Free math worksheets

- 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.

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

The commission is \(\$75\).

- 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.

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

\(\$3.00\)

- 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.

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

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

- 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.

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

The markup is \(\$6.00\).

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

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

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

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

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 \(\$396 - \$395 = \$1\).

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

- Use the old price as the base value. - Divide the new price by the old price. - A price increase means the new price is more than \(100\%\) of the old price.

1. The ratio of the new price to the old price is \(\frac{\$5.00}{\$4.00} = 1.25\). 2. Therefore, the new price is \(125\%\) of the old price. 3. The percent increase is \(125\% - 100\% = 25\%\).

The new price is \(125\%\) of the old price, so the price increased by \(25\%\).

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

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

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

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

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.

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

- 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.

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

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

- 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.

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

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

- 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.

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

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

- 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.

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 \(\$45.00\).

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

- 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.

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, \(\$24.00\).

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

- 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.

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.

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.

- 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.

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.

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

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

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 \(\$32 - \$30 = \$2\).

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

- 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.

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 \(\$400\).

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

- 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.

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.

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

- 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.

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 \(\frac{48}{120} = 0.40 = 40\%\).

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

- 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.

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.”

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.”

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

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.

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 \(5\%\).

- 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?

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.

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

- 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.

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.

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

- 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.

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 \(\frac{18}{72} = \frac{1}{4} = 0.25 = 25\%\).

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

- Calculate each option step by step. - In Option 2, identify the base for the second discount. - Compare the two final prices.

1. For Option 1, the final price is \(\$450 \cdot 0.80 = \$360\). 2. For Option 2, the price after the first discount is \(\$450 \cdot 0.90 = \$405\). 3. After the second discount, the final price is \(\$405 \cdot 0.90 = \$364.50\). 4. Option 1 costs \(\$364.50 - \$360 = \$4.50\) less. 5. In Option 2, the second \(10\%\) discount is based on the already reduced price, so two \(10\%\) discounts do not equal one \(20\%\) discount.

Option 1: \(\$360\) Option 2: \(\$364.50\) Option 1 costs less by \(\$4.50\).

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

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

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

- 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.

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 \(\frac{46}{1200} \cdot 100\% \approx 3.83\%\).

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

- 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.

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: \(\frac{262.50}{2012.50} \approx 0.13043 = 13.04\%\).

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

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

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 \(\frac{15}{405} \approx 0.0370 = 3.7\%\).

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

- 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.

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 \(\$690.00 - \$552.00 = \$138.00\).

The savings are \(\$138.00\).

- 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.

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 \(\$644 - \$575 = \$69\).

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

- 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.

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

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

- 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.

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 \(\$1.68/\text{L} - \$1.40/\text{L} = \$0.28/\text{L}\).

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

- 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.

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.

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

- 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.

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: \(G=0.80np+5.90\).

\(G=0.80np+5.90\) dollars

- 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.

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, \(\$800.00 \cdot 0.80 \cdot 0.85 = \$544.00\).

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

- 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.

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: \(c=63 \div 1.40=45\).

The store paid \(\$45.00\).

- 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.

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 \(60.00+4.80=64.80\).

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

- 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.

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

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

- 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.

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

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

- 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.

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 \(\$72.00+\$4.32=\$76.32\).

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

- 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.

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 \(\$5.40-\$4.80=\$0.60\).

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

- 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.

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

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

- 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.

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 \(\$54.00-\$50.00=\$4.00\).

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

- 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.

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 \(\$72.00-\$60.00=\$12.00\).

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

- 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.

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.

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) \(\$9.33\)

- 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 \(1\).

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 \(\$100\).

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.

- 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.

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 \(\$102.00 - \$86.70 = \$15.30\).

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

- 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.

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 \(\frac{126}{450} = 0.28 = 28\%\).

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

- First find the correct original price for each item. - Then perform the incorrect calculations described in the claim. - Compare each incorrect result with the corresponding correct original price. - The size of a percent amount depends on both the percent and the base amount.

1. At Northline, \(\$125\) is \(125\%\) of the original price. The original price was \(\$125 \div 1.25 = \$100\). 2. At Grand Avenue, \(\$150\) is \(75\%\) of the original price. The original price was \(\$150 \div 0.75 = \$200\). 3. The incorrect calculation for Northline gives \(25\%\) of \(\$125\), or \(\$31.25\). Subtracting gives \(\$125 - \$31.25 = \$93.75\), which is \(\$6.25\) below the correct original price. 4. The incorrect calculation for Grand Avenue gives \(25\%\) of \(\$150\), or \(\$37.50\). Adding gives \(\$150 + \$37.50 = \$187.50\), which is \(\$12.50\) below the correct original price. 5. The errors are not equal. Reversing a \(25\%\) increase requires division by \(1.25\), while reversing a \(25\%\) decrease requires division by \(0.75\). The new prices are also different.

a) Northline: \(\$100\); Grand Avenue: \(\$200\) b) Zuri's claim is incorrect. The error is \(\$6.25\) for Northline and \(\$12.50\) for Grand Avenue.

- 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.

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.

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

- Find the rebate on an item priced at exactly \(\$500\). - Compare that amount with the given rebate. - Separate the total rebate into the first-tier rebate and the rebate on the amount above \(\$500\).

1. A \(\$500\) item earns a rebate of \(\$500 \cdot 0.05 = \$25\). Since \(\$37 > \$25\), the bag cost more than \(\$500\). 2. The rebate from the portion above \(\$500\) is \(\$37 - \$25 = \$12\). 3. Let \(x\) be the amount of the price above \(\$500\). Then \(0.03x = 12\), so \(x = \frac{12}{0.03} = 400\). 4. The bag's price was \(\$500 + \$400 = \$900\).

The bag's price was \(\$900\).

- Calculate both rebates separately in parts a and b. - For a price above \(\$500\), split Plan A into two tiers. - For part c, write an equation that sets the two rebate amounts equal.

1. For part a, Plan A gives \(\$400 \cdot 0.05 = \$20\), while Plan B gives \(\$400 \cdot 0.04 = \$16\). Plan A is better. 2. For part b, Plan A gives \(\$25 + (\$2000 - \$500)\cdot 0.03 = \$25 + \$45 = \$70\). Plan B gives \(\$2000 \cdot 0.04 = \$80\). Plan B is better. 3. For part c, the equal-rebate price must be above \(\$500\). Let \(x\) be the price. Solve \(25 + 0.03(x - 500) = 0.04x\). 4. Simplifying gives \(10 + 0.03x = 0.04x\), so \(10 = 0.01x\) and \(x = 1000\). Each plan then gives a \(\$40\) rebate.

a) Plan A gives the larger rebate: \(\$20\) instead of \(\$16\). b) Plan B gives the larger rebate: \(\$80\) instead of \(\$70\). c) The rebates are equal at an item price of \(\$1000\); each rebate is \(\$40\).

- 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.

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.

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

- 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?

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.

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