Exam 1 Review: Percentage Calculations and Price Comparisons

Exam 1 Review Report: Key Concepts in Percentage Calculations

This review covers fundamental percentage calculations including discounts, markups, percentage increases/decreases, and finding original amounts.

Question 1: Finding a Percentage of a Total Amount (Discount Calculation)

  • Concept: Calculating a specific percentage of a given number, often applied to discounts, sales tax, or commission.
  • Problem: An item is regularly priced at 7070. It is on sale for 60%60\% off the regular price. How much (in dollars) is discounted from the regular price?
  • Solution Strategy: To find the discount amount, multiply the regular price by the discount percentage (expressed as a decimal).
    • Step 1: Convert percentage to decimal: 60%=0.6060\% = 0.60
    • Step 2: Calculate the discount: Multiply the regular price by the decimal equivalent of the discount percentage.
      Discount Amount=Regular Price×Discount Percentage Discount Amount=$70×0.60 Discount Amount=$42\text{Discount Amount} = \text{Regular Price} \times \text{Discount Percentage} \ \text{Discount Amount} = \$70 \times 0.60 \ \text{Discount Amount} = \$42
  • Answer: The amount discounted from the regular price is 4242.

Question 2: Comparing Discounts

  • Concept: Evaluating different discount offers to determine which results in a lower final price. This involves comparing a fixed rebate with a percentage discount.
  • Problem: David has two coupons for pants regularly priced at 4545.
    • Coupon A: 1010 rebate on 4545 pants.
    • Coupon B: 15%15\% off of 4545 pants.
      Choose the coupon that gives the lower price. Then, determine the price difference between the two coupons.
  • Solution Strategy: Calculate the final price for each coupon and then compare.
    • Coupon A Calculation (Rebate): A rebate directly reduces the price by a fixed amount.
      Price with Coupon A=Regular Price−Rebate Amount Price with Coupon A=$45−$10 Price with Coupon A=$35\text{Price with Coupon A} = \text{Regular Price} - \text{Rebate Amount} \ \text{Price with Coupon A} = \$45 - \$10 \ \text{Price with Coupon A} = \$35
    • Coupon B Calculation (Percentage Discount): A percentage discount reduces the price by a percentage of the original price.
      • Step 1: Calculate the discount amount:
        Discount Amount=Regular Price×Discount Percentage Discount Amount=$45×0.15 Discount Amount=$6.75\text{Discount Amount} = \text{Regular Price} \times \text{Discount Percentage} \ \text{Discount Amount} = \$45 \times 0.15 \ \text{Discount Amount} = \$6.75
      • Step 2: Calculate the final price:
        Price with Coupon B=Regular Price−Discount Amount Price with Coupon B=$45−$6.75 Price with Coupon B=$38.25\text{Price with Coupon B} = \text{Regular Price} - \text{Discount Amount} \ \text{Price with Coupon B} = \$45 - \$6.75 \ \text{Price with Coupon B} = \$38.25
    • Comparison: Compare the final prices:
      • Price with Coupon A: 3535
      • Price with Coupon B: 38.2538.25
        Coupon A results in a lower price.
    • Difference Calculation: Subtract the lower price from the higher price.
      Difference=Price with Coupon B−Price with Coupon A Difference=$38.25−$35 Difference=$3.25\text{Difference} = \text{Price with Coupon B} - \text{Price with Coupon A} \ \text{Difference} = \$38.25 - \$35 \ \text{Difference} = \$3.25
  • Answer: Coupon A gives the lower price. The price with Coupon A is 3.253.25 less than the price with Coupon B.

