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Discounts, markups, tax, and tip

Learning goals

Identifies the original amount and applicable percent rate.
Finds the amount of a discount, markup, tax, or tip from its percent.
Determines the final amount after the increase or decrease.
Distinguishes 'increase/decrease by' from 'increase/decrease to'.
Compares multi-step offers or charges when relevant.

Difficulty: 1 – 5

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Discounts, markups, tax, and tip 1

Nia earns a \(6\%\) commission on sales as a student representative. During one week, Nia sells \(\$1250\) worth of merchandise. How much commission does Nia earn?

Hints

- Commission is based on a percentage of the sales amount. - Decide what quantity the given percent is taken from. - Check that the commission is only a portion of the total sales.

Solution

1. Convert the commission rate to a decimal: \(6\%=0.06\). 2. Compute \(1250\cdot0.06=75\).

Answer

The commission is \(\$75\).

Discounts, markups, tax, and tip 2

Lucia's café check is \(\$20.00\). She wants to leave a \(15\%\) tip based on the check. How much is the tip?

Hints

- Identify the amount on which the tip is based. - Convert the tip percentage to a decimal. - The question asks for the tip amount only, not the total check.

Solution

1. The tip is \(15\%\) of \(\$20.00\). 2. \(0.15\cdot20=3\). 3. The tip is \(\$3.00\).

Answer

\(\$3.00\)

Discounts, markups, tax, and tip 3

At Maple Street Music, a set of drumsticks costs \(\$35.00\) before tax. The sales tax rate is \(6\%\). How much sales tax is charged?

Hints

- Identify the price before tax. - The tax amount is a percent of that price. - Check that the tax is much smaller than the original price.

Solution

1. Convert the tax rate to a decimal: \(6\%=0.06\). 2. Compute \(35.00\cdot0.06=2.10\).

Answer

The sales tax is \(\$2.10\).

Discounts, markups, tax, and tip 4

Northside Outfitters pays \(\$30.00\) for a hiking daypack and uses a \(20\%\) markup. How many dollars is the markup?

Hints

- The markup is a percent of the store's cost. - Find only the markup amount, not the final selling price. - Check that the markup is one part of the original \(\$30.00\) cost.

Solution

1. Convert the markup rate to a decimal: \(20\%=0.20\). 2. Compute \(30.00\cdot0.20=6.00\).

Answer

The markup is \(\$6.00\).

Discounts, markups, tax, and tip 5

A mountain bike normally costs \(\$320\). It is on sale for \(15\%\) off. a) Find the discount amount. b) Find the sale price.

Hints

- The original price represents \(100\%\). - Find the discount amount, then subtract it from the original price.

Solution

1. The discount is \(\$320 \cdot 0.15 = \$48\). 2. The sale price is \(\$320 - \$48 = \$272\).

Answer

a) \(\$48\) b) \(\$272\)

Discounts, markups, tax, and tip 6

A mountain bike has a list price of \(\$450\). Trailhead Bikes offers \(12\%\) off the list price. Summit Cycle offers a flat \(\$55\) discount. Which offer costs less, and what is the difference between the final prices?

Hints

- Convert Trailhead Bikes' percent discount to dollars. - Find both final prices before comparing them.

Solution

1. Trailhead Bikes' discount is \(\$450 \cdot 0.12 = \$54\), so its final price is \(\$450 - \$54 = \$396\). 2. Summit Cycle's final price is \(\$450 - \$55 = \$395\). 3. Summit Cycle costs less by \(\$396 - \$395 = \$1\).

Answer

Summit Cycle costs less by \(\$1\).

Discounts, markups, tax, and tip 7

A smartphone originally cost \(\$250\) and is now on clearance for \(\$215\). What percent of the original price must now be paid? Also find the percent discount.

Hints

- Use the original price as \(100\%\). - Divide the sale price by the original price. - The percent paid and the percent discounted add to \(100\%\).

Solution

1. The sale price as a percent of the original price is \(\frac{215}{250} = 0.86 = 86\%\). 2. The discount is \(100\% - 86\% = 14\%\).

Answer

The sale price is \(86\%\) of the original price, so the discount is \(14\%\).

Discounts, markups, tax, and tip 8

Two bicycle stores offer discounts on different models. Greenway Cycles takes \(\$90\) off a bicycle originally priced at \(\$450\). Ridgeline Bikes takes \(\$132\) off a bicycle originally priced at \(\$600\). Which store gives the greater percent discount? Support your answer with calculations.

Hints

- Divide each dollar discount by its original price. - Identify the original price as the base in each case. - Compare the resulting percents.

Solution

1. Greenway Cycles' percent discount is \(\frac{90}{450} = 0.20 = 20\%\). 2. Ridgeline Bikes' percent discount is \(\frac{132}{600} = 0.22 = 22\%\). 3. Since \(22\% > 20\%\), Ridgeline Bikes gives the greater percent discount.

Answer

Ridgeline Bikes gives the greater discount: \(22\%\), compared with \(20\%\) at Greenway Cycles.

Discounts, markups, tax, and tip 9

A pair of running shoes costs \(\$42.50\) before tax. The sales tax rate is \(8\%\). Find the sales tax and the total price.

Hints

- Identify the amount to which the tax rate applies. - Find the tax amount before finding the final price. - Check that the total is slightly greater than the listed price.

Solution

1. The sales tax is \(0.08\cdot42.50=3.40\). 2. Add the tax to the price: \(42.50+3.40=45.90\).

Answer

The sales tax is \(\$3.40\), and the total price is \(\$45.90\).

Discounts, markups, tax, and tip 10

Diego's restaurant check is \(\$62.50\) before tip. He leaves an \(18\%\) tip based on that amount. How much is the tip, and how much is the check including the tip?

Hints

- The tip is a percent of the amount stated in the problem. - Keep the tip amount separate from the original check until the final step. - Use the size of an \(18\%\) tip to estimate whether your result is reasonable.

Solution

1. The tip is \(0.18\cdot62.50=11.25\). 2. The total including tip is \(62.50+11.25=73.75\).

Answer

The tip is \(\$11.25\), and the total including tip is \(\$73.75\).

Discounts, markups, tax, and tip 11

A store pays \(\$32.00\) for a backpack from its supplier and marks up the cost by \(45\%\). What retail price does the store set before tax?

Hints

- A markup increases the store's cost rather than reducing it. - Decide what amount the markup percentage is based on. - The retail price should be greater than the supplier cost.

Solution

1. The markup is \(0.45\cdot32=14.40\). 2. Add the markup to the cost: \(32.00+14.40=46.40\).

Answer

The retail price is \(\$46.40\).

Discounts, markups, tax, and tip 12

A backpack is marked \(\$45.00\). After a store discount, its price before tax is \(\$36.00\). The receipt says the discount was \(20\%\). What dollar amount is the base for the \(20\%\) discount? Verify your choice by finding the discount amount and checking that it is \(20\%\) of the base.

