question 28Question →

An IT director is upgrading a company's data center and orders 4 new enterprise servers. The original retail price for each server is $4,500\$4,500. The hardware supplier offers a 20%20\% corporate discount on the total order. Additionally, because the company pays the invoice in full within 10 days, a 5%5\% early-payment discount is applied to the already reduced price. What is the final total price, in dollars, paid for the servers?

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Answer: 13680

Answer

13680
The correct total is found by first determining the initial total order value ($18,000\$18,000), reducing it by the initial 20%20\% discount to find the new subtotal ($14,400\$14,400), and then reducing that subtotal by the additional 5%5\% discount to arrive at the final cost ($13,680\$13,680).

Step-by-Step Solution

1
Calculate the total original price of the 4 servers.
4×$4,500=$18,0004 \times \$4,500 = \$18,000
The discounts apply to the total order, so the total initial cost must be determined first.
2
Apply the 20% corporate discount to the total original price.
\$18,000 \times 0.20 = \$3,600 \text{ discount; } \$18,000 - \$3,600 = \$14,400
The first discount reduces the total cost before any other discounts are considered.
3
Apply the 5% early-payment discount to the new, reduced price.
\$14,400 \times 0.05 = \$720 \text{ discount; } \$14,400 - \$720 = \$13,680
Successive discounts are applied sequentially to the most recently calculated price, not the original price.

Key Concept

Calculating successive discounts on a total order amount.
Estimated Time:1m 30s
Question 29Question →

A mechanic buys a new diagnostic tool for their shop. The tool is first marked down by 20%20\% for a storewide clearance. The mechanic then applies a special promotional code for an additional 15%15\% off the clearance price. Finally, a 6%6\% sales tax is applied to the discounted price.

If the total amount the mechanic paid, including the sales tax, was $864.96\$864.96, calculate the original price of the diagnostic tool before any discounts were applied.

Fill in the blanks below

The original price of the diagnostic tool was $\$$.
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Answer

The original price of the diagnostic tool was $1,200.
To find the original price, we must reverse each transaction step starting from the final price. First, we remove the 6%6\% sales tax by dividing $864.96\$864.96 by 1.061.06 to get $816.00\$816.00. Next, we reverse the 15%15\% discount by dividing $816.00\$816.00 by 0.850.85 (since the discounted price represents 85%85\% of the previous price), resulting in $960.00\$960.00. Finally, we reverse the 20%20\% clearance discount by dividing $960.00\$960.00 by 0.800.80, which yields the original price of $1,200.00\$1,200.00.

Step-by-Step Solution

1
Remove the 6%6\% sales tax from the final total to find the price of the tool after all discounts.
$864.96÷1.06=$816.00\$864.96 \div 1.06 = \$816.00
The final total represents 106%106\% of the discounted price. Dividing by 1.061.06 mathematically reverses the tax application.
2
Reverse the 15%15\% promotional discount to find the initial clearance price.
$816.00÷0.85=$960.00\$816.00 \div 0.85 = \$960.00
The price after the promo code was applied represents 85%85\% (100%−15%100\% - 15\%) of the clearance price. Dividing by 0.850.85 finds the value before this second discount.
3
Reverse the initial 20%20\% clearance markdown to determine the original price.
$960.00÷0.80=$1,200.00\$960.00 \div 0.80 = \$1,200.00
The clearance price represents 80%80\% (100%−20%100\% - 20\%) of the original price. Dividing by 0.800.80 finds the starting value.

Key Concept

Reversing successive percentage changes (chain discounts and taxes) using inverse operations to find an original value.
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