Question 3: Comparing Deals with Markups and Discounts

  • Concept: Analyzing multi-step pricing scenarios involving both percentage markups (an increase in price from a wholesale cost) and percentage discounts to find the best deal.
  • Problem: Greg is comparing prices for an electric stove with a wholesale price of 539539 at two different stores. Ignoring tax:
    • (a) Department Store: Marks up the wholesale price by 50\%$, then offers a 20\%discountoffthein−store(marked−up)pricebecauseofacustomerloyaltyprogram.HowmuchwouldGregpay?</li><li><strong>(b)Superstore:</strong>Marksupthewholesalepricebydiscount off the in-store (marked-up) price because of a customer loyalty program. How much would Greg pay?</li> <li><strong>(b) Superstore:</strong> Marks up the wholesale price by30\%$. Greg is not a loyalty program member, so he pays the full in-store price. How much would Greg pay?
    • (c) Select the true statement: Which store offers the lower price?
  • Solution Strategy: Calculate the final price for each store following the described steps.
    • (a) Department Store Calculation:
      • Step 1: Calculate the markup amount:
        Markup Amount=Wholesale Price×Markup Percentage Markup Amount=$539×0.50 Markup Amount=$269.50\text{Markup Amount} = \text{Wholesale Price} \times \text{Markup Percentage} \ \text{Markup Amount} = \$539 \times 0.50 \ \text{Markup Amount} = \$269.50
      • Step 2: Calculate the in-store price (after markup):
        In-store Price=Wholesale Price+Markup Amount In-store Price=$539+$269.50 In-store Price=$808.50\text{In-store Price} = \text{Wholesale Price} + \text{Markup Amount} \ \text{In-store Price} = \$539 + \$269.50 \ \text{In-store Price} = \$808.50
      • Step 3: Calculate the discount amount: The discount is applied to the in-store price.
        Discount Amount=In-store Price×Discount Percentage Discount Amount=$808.50×0.20 Discount Amount=$161.70\text{Discount Amount} = \text{In-store Price} \times \text{Discount Percentage} \ \text{Discount Amount} = \$808.50 \times 0.20 \ \text{Discount Amount} = \$161.70
      • Step 4: Calculate the final price for Greg:
        Final Price (Dept. Store)=In-store Price−Discount Amount Final Price (Dept. Store)=$808.50−$161.70 Final Price (Dept. Store)=$646.80\text{Final Price (Dept. Store)} = \text{In-store Price} - \text{Discount Amount} \ \text{Final Price (Dept. Store)} = \$808.50 - \$161.70 \ \text{Final Price (Dept. Store)} = \$646.80
    • (b) Superstore Calculation:
      • Step 1: Calculate the markup amount:
        Markup Amount=Wholesale Price×Markup Percentage Markup Amount=$539×0.30 Markup Amount=$161.70\text{Markup Amount} = \text{Wholesale Price} \times \text{Markup Percentage} \ \text{Markup Amount} = \$539 \times 0.30 \ \text{Markup Amount} = \$161.70
      • Step 2: Calculate the final price for Greg (in-store price): Since no discount is applied.
        Final Price (Superstore)=Wholesale Price+Markup Amount Final Price (Superstore)=$539+$161.70 Final Price (Superstore)=$700.70\text{Final Price (Superstore)} = \text{Wholesale Price} + \text{Markup Amount} \ \text{Final Price (Superstore)} = \$539 + \$161.70 \ \text{Final Price (Superstore)} = \$700.70
    • (c) Comparison:
      • Department Store: 646.80646.80
      • Superstore: 700.70700.70
        The department store offers the lower price.
  • Answer:
    • (a) Greg would pay 646.80646.80 at the department store.
    • (b) Greg would pay 700.70700.70 at the superstore.
    • (c) Greg would pay less for the electric stove at the department store.

Question 4: Finding the Original Amount Given a Percentage Increase

  • Concept: Determining an initial value before a percentage increase was applied, given the final value and the percentage increase.
  • Problem: A house has increased in value by 35%35\% since it was purchased. If the current value is 513,000513,000, what was the value when it was purchased?
  • Solution Strategy: If the value increased by 35%35\%, the current value represents 100%+35%=135%100\% + 35\% = 135\% of the original value. Let PP be the original purchase value.
    • Formula: ( \text{Current Value} = \text{Original Value} \times (1 + \text{Percentage Increase}) )
    • Step 1: Set up the equation:
      $513,000=P×(1+0.35) $513,000=P×1.35\$513,000 = P \times (1 + 0.35) \ \$513,000 = P \times 1.35
    • Step 2: Solve for P (Original Value): Divide the current value by the factor representing the increase.
      P=$513,0001.35 P=$380,000P = \frac{\$513,000}{1.35} \ P = \$380,000
  • Answer: The value of the house when it was purchased was 380,000380,000.

Question 5: Finding the Original Price Given the Sale Price and Percentage "Of" Original Price

  • Concept: Calculating the original price when the final (sale) price is given as a specific percentage of the original price.
  • Problem: At a sale this week, a sofa is being sold for 235.20235.20. This is 28%28\% of the original price. What is the original price?
  • Solution Strategy: If the sale price is 28%28\% of the original price, the sale price is simply the original price multiplied by 0.280.28. Let OO be the original price.
    • Formula: ( \text{Sale Price} = \text{Original Price} \times \text{Percentage (as decimal)} )
    • Step 1: Set up the equation:
      $235.20=O×0.28\$235.20 = O \times 0.28
    • Step 2: Solve for O (Original Price): Divide the sale price by the decimal equivalent of the percentage.
      O=$235.200.28 O=$840O = \frac{\$235.20}{0.28} \ O = \$840
  • Answer: The original price of the sofa was 840840.