Hints

- First compare the marked price with the price after the discount. - The percent discount should reproduce the dollar amount that was taken off. - Test the starting price as the percent base and check whether the calculation matches the discount amount.

Solution

1. The discount amount is \(\$45.00-\$36.00=\$9.00\). 2. The marked price is the starting amount, so test it as the base: \(45.00\cdot0.20=9.00\). 3. Therefore, the base for the discount is \(\$45.00\).

Answer

The base is \(\$45.00\). The discount is \(\$9.00\), and \(20\%\) of \(\$45.00\) is \(\$9.00\).

Discounts, markups, tax, and tip 13

At Pine Street Books, a notebook is marked \(\$32.00\). A coupon reduces the price by \(\$8.00\) before tax. The receipt then shows \(\$1.80\) in sales tax at a rate of \(7.5\%\). Was the sales tax calculated from the marked price or from the discounted price? State the tax base and verify your answer with a calculation.

Hints

- First determine the price after the coupon but before tax. - Test each plausible price by multiplying it by the tax rate. - The correct base must reproduce the \(\$1.80\) tax shown on the receipt.

Solution

1. The discounted price is \(\$32.00-\$8.00=\$24.00\). 2. Tax on the discounted price is \(24.00\cdot0.075=1.80\), which matches the receipt. 3. Tax on the marked price would be \(32.00\cdot0.075=2.40\), which does not match the receipt. 4. Therefore, the sales-tax base is the discounted price, \(\$24.00\).

Answer

The tax base is the discounted price, \(\$24.00\). Since \(24.00\cdot0.075=1.80\), it matches the tax shown on the receipt.

Discounts, markups, tax, and tip 14

A bookstore sells a puzzle for \(\$50\) before tax. The receipt shows \(\$3\) in sales tax. What percent sales tax was charged? Show how you know.

Hints

- Think about how the tax amount compares with the price before tax. - Which amount represents the whole price before tax? - Write the comparison as a percent.

Solution

1. Compare the \(\$3\) tax with the \(\$50\) price before tax. 2. Calculate \(3\div50=0.06=6\%\).

Answer

\(6\%\), since \(3\div50=0.06\).

Discounts, markups, tax, and tip 15

A restaurant bill lists \(\$24\) for food and \(\$1.20\) in sales tax. Mateo leaves a \(15\%\) tip based on the food price before tax. His friend would tip \(15\%\) of the total after tax. How much would each person tip? How much more would Mateo's friend leave?

Hints

- Compare the amounts on which the two people calculate their tips. - How does adding tax change the amount used to calculate a percentage? - Subtract the tips, not the two bills, to find the extra amount.

Solution

1. Mateo tips \(0.15\cdot24=3.60\) dollars. 2. His friend tips \(0.15(24+1.20)=3.78\) dollars. 3. The difference is \(3.78-3.60=0.18\) dollar. The second tip includes the tax in its base.

Answer

Mateo: \(\$3.60\). Friend: \(\$3.78\). The friend leaves \(\$0.18\) more because the tax is included in the amount used for that tip.

Discounts, markups, tax, and tip 16

A craft shop pays \(\$40\) for a lamp and sells it for \(\$54\). Eli says the shop's percent markup is \(14\div54\). Liv says it is \(14\div40\). Who is correct? Find the percent markup and explain which price is the base.

Hints

- Find how much higher the selling price is than the cost. - What does a shop mean when it says it marked up a product? - Which price existed before the shop added its increase?

Solution

1. The shop added \(54-40=14\) dollars to its cost. 2. Markup is measured against the store's cost, \(\$40\), not the sale price, so Liv uses the correct calculation. 3. The percent markup is \(14\div40=0.35=35\%\).

Answer

Liv is correct. The markup is \(\frac{54-40}{40}=0.35=35\%\), since markup compares the increase with the shop's purchase cost.

Discounts, markups, tax, and tip 17

A jacket's regular price is \(\$80\). It is on sale for \(25\%\) off. The store then charges \(8\%\) sales tax. What does the customer pay? A classmate calculates tax on the original \(\$80\). Explain whether that is correct.

Hints

- Which adjustment happens first? - What is the actual jacket price when the cashier adds the tax? - Add the tax only after finding the sale price.

Solution

1. The sale price is \(80(1-0.25)=60\) dollars. 2. Tax is based on \(\$60\): \(0.08\cdot60=4.80\) dollars. 3. The final total is \(60+4.80=64.80\) dollars.

Answer

\(\$64.80\). The classmate is incorrect: after the discount the jacket costs \(\$60\), so the \(8\%\) tax is \(\$4.80\), not \(\$6.40\).

Discounts, markups, tax, and tip 18

A family orders \(\$48\) of food at a restaurant. The sales tax on the food is \(\$3.84\). They leave a tip of \(20\%\) of the food price before tax. Find the amount they pay in all, including the tax and tip.

Hints

- Tax is already given, so do not calculate it again. - The tip is based on the food amount alone. - Add all three amounts after finding the tip.

Solution

1. Tip is \(0.20\cdot48=9.60\) dollars. 2. Add food, tax, and tip: \(48+3.84+9.60=61.44\) dollars.

Answer

\(\$61.44\), including the \(\$48\) food cost, \(\$3.84\) tax, and \(\$9.60\) tip.

Discounts, markups, tax, and tip 19

At a science museum, admission costs \(\$8\) and a gift-shop T-shirt costs \(\$20\). Sales tax is \(5\%\) on the shirt, but admission is tax-free. The receipt shows a total of \(\$29.40\). Is it correct? Find the right total and explain the mistake.

Hints

- How much extra does the receipt charge beyond the two listed prices? - What should you compare that extra amount with? - Which price is subject to sales tax in this purchase?

Solution

1. The two prices total \(8+20=28\) dollars. The receipt includes \(29.40-28=1.40\) dollars in tax, which is \(5\%\) of the full \(\$28\). 2. Only the shirt is taxable, so the correct tax is \(0.05\cdot20=1\) dollar. 3. The correct total is \(8+20+1=29\) dollars. The receipt incorrectly includes tax on admission.

Answer

No. The receipt includes \(\$1.40\) in tax, as though admission were taxable. Only the shirt is taxed, so the tax should be \(\$1.00\) and the correct total is \(\$29.00\).

Discounts, markups, tax, and tip 20

A group orders a pizza for \(\$7.50\), pasta for \(\$8.20\), a large salad for \(\$5.40\), and drinks for \(\$4.80\). The restaurant offers a \(10\%\) discount on the entire bill when the order is at least \(\$30\). The group is considering adding a dessert for \(\$4.50\). Will the final bill with the dessert be greater or less than the bill without it? Show the exact amounts.

Hints

- Find the bill without dessert and check whether it qualifies for the discount. - Add the dessert, apply the discount to the new subtotal, and compare the two final bills.

Solution

1. Without dessert, the bill is \(\$7.50 + \$8.20 + \$5.40 + \$4.80 = \$25.90\). This is below \(\$30\), so there is no discount. 2. With dessert, the subtotal is \(\$25.90 + \$4.50 = \$30.40\), which qualifies for the discount. 3. The discount is \(\$30.40 \cdot 0.10 = \$3.04\). 4. The discounted bill is \(\$30.40 - \$3.04 = \$27.36\). 5. Since \(\$27.36 > \$25.90\), the group pays \(\$27.36 - \$25.90 = \$1.46\) more with dessert.

Answer

The bill is greater with dessert. Without dessert it is \(\$25.90\); with dessert and the discount it is \(\$27.36\), which is \(\$1.46\) more.

Discounts, markups, tax, and tip 21

A sporting-goods store offers discounts on three balls. For which ball is the dollar savings greatest? - Soccer ball: \(20\%\) off \(\$35.00\) - Basketball: \(15\%\) off \(\$48.00\) - Handball: \(25\%\) off \(\$28.00\) Find the savings for each ball and compare them.

Hints

- The question asks for the amount saved, not the sale price. - Find each discount amount using the original price as the base. - Compare the three dollar savings after calculating them.

Solution

1. Soccer ball: \(\$35.00\cdot0.20=\$7.00\). 2. Basketball: \(\$48.00\cdot0.15=\$7.20\). 3. Handball: \(\$28.00\cdot0.25=\$7.00\). 4. Since \(\$7.20>\$7.00\), the basketball gives the greatest savings.

Answer

Soccer ball: \(\$7.00\) saved. Basketball: \(\$7.20\) saved. Handball: \(\$7.00\) saved. The basketball gives the greatest savings.

Discounts, markups, tax, and tip 22

A name-brand soccer ball has an original price of \(\$40\). Two promotions are available. Promotion A: \(25\%\) off the original price. Promotion B: A sale price of \(\$32\). Which promotion costs less, and by how much?

Hints

- Find the final price under each promotion. - A \(25\%\) discount leaves \(75\%\) of the original price to pay.

Solution

1. Under Promotion A, the price is \(75\%\) of the original price: \(\$40 \cdot 0.75 = \$30\). 2. Promotion B costs \(\$32\). 3. Promotion A costs less, and the difference is \(\$32 - \$30 = \$2\).

Answer

Promotion A costs less by \(\$2\).

Discounts, markups, tax, and tip 23

A tablet originally costs \(\$400\). During a promotion, the price is reduced by \(20\%\). After the promotion, the reduced price is increased by \(20\%\). a) Find the promotional price. b) Find the final price after the increase. c) Compare the final price with the original \(\$400\) price. Explain why the final price is not \(\$400\).

Hints

- Apply the two changes in order. - Identify the base amount for the second percent calculation. - The same percent of two different base amounts gives different dollar amounts.

Solution

1. The promotional price is \(\$400 \cdot 0.80 = \$320\). 2. The later increase is based on \(\$320\), so the final price is \(\$320 \cdot 1.20 = \$384\). 3. The final price is lower than the original price because the \(20\%\) increase is calculated from the smaller discounted price, not from \(\$400\).

Answer

a) \(\$320\) b) \(\$384\) c) The final price is \(\$16\) below the original price because the increase uses the reduced price as its base.

Discounts, markups, tax, and tip 24

Two friends buy bicycles on sale. Lara buys a bicycle that was discounted by \(20\%\) and now costs \(\$320\). Tom buys a bicycle that originally cost \(\$500\) and is now sold for \(\$375\). a) What was Lara's bicycle's original price? b) Who received the greater percent discount compared with the original price?

Hints

- After a \(20\%\) discount, what percent of the original price remains? - For Tom, first find the dollar savings. - Compare the percent discounts, not only the dollar amounts saved.

Solution

1. Lara's sale price is \(80\%\) of the original price. The original price was \(\$320 \div 0.80 = \$400\). 2. Tom saved \(\$500 - \$375 = \$125\). 3. Tom's percent discount was \(\frac{125}{500} = 0.25 = 25\%\). 4. Since \(25\% > 20\%\), Tom received the greater percent discount.

Answer

a) \(\$400\) b) Tom, with a \(25\%\) discount compared with Lara's \(20\%\) discount

Discounts, markups, tax, and tip 25

A toy store puts a remote-control car on sale. Its original price is \(\$120\). The price is first reduced by \(25\%\), and then the reduced price is lowered by another \(20\%\). a) What is the price after the second discount? b) What is the overall percent savings compared with the original price?

Hints

- Apply the first discount before finding the second. - Use the reduced price as the base for the second discount. - Compare the total dollar savings with the original price.

Solution

1. After the first discount, the price is \(\$120 \cdot 0.75 = \$90\). 2. After the second discount, the price is \(\$90 \cdot 0.80 = \$72\). 3. The total savings are \(\$120 - \$72 = \$48\). 4. Relative to the original price, the savings are \(\frac{48}{120} = 0.40 = 40\%\).

Answer

a) \(\$72\) b) \(40\%\)

Discounts, markups, tax, and tip 26

A bakery advertisement says: “Our old deal gave one free roll in every group of four identical rolls. Our new deal is even better: one free roll in every group of five identical rolls! You save \(25\%\).” Evaluate the advertisement. Identify its mathematical errors and rewrite it so that it is correct and logical.

Hints

- Convert one fourth and one fifth to percentages. - A better offer must have a greater free-item share. - Make sure the claimed percent matches the current offer.

Solution

1. Under the old deal, one of every four equal-priced rolls is free, so the savings rate is \(\frac{1}{4} = 25\%\). 2. Under the new deal, one of every five equal-priced rolls is free, so the savings rate is \(\frac{1}{5} = 20\%\). 3. The new deal is worse, not better, and its savings rate is \(20\%\), not \(25\%\). 4. A correct improved-deal advertisement is: “Our old deal gave one free roll in every group of five identical rolls, a \(20\%\) savings. Our new deal gives one free roll in every group of four identical rolls, a \(25\%\) savings.”

Answer

The advertisement is incorrect because one free item in a group of five is a \(20\%\) savings, which is less than the old \(25\%\) savings. A correct version is: “Our old deal gave one free roll in every group of five identical rolls, a \(20\%\) savings. Our new deal gives one free roll in every group of four identical rolls, a \(25\%\) savings.”

Discounts, markups, tax, and tip 27

Loft & Pine Furnishings advertises, “Clearance sale—everything reduced by up to \(70\%\)!” The sale details show that only scented candles are \(70\%\) off, while cabinets, tables, and sofas are only \(5\%\) off. Explain how the percent wording creates a misleading impression and why the slogan can still be technically true.

Hints

- Focus on the meaning of “up to.” - Decide whether the statement describes all items, most items, or only the maximum possible discount.

Solution

1. The phrase “up to \(70\%\)” states only the greatest discount, not the discount on every item or on most items. 2. The claim is technically true because at least one category has a \(70\%\) discount and no discount is greater than \(70\%\). 3. It can mislead shoppers because the large \(70\%\) figure is emphasized, while expensive furniture is discounted by only \(5\%\). Someone may incorrectly expect the larger discount to apply broadly.

Answer

The wording is technically true because \(70\%\) is the maximum discount. It is potentially misleading because the maximum applies only to a small category, while the furniture is reduced by just \(5\%\).

Discounts, markups, tax, and tip 28

After a \(20\%\) discount, a tablet costs \(\$480\). Leon says, “It must have cost \(\$576\) before the discount because \(20\%\) of \(\$480\) is \(\$96\), and \(\$480 + \$96 = \$576\).” Explain the error in Leon's reasoning and find the actual price before the discount.

Hints

- What amount is the \(20\%\) discount based on? - After a \(20\%\) discount, what percent of the original price remains? - The original price must be greater than the sale price. - Can a percent of the new price be added back to reverse the discount directly?

Solution

1. After a \(20\%\) discount, the sale price is \(100\% - 20\% = 80\%\) of the original price. 2. Divide the sale price by \(0.80\): \(\$480 \div 0.80 = \$600\). 3. Leon calculated \(20\%\) of the discounted price. However, the discount was \(20\%\) of the original price, which is a different and larger base amount.

Answer

The original price was \(\$600\). Leon's error was finding \(20\%\) of the discounted price instead of \(20\%\) of the original price.

Discounts, markups, tax, and tip 29

A skateboard originally costs \(\$120\). Because of high demand, the price is increased by \(25\%\). One month later, the new price is decreased by \(20\%\). a) Find the price after the increase. b) Find the final price after the decrease. c) Compare the final price with the original \(\$120\) price. What do you notice?

Hints

- Apply the increase before the decrease. - The \(20\%\) decrease is based on the increased price. - Compare the dollar amount added with the dollar amount later subtracted.

Solution

1. After the increase, the price is \(\$120 \cdot 1.25 = \$150\). 2. After the decrease, the price is \(\$150 \cdot 0.80 = \$120\). 3. The final price is equal to the original price. The \(20\%\) decrease of \(\$150\) is \(\$30\), which exactly reverses the \(\$30\) increase.

Answer

a) \(\$150\) b) \(\$120\) c) The final price is the same as the original price.

Discounts, markups, tax, and tip 30

A pair of athletic shoes was discounted in two steps. The original price was \(\$80\). First, the price was reduced by \(10\%\). After a second discount, the shoes cost \(\$54\). a) What was the price after the first discount? b) How many dollars were taken off in the second discount? c) What percent of the price after the first discount was the second discount?

Hints

- Find the intermediate price after the first discount. - Subtract the final price from the intermediate price. - Compare the second dollar discount with the intermediate price.

Solution

1. After the first discount, the price was \(\$80 \cdot 0.90 = \$72\). 2. The second discount was \(\$72 - \$54 = \$18\). 3. Relative to the \(\$72\) intermediate price, the second discount was \(\frac{18}{72} = \frac{1}{4} = 0.25 = 25\%\).

Answer

a) \(\$72\) b) \(\$18\) c) \(25\%\)

Discounts, markups, tax, and tip 31

A home-improvement store sells a patio furniture set for \(\$450\). You may choose one of two discounts. Option 1: One \(20\%\) discount on the original price. Option 2: A \(10\%\) discount followed by another \(10\%\) discount on the reduced price. Find the final price for each option. Which option costs less? Explain why the results are not equal.

Hints

- Apply the discounts in each option in the order stated. - For Option 2, what amount is used as the base for the second \(10\%\)? - Compare both final prices and the amounts discounted in each step.

Solution

1. With Option 1, the final price is \(450\cdot0.80=\$360\). 2. With Option 2, the first discount gives \(450\cdot0.90=\$405\). The second gives \(405\cdot0.90=\$364.50\). 3. Option 1 costs \(364.50-360=\$4.50\) less. 4. In Option 2 the second \(10\%\) reduction is taken from the smaller, already discounted price, so the two discounts do not add to \(20\%\) of the original price.

Answer

Option 1: \(\$360\). Option 2: \(\$364.50\). Option 1 costs \(\$4.50\) less because Option 2's second discount is calculated from the reduced \(\$405\) price, not from the original \(\$450\).

Discounts, markups, tax, and tip 32

Mirela receives a \(15\%\) loyalty discount on a bicycle. The discount saves exactly \(\$72\). What was the bicycle's original price?

Hints

- The discount amount is a percent of the original price. - Use the \(15\%\) part to find \(100\%\).

Solution

1. The \(\$72\) savings is \(15\%\) of the original price. 2. Divide the discount amount by the discount rate: \(\$72 \div 0.15 = \$480\).

Answer

The bicycle's original price was \(\$480\).

Discounts, markups, tax, and tip 33

A store offers a tiered rebate: \(5\%\) of the first \(\$500\) of an item’s price, plus \(3\%\) of any portion above \(\$500\). a) Find the rebate on a smartphone priced at \(\$450\). b) Find the rebate on a camera priced at \(\$1200\). c) For the camera in part b), what percent of the full price is the total rebate? Round to the nearest hundredth of a percent.

Hints

- Determine which portion of each price is assigned to each rebate rate. - For a price above \(\$500\), split the calculation into two parts. - To find an overall percent, compare the total rebate with the full price.

Solution

1. For part a, the entire price is in the first tier, so the rebate is \(\$450 \cdot 0.05 = \$22.50\). 2. For part b, the rebate on the first \(\$500\) is \(\$500 \cdot 0.05 = \$25\). The amount above \(\$500\) is \(\$1200 - \$500 = \$700\), and its rebate is \(\$700 \cdot 0.03 = \$21\). The total rebate is \(\$25 + \$21 = \$46\). 3. For part c, the rebate as a percent of the full price is \(\frac{46}{1200} \cdot 100\% \approx 3.83\%\).

Answer

a) The rebate is \(\$22.50\). b) The rebate is \(\$46\). c) The rebate is approximately \(3.83\%\) of the camera’s price.

Discounts, markups, tax, and tip 34

After a \(15\%\) price increase, an electric bicycle costs \(\$2012.50\). a) Find the original price before the increase. b) By how many dollars did the price increase? c) The seller wants to reduce the current price back to the original price. By what percent must the current price be reduced? Round to the nearest hundredth of a percent.

Hints

- After a \(15\%\) increase, what percent of the original price is the current price? - For part c), use the current price as the base amount. - The dollar increase and the dollar decrease needed to return are equal, but the percent bases are different.

Solution

1. The current price is \(115\%\) of the original price. The original price was \(\$2012.50 \div 1.15 = \$1750\). 2. The price increase was \(\$2012.50 - \$1750 = \$262.50\). 3. To return to the original price, the reduction is based on the current price: \(\frac{262.50}{2012.50} \approx 0.13043 = 13.04\%\).

Answer

a) \(\$1750\) b) \(\$262.50\) c) Approximately \(13.04\%\)

Discounts, markups, tax, and tip 35

A tablet is offered by two online stores. PixelPort: Original price \(\$450\), with a \(10\%\) discount. TechHarbor: Original price \(\$520\), with a \(25\%\) discount. a) Which store has the lower sale price? b) By what percent is the lower price less than the higher price? Round to the nearest tenth of a percent.

Hints

- Find both sale prices first. - For “percent lower than,” use the higher comparison price as the base.

Solution

1. PixelPort's sale price is \(\$450 \cdot 0.90 = \$405\). 2. TechHarbor's sale price is \(\$520 \cdot 0.75 = \$390\), so TechHarbor is less expensive. 3. The price difference is \(\$405 - \$390 = \$15\). 4. Relative to the higher price, the lower price is less by \(\frac{15}{405} \approx 0.0370 = 3.7\%\).

Answer

a) TechHarbor, at \(\$390\) b) Approximately \(3.7\%\) lower

Discounts, markups, tax, and tip 36

A mountain bike is discounted by \(20\%\) and sells for \(\$552.00\). How many dollars are saved compared with the original price?

Hints

- After a \(20\%\) discount, what percent of the original price remains? - Find the original price before calculating the dollar savings. - The question asks for the difference between the original and sale prices.

Solution

1. The sale price is \(80\%\) of the original price. 2. The original price was \(\$552.00 \div 0.80 = \$690.00\). 3. The savings were \(\$690.00 - \$552.00 = \$138.00\).

Answer

The savings are \(\$138.00\).

Discounts, markups, tax, and tip 37

Blue Ridge Cycles increases the prices of all touring bicycles by \(12\%\). One model costs \(\$644\) after the increase. a) Find the original price. b) By how many dollars did the price increase?

Hints

- The new price is greater than \(100\%\) of the original price. - Divide by the increase multiplier to recover the original price. - Subtract the original price from the new price.

Solution

1. The new price is \(112\%\) of the original price, so the multiplier is \(1.12\). 2. The original price was \(\$644 \div 1.12 = \$575\). 3. The price increased by \(\$644 - \$575 = \$69\).

Answer

a) \(\$575\) b) \(\$69\)

Discounts, markups, tax, and tip 38

A laptop is discounted by \(15\%\) and now costs \(\$680.00\). What was the original price?

Hints

- What percent of the original price remains after a \(15\%\) discount? - The original price must be greater than the sale price. - Divide the sale price by the remaining-price multiplier.

Solution

1. After the discount, \(85\%\) of the original price remains. 2. The original price was \(\$680.00 \div 0.85 = \$800.00\).

Answer

The original price was \(\$800.00\).

Discounts, markups, tax, and tip 39

Sun Orchard Juice offers a promotional bottle labeled “\(20\%\) more for the same price.” The bottle now contains \(1.5\,\text{L}\) of juice and costs \(\$2.10\). a) How much juice was in the standard bottle? b) Find the price per liter for each bottle. By how many dollars per liter did the unit price decrease?

Hints

- The promotional amount is \(120\%\) of the standard amount. - Divide price by volume to find price per liter. - Find the standard amount before comparing unit prices.

Solution

1. The promotional amount is \(120\%\) of the standard amount. The standard bottle contained \(1.5\,\text{L} \div 1.20 = 1.25\,\text{L}\). 2. The standard bottle's unit price was \(\$2.10 \div 1.25\,\text{L} = \$1.68/\text{L}\). 3. The promotional bottle's unit price is \(\$2.10 \div 1.5\,\text{L} = \$1.40/\text{L}\). 4. The unit price decreased by \(\$1.68/\text{L} - \$1.40/\text{L} = \$0.28/\text{L}\).

Answer

a) \(1.25\,\text{L}\) b) Standard: \(\$1.68/\text{L}\); promotional: \(\$1.40/\text{L}\); decrease: \(\$0.28/\text{L}\)

Discounts, markups, tax, and tip 40

Circuit Corner advertises a “double discount” on a television. First, the price is reduced by \(15\%\). At checkout, the reduced price receives an additional \(10\%\) discount. Farah says, “Great—I will save exactly one-fourth of the original price.” Check Farah's claim and find the actual overall percent discount.

Hints

- Write one-fourth as a percent. - Find the percent of the original price remaining after each discount. - The second discount is based on the already reduced price.

Solution

1. After the first discount, \(85\%\) of the original price remains, giving a multiplier of \(0.85\). 2. After the second discount, \(90\%\) of the reduced price remains, giving a multiplier of \(0.90\). 3. The combined multiplier is \(0.85 \cdot 0.90 = 0.765\), so the final price is \(76.5\%\) of the original price. 4. The overall discount is \(100\% - 76.5\% = 23.5\%\). 5. One-fourth is \(25\%\), so Farah's claim is incorrect.

Answer

Farah's claim is incorrect. The actual overall discount is \(23.5\%\), not \(25\%\).

Discounts, markups, tax, and tip 41

ThreadCart sells T-shirts for \(\$p\) each. For an order of more than \(10\) shirts, ThreadCart gives a \(20\%\) discount on the merchandise. A one-time shipping charge of \(\$5.90\) is then added. Write and simplify an expression for the total cost \(G\) of an order of \(n\) shirts, where \(n>10\).

Hints

- First write the merchandise cost before the discount. - After a \(20\%\) discount, what percent of the original merchandise cost remains? - The shipping charge is added once, not once per shirt. - Write the percent as a decimal factor.

Solution

1. Before the discount, the shirts cost \(np\) dollars. 2. A \(20\%\) discount means \(80\%\), or \(0.80\), of that amount remains: \(0.80np\). 3. Add the one-time shipping charge: \(G=0.80np+5.90\).

Answer

\(G=0.80np+5.90\) dollars

Discounts, markups, tax, and tip 42

A store reduces an item’s original price \(x\) by \(r\%\). Write an expression for the sale price. Then evaluate it for: 1) \(x = \$45\), \(r = 20\) 2) \(x = \$129\), \(r = 15\)

Hints

- A discount subtracts part of the original price. - After a discount of \(r\%\), what percent of the original price remains? - Follow the order of operations when substituting values.

Solution

1. A discount of \(r\%\) leaves \(1 - \frac{r}{100}\) of the original price, so the sale price is \(x\left(1 - \frac{r}{100}\right)\). An equivalent expression is \(x - \frac{xr}{100}\). 2. For case 1, \(\$45\left(1 - \frac{20}{100}\right) = \$45 \cdot 0.80 = \$36\). 3. For case 2, \(\$129\left(1 - \frac{15}{100}\right) = \$129 \cdot 0.85 = \$109.65\).

Answer

Expression: \(x\left(1 - \frac{r}{100}\right)\) 1) \(\$36\) 2) \(\$109.65\)

Discounts, markups, tax, and tip 43

During a sale, the price of a laptop was reduced twice. First, the original price was reduced by \(20\%\). Because the laptop still did not sell, the reduced price was then lowered by another \(15\%\). The laptop now costs \(\$544.00\). What was its original price before the two discounts?

Hints

- Work backward from the final price. - After a \(15\%\) discount, what percent of the previous price remains? - Remember that the second discount is applied to the already reduced price. - You can check your answer by applying both discounts to the original price.

Solution

1. The second discount left \(85\%\) of the price before that discount. Therefore, the price before the second discount was \(\$544.00 \div 0.85 = \$640.00\). 2. The first discount left \(80\%\) of the original price. Therefore, the original price was \(\$640.00 \div 0.80 = \$800.00\). 3. As a check, \(\$800.00 \cdot 0.80 \cdot 0.85 = \$544.00\).

Answer

The original price was \(\$800.00\).

Discounts, markups, tax, and tip 44

A desk lamp sells for \(\$63.00\) before tax. That price includes a \(40\%\) markup on the store's cost. What did the store pay for the lamp?

Hints

- Work backward from the marked-up price. - Ask what percent of the original cost the final selling price represents. - Check that the store's cost is less than the selling price.

Solution

1. A \(40\%\) markup means the selling price is \(140\%\) of the store's cost. 2. Let the store's cost be \(c\). Then \(1.40c=63\). 3. Solve: \(c=63 \div 1.40=45\).

Answer

The store paid \(\$45.00\).

Discounts, markups, tax, and tip 45

A jacket is listed at \(\$80.00\). A store applies a \(25\%\) discount, then charges \(8\%\) sales tax on the discounted price. What is the final price?

Hints

- Pay attention to the order in which the discount and tax are applied. - The tax is based on the price after the discount, not the original listed price. - Check the final amount against both the original and discounted prices.

Solution

1. After the \(25\%\) discount, the price is \(80.00\cdot0.75=60.00\). 2. The sales tax is \(0.08\cdot60.00=4.80\). 3. The final price is \(60.00+4.80=64.80\).

Answer

The final price is \(\$64.80\).

Discounts, markups, tax, and tip 46

Evan earns \(\$180\) in base pay for a weekend plus a \(4.5\%\) commission on sales. Evan's weekend sales total \(\$2400\). What is Evan's total pay for the weekend?

Hints

- Separate the fixed part of the pay from the part that depends on sales. - Find the sales-dependent amount before combining the two parts. - Make sure the final answer includes both sources of pay.

Solution

1. The commission is \(2400\cdot0.045=108\) dollars. 2. Add the base pay: \(180+108=288\).

Answer

The total pay is \(\$288\).

Discounts, markups, tax, and tip 47

Rafael earns a \(7\%\) commission and receives \(\$126\) in commission for one month. What were Rafael's commissionable sales for that month?

Hints

- The commission amount is a known fraction of the unknown sales total. - Work backward from the part to the whole. - Check that taking the stated percent of your sales total reproduces the commission.

Solution

1. Let the commissionable sales be \(s\). Then \(0.07s=126\). 2. Solve: \(s=126 \div 0.07=1800\).

Answer

The commissionable sales were \(\$1800\).

Discounts, markups, tax, and tip 48

Two museum tickets cost \(\$45.00\) each. A promo code reduces the ticket subtotal by \(20\%\). The ticketing site then charges a service fee equal to \(6\%\) of the discounted subtotal. What is the final amount charged before any sales tax?

Hints

- Apply each change to the amount it is actually based on. - The fee is calculated after the promotional reduction, not from the original subtotal. - Keep the discounted subtotal and the fee as separate quantities until the final step.

Solution

1. The original subtotal is \(2\cdot\$45.00=\$90.00\). 2. After the \(20\%\) discount, the subtotal is \(\$90.00\cdot0.80=\$72.00\). 3. The service fee is \(\$72.00\cdot0.06=\$4.32\). 4. The final charge is \(\$72.00+\$4.32=\$76.32\).

Answer

The final amount charged is \(\$76.32\).

Discounts, markups, tax, and tip 49

For a \(\$60.00\) event registration, EventNest charges an \(8\%\) service fee. RegisterNow charges a \(\$3.00\) fixed fee plus a \(4\%\) service fee based on the registration price. Which site has the lower total fee, and by how much?

Hints

- Write down every fee component for each site before comparing them. - One site has only a percentage-based fee, while the other combines two kinds of fees. - Compare the final fee amounts, not just the listed percentages.

Solution

1. EventNest's fee is \(\$60.00\cdot0.08=\$4.80\). 2. RegisterNow's percentage fee is \(\$60.00\cdot0.04=\$2.40\), so its total fee is \(\$3.00+\$2.40=\$5.40\). 3. The difference is \(\$5.40-\$4.80=\$0.60\).

Answer

EventNest has the lower fee by \(\$0.60\).

Discounts, markups, tax, and tip 50

Mei earns a \(5\%\) commission on net sales after customer returns are subtracted. During one week, Mei records \(\$3200\) in sales and \(\$400\) in returns. How much commission does Mei earn?

Hints

- Determine which sales amount the commission is actually based on. - Account for the returns before applying the commission rate. - Check that the commission is a small fraction of the net sales amount.

Solution

1. Net sales are \(\$3200-\$400=\$2800\). 2. The commission is \(\$2800\cdot0.05=\$140\).

Answer

Mei earns \(\$140\) in commission.

Discounts, markups, tax, and tip 51

A purchase total is \(\$54.00\) after \(8\%\) sales tax is added. There are no discounts or other fees. a) What was the price before tax? b) How much sales tax was charged?

Hints

- The final total represents more than \(100\%\) of the pretax price. - Think about what decimal multiplier corresponds to adding \(8\%\). - Once the pretax price is known, compare it with the final total to find the tax amount.

Solution

1. After \(8\%\) tax, the total is \(108\%\) of the pretax price, so the multiplier is \(1.08\). 2. Divide by the multiplier: \(54.00\div1.08=50.00\). The pretax price was \(\$50.00\). 3. The tax was \(\$54.00-\$50.00=\$4.00\).

Answer

a) \(\$50.00\) b) \(\$4.00\)

Discounts, markups, tax, and tip 52

A restaurant total is \(\$72.00\) after a \(20\%\) tip is added to the pre-tip check. No other charges are included. a) What was the pre-tip check amount? b) How much was the tip?

Hints

- The final total includes both the original check and a percent of that check. - Represent the final total as a percent of the pre-tip amount. - After finding the original check, use the difference to find the tip.

Solution

1. After a \(20\%\) tip, the total is \(120\%\) of the pre-tip check, so the multiplier is \(1.20\). 2. Divide by the multiplier: \(72.00\div1.20=60.00\). The pre-tip check was \(\$60.00\). 3. The tip was \(\$72.00-\$60.00=\$12.00\).

Answer

a) \(\$60.00\) b) \(\$12.00\)

Discounts, markups, tax, and tip 53

At Riverside Grill, a receipt shows a meal subtotal of \(\$48.00\), sales tax of \(\$3.84\), and a suggested \(18\%\) tip of \(\$8.64\). Cam argues that the suggested tip should instead be \(18\%\) of \(\$51.84\), the amount due before any tip is added. a) Based on the numbers on the receipt, which amount did the restaurant use as the base for the suggested tip? Verify with a calculation. b) Explain the error in Cam's reasoning. c) Find Cam's proposed tip, rounded to the nearest cent.

Hints

- Test the meal subtotal against the stated \(18\%\) rate and compare the result with the suggested tip on the receipt. - Then compare that verified base with the amount Cam uses. - For Cam's proposal, apply the same rate to his chosen base and round only at the end.

Solution

1. Check the meal subtotal: \(48.00\cdot0.18=8.64\), which exactly matches the suggested tip on the receipt. Therefore, the restaurant used \(\$48.00\) as the tip base. 2. Cam uses \(\$51.84\), which includes the \(\$3.84\) sales tax. His reasoning changes the base from the one evidenced by the receipt. 3. Cam's proposed tip is \(51.84\cdot0.18=9.3312\approx9.33\), or \(\$9.33\) to the nearest cent.

Answer

a) The restaurant used \(\$48.00\) as the tip base because \(48.00\cdot0.18=8.64\). b) Cam includes the sales tax in the base even though the receipt's suggested-tip amount corresponds to the meal subtotal. c) \(\$9.33\)

Discounts, markups, tax, and tip 54

Boardwalk Skate Shop pays \(\$80\) wholesale for a skateboard. The shop marks up the wholesale price by \(25\%\). Later, it discounts the marked price by \(20\%\). a) Find the marked price. b) Find the sale price after the discount. c) Compare the sale price with the original wholesale price and explain why the \(25\%\) increase and \(20\%\) decrease produce that result.

Hints

- Apply the markup to the wholesale price before considering the discount. - The discount is based on the marked price, not on the wholesale price. - Compare the product of the two percent multipliers with \(1\).

Solution

1. The marked price is \(\$80\cdot1.25=\$100\). 2. The sale price is \(\$100\cdot0.80=\$80\). 3. The overall multiplier is \(1.25\cdot0.80=1\), so the sale price equals the wholesale price. 4. The two percent changes use different bases: the \(25\%\) markup is based on \(\$80\), while the \(20\%\) discount is based on \(\$100\).

Answer

a) \(\$100\) b) \(\$80\) c) The sale price equals the \(\$80\) wholesale price because \(1.25\cdot0.80=1\); the two percentages are taken from different bases.

Discounts, markups, tax, and tip 55

A restaurant bill has \(\$60\) in food, \(8\%\) sales tax on the food, and an \(18\%\) tip on the food price before tax. The customer has a \(\$15\) gift card to use toward the payment. a) Find the tax and the tip separately. b) How much money must the diner pay after using the gift card? c) A friend subtracts the gift card before working out the tax and tip. Explain why that is incorrect.

Hints

- Both percentages are defined using the food price before tax. - Find the full bill before applying the gift card. - Does a payment method change the price of the food?

Solution

1. Sales tax is \(0.08\cdot60=4.80\) dollars. 2. The tip is \(0.18\cdot60=10.80\) dollars. 3. The bill before the gift card is \(60+4.80+10.80=75.60\) dollars. 4. The remaining payment is \(75.60-15=60.60\) dollars. 5. The gift card pays part of the final bill; it does not lower the food price on which tax and tip were calculated.

Answer

a) Tax: \(\$4.80\); tip: \(\$10.80\). b) \(\$60.60\). c) A gift card pays part of the bill; it does not lower the food price used to calculate tax and tip.

Discounts, markups, tax, and tip 56

A group of \(12\) people is choosing between two movie-ticket discounts. A regular ticket costs \(\$8.50\). Offer A: A \(15\%\) group discount on the total price for groups of at least \(10\). Offer B: A package of exactly \(4\) tickets for \(\$30\). Which offer costs less for the group? How much does the group save with the less expensive offer compared with regular prices?

Hints

- Find the regular cost for all \(12\) tickets. - Apply the percent discount for Offer A. - Determine how many four-ticket packages are required for Offer B, then compare the totals.

Solution

1. The regular price is \(12 \cdot \$8.50 = \$102.00\). 2. Under Offer A, the discount is \(\$102.00 \cdot 0.15 = \$15.30\), so the price is \(\$102.00 - \$15.30 = \$86.70\). 3. Under Offer B, the group needs \(12 \div 4 = 3\) packages, costing \(3 \cdot \$30 = \$90.00\). 4. Since \(\$86.70 < \$90.00\), Offer A costs less. 5. The savings compared with regular prices are \(\$102.00 - \$86.70 = \$15.30\).

Answer

Offer A costs less at \(\$86.70\). Offer B costs \(\$90.00\). The group saves \(\$15.30\) compared with the regular price of \(\$102.00\).

Discounts, markups, tax, and tip 57

A mountain bike's price was reduced twice. First, the original price was reduced by \(20\%\). When the bike still did not sell, the reduced price was lowered by another \(10\%\). The bike now costs \(\$324\). a) What was the price after the first discount but before the second? b) What was the original price? c) By what percent was the original price reduced overall?

Hints

- Work backward from the final price. - After a \(10\%\) discount, \(90\%\) of the previous price remains. - The second discount is based on the already reduced price. - Compare the total dollar savings with the original price.

Solution

1. The final price is \(90\%\) of the price after the first discount. That price was \(\$324 \div 0.90 = \$360\). 2. The price after the first discount is \(80\%\) of the original price. The original price was \(\$360 \div 0.80 = \$450\). 3. The total savings were \(\$450 - \$324 = \$126\). 4. Relative to the original price, the overall discount was \(\frac{126}{450} = 0.28 = 28\%\).

Answer

a) \(\$360\) b) \(\$450\) c) \(28\%\)

Discounts, markups, tax, and tip 58

Two department stores change prices on winter clothing. - Northline Department Store increases the price of a jacket by \(25\%\) to \(\$125\). - Grand Avenue Department Store decreases the price of a coat by \(25\%\) to \(\$150\). a) Find the price of each item before the change. b) Zuri tries to work backward. She subtracts \(25\%\) of Northline's new price and adds \(25\%\) of Grand Avenue's new price. She thinks both answers will be off by the same amount because both changes were \(25\%\). Check Zuri's answers. How far off is each one? Is she right?

Hints

- First find the original price for each item by using the percent that remains or is reached. - Then perform the incorrect reverse calculations in the claim. - Compare each result with the correct original price. - Think about why equal percentages of different amounts are unequal.

Solution

1. Northline's original price was \(125\div1.25=\$100\). 2. Grand Avenue's original price was \(150\div0.75=\$200\). 3. The incorrect Northline calculation gives \(125-0.25\cdot125=\$93.75\), which is \(\$6.25\) below the actual original. 4. The incorrect Grand Avenue calculation gives \(150+0.25\cdot150=\$187.50\), which is \(\$12.50\) below the actual original. 5. The errors differ because the \(25\%\) must be taken from the original prices, not from the two new prices.

Answer

a) Northline: \(\$100\); Grand Avenue: \(\$200\). b) Zuri is incorrect. Her method gives \(\$93.75\) instead of \(\$100\) (error \(\$6.25\)) and \(\$187.50\) instead of \(\$200\) (error \(\$12.50\)). The changes use different original price bases.

Discounts, markups, tax, and tip 59

A movie theater offers two annual discount cards. A regular ticket costs \(\$12\). - The Saver Card costs \(\$24\) per year and gives \(25\%\) off each ticket. - The Premium Card costs \(\$72\) per year and gives \(50\%\) off each ticket. a) Find the total annual cost for someone who sees \(10\) movies with each card. b) Starting with which movie visit is the Premium Card less expensive overall than the Saver Card?

Hints

- First find the discounted ticket price for each card. - Include both the annual card fee and ticket costs. - For part b), compare the extra card fee with the amount saved per visit.

Solution

1. A Saver Card ticket costs \(\$12 \cdot 0.75 = \$9\). A Premium Card ticket costs \(\$12 \cdot 0.50 = \$6\). 2. For \(10\) visits, the Saver Card costs \(\$24 + 10 \cdot \$9 = \$114\). 3. For \(10\) visits, the Premium Card costs \(\$72 + 10 \cdot \$6 = \$132\). 4. The Premium Card costs \(\$72 - \$24 = \$48\) more initially but saves \(\$9 - \$6 = \$3\) per visit. 5. The cards cost the same after \(\$48 \div \$3 = 16\) visits. Therefore, the Premium Card becomes less expensive at the \(17\)th visit.

Answer

a) Saver Card: \(\$114\); Premium Card: \(\$132\) b) The Premium Card is less expensive starting with the \(17\)th visit.

Discounts, markups, tax, and tip 60

A store offers a tiered rebate: \(5\%\) of the first \(\$500\) of an item's price, plus \(3\%\) of any portion above \(\$500\). Samir receives a rebate of exactly \(\$37\) on an electric scooter. What was the scooter's price? Explain your reasoning.

Hints

- Determine the rebate at exactly the first-tier limit. - Which part of the given rebate must come from the amount beyond that limit? - Work backward from the extra rebate using the second rate.

Solution

1. The first \(\$500\) earns a rebate of \(0.05\cdot500=\$25\). Because \(\$37>\$25\), some of the price is above the first tier. 2. The additional rebate is \(37-25=\$12\). 3. This equals \(3\%\) of the excess price, so the excess is \(12\div0.03=\$400\). 4. The original price was \(500+400=\$900\).

Answer

\(\$900\). The first \(\$500\) gives a \(\$25\) rebate; the remaining \(\$12\) comes from \(3\%\) of \(\$400\). Thus the price is \(\$500+\$400\).

Discounts, markups, tax, and tip 61

A store is considering two rebate plans. Plan A: \(5\%\) of the first \(\$500\) of an item's price, plus \(3\%\) of any portion above \(\$500\). Plan B: \(4\%\) of the item's full price. Determine which plan gives the larger rebate when the item costs: a) \(\$400\) b) \(\$2000\) c) For an item priced above \(\$500\), at what price do the plans give exactly the same rebate?

Hints

- Calculate both plans separately for the first two prices. - When the price is above \(\$500\), what portion is discounted at each rate in Plan A? - For part c), equate the two rebates within the stated price range.

Solution

1. At \(\$400\), Plan A gives \(0.05\cdot400=\$20\) and Plan B gives \(0.04\cdot400=\$16\). Plan A gives more. 2. At \(\$2000\), Plan A gives \(0.05\cdot500+0.03\cdot1500=25+45=\$70\). Plan B gives \(0.04\cdot2000=\$80\). Plan B gives more. 3. For price \(x>500\), set \(25+0.03(x-500)=0.04x\). 4. This simplifies to \(10+0.03x=0.04x\), so \(x=1000\). Each plan then gives \(\$40\).

Answer

a) Plan A: \(\$20\), compared with Plan B's \(\$16\). b) Plan B: \(\$80\), compared with Plan A's \(\$70\). c) At \(\$1000\), each rebate is \(\$40\).

Discounts, markups, tax, and tip 62

Morning Trail Cereal reduces a package from \(24\,\text{oz}\) to \(20\,\text{oz}\) but keeps the price at \(\$4.50\). Consumer advocate Imani says, “That is a hidden price increase of \(20\%\).” Morning Trail replies, “We reduced the amount by less than \(17\%\).” Determine whether both statements are mathematically correct. Find the percent decrease in package size and the percent increase in price per \(4\,\text{oz}\).

Hints

- Use the original package size as the base for the size decrease. - Find the old and new prices for the same \(4\,\text{oz}\) amount. - Compare the unit-price increase with the old unit price.

Solution

1. The package size decreased by \(24 - 20 = 4\,\text{oz}\). Relative to the original size, \(\frac{4}{24} = \frac{1}{6} \approx 16.67\%\). Therefore, Morning Trail's statement is correct. 2. The old price per \(4\,\text{oz}\) was \(\$4.50 \div 6 = \$0.75\). 3. The new price per \(4\,\text{oz}\) is \(\$4.50 \div 5 = \$0.90\). 4. The unit price increased by \(\$0.90 - \$0.75 = \$0.15\). Relative to the old unit price, \(\frac{0.15}{0.75} = 0.20 = 20\%\). Therefore, Imani's statement is also correct.

Answer

Both statements are correct. The package size decreased by approximately \(16.67\%\), while the price per \(4\,\text{oz}\) increased by \(20\%\).

Discounts, markups, tax, and tip 63

GamePort considers two pricing strategies for a game console. Strategy A: Increase the price by \(15\%\), then decrease the new price by \(15\%\) the next month. Strategy B: Decrease the price by \(15\%\), then increase the new price by \(15\%\) the next month. Compare the final prices. Does either strategy give a lower final price? Justify your answer.

Hints

- Write each price change as a decimal multiplier. - Compare the products of the two multipliers. - Does changing the order of two factors change their product?

Solution

1. Strategy A has an overall multiplier of \(1.15 \cdot 0.85 = 0.9775\). 2. Strategy B has an overall multiplier of \(0.85 \cdot 1.15 = 0.9775\). 3. The multipliers are equal because multiplication is commutative. 4. Both final prices are \(97.75\%\) of the original price, which is a \(2.25\%\) decrease.

Answer

Neither strategy is lower. Both give the same final price, \(2.25\%\) below the original price.

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