Solved Exercises on Discounts and Markups in Grade 7

Master discounts and markups: calculating discounts, markups, and real-world applications through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Basic Discount Calculation
Exercise 1
A store offers a 25% discount on a $120 jacket. What is the sale price? How much money is saved?
Definition:

Discount: A reduction in the original price of an item, expressed as a percentage of the original price.

Method for calculating discounts:
  1. Convert the discount percentage to a decimal by dividing by 100
  2. Multiply the original price by the decimal to find the discount amount
  3. Subtract the discount amount from the original price to get the sale price
  4. Alternatively: Sale Price = Original Price × (1 - decimal)
Discount Amount
$120 × 0.25 = $30
Sale Price
$120 - $30 = $90
Step 1: Convert discount percentage to decimal

25% = 25 ÷ 100 = 0.25

Step 2: Calculate the discount amount

Discount amount = Original price × Decimal

Discount amount = $120 × 0.25 = $30

Step 3: Calculate the sale price

Sale price = Original price - Discount amount

Sale price = $120 - $30 = $90

Discount: $30 | Sale price: $90
Final answer:

The discount is $30, and the sale price is $90. The customer saves $30.

Applied rules:

• Decimal Conversion: Percent ÷ 100 = Decimal

• Discount Formula: Sale Price = Original × (1 - decimal)

• Savings: Original - Sale = Savings

Key Concept:

When calculating discounts, you're finding what portion of the original price is reduced. The sale price is what remains after the discount is applied.

2 Markup Calculation
Exercise 2
A store buys a bicycle for $150 and marks it up by 40% to sell. What is the selling price? How much profit is made?
Definition:

Markup: An increase in the original price of an item, expressed as a percentage of the original price, to determine the selling price.

Markup Amount
$150 × 0.40 = $60
Selling Price
$150 + $60 = $210
Step 1: Convert markup percentage to decimal

40% = 40 ÷ 100 = 0.40

Step 2: Calculate the markup amount

Markup amount = Original price × Decimal

Markup amount = $150 × 0.40 = $60

Step 3: Calculate the selling price

Selling price = Original price + Markup amount

Selling price = $150 + $60 = $210

Markup: $60 | Selling price: $210
Final answer:

The markup is $60, and the selling price is $210. The store makes a $60 profit.

Applied rules:

• Decimal Conversion: Percent ÷ 100 = Decimal

• Markup Formula: Selling Price = Original × (1 + decimal)

• Profit: Selling Price - Original = Profit

$150
Cost Price
→
+40%
Markup
→
$210
Selling Price
3 Discount After Markup
Exercise 3
A retailer buys a TV for $400, marks it up by 25%, then offers a 15% discount on the marked-up price. What is the final selling price? What is the net profit?
Definition:

Sequential Pricing: When both markup and discount are applied to an item, the markup is applied first to determine the marked price, then the discount is applied to the marked price.

Marked Price
$400 × 1.25 = $500
Final Price
$500 × 0.85 = $425
Net Profit
$425 - $400 = $25
Step 1: Calculate the marked price after markup

Marked price = Cost price × (1 + markup decimal)

Marked price = $400 × (1 + 0.25) = $400 × 1.25 = $500

Step 2: Calculate the final price after discount

Final price = Marked price × (1 - discount decimal)

Final price = $500 × (1 - 0.15) = $500 × 0.85 = $425

Step 3: Calculate the net profit

Net profit = Final selling price - Cost price

Net profit = $425 - $400 = $25

Final price: $425 | Net profit: $25
Final answer:

The final selling price is $425. The net profit is $25.

Applied rules:

• Sequential Application: Apply markup first, then discount

• Compound Effect: Discount is applied to the already-marked-up price

• Profit Calculation: Final price - Cost price = Profit

Key Concept:

When both markup and discount are applied, the discount is taken off the marked-up price, not the original price. The net effect may be positive or negative depending on the percentages.

Discounts and Markups: Rules and Methods
\(\text{Discount}: \text{Sale Price} = \text{Original Price} \times (1 - \frac{\text{Discount}}{100})\) \(\text{Markup}: \text{Selling Price} = \text{Original Price} \times (1 + \frac{\text{Markup}}{100})\)
Discount and Markup Formulas
Discount
\(P \times (1 - \frac{d}{100})\)
Price after discount
Markup
\(P \times (1 + \frac{m}{100})\)
Price after markup
Profit
\(SP - CP\)
Selling Price - Cost Price
Key definitions:

Discount: A reduction in the original price of an item, expressed as a percentage of the original price.

Markup: An increase in the original price of an item, expressed as a percentage of the original price, to determine the selling price.

Cost Price: The original price at which an item is purchased.

Selling Price: The price at which an item is sold to customers.

Discount and markup methodology:
  1. Identify Original Price: Determine the starting value (cost price)
  2. Convert Percentage to Decimal: Divide percentage by 100
  3. Determine Operation: Subtract for discounts, add for markups
  4. Calculate New Price: Apply the appropriate formula
  5. Calculate Profit/Loss: If needed, find difference between selling and cost prices
Tip 1: For discounts, multiply by (1 - decimal). For markups, multiply by (1 + decimal).
Tip 2: Always apply percentages to the correct base price.
Tip 3: When both are applied, markup first, then discount.
Tip 4: Verify that discount prices are lower than original, markup prices are higher.
Real-Life Applications: Retail pricing, sales promotions, wholesale to retail pricing, profit margins, clearance sales.
Common Mistakes: Applying percentages to wrong base, mixing up discount and markup formulas, not following sequence.
Discount and Markup Rules:

• Discount Rule: Sale Price = Original × (1 - Discount%)

• Markup Rule: Selling Price = Original × (1 + Markup%)

• Sequential Rule: Apply markup first, then discount if both exist

• Verification: Discount prices should be lower, markup prices should be higher

Solution: Exercises 4 to 5
4 Comparison Problem
Exercise 4
Store A sells a $200 item with a 30% markup. Store B sells the same item with a 25% discount from the original price. Which store has the higher final price? By how much?
Definition:

Price Comparison: Understanding how markups and discounts affect final prices differently.

Store A
$200 × 1.30 = $260
Store B
$200 × 0.75 = $150
Difference
$260 - $150 = $110
Step 1: Calculate Store A's selling price

Price after markup = $200 × (1 + 0.30) = $200 × 1.30 = $260

Step 2: Calculate Store B's selling price

Price after discount = $200 × (1 - 0.25) = $200 × 0.75 = $150

Step 3: Compare the prices

Difference = $260 - $150 = $110

Store A has the higher price by $110

Store A: $260 | Store B: $150 | Difference: $110
Final answer:

Store A has the higher final price at $260. Store A's price is $110 higher than Store B's price of $150.

Applied rules:

• Markup Application: Multiply by (1 + markup decimal)

• Discount Application: Multiply by (1 - discount decimal)

• Price Comparison: Direct comparison of final prices

📊
Store A
Markup: +30%
📉
Store B
Discount: -25%
⚖️
Comparison
$260 vs $150
5 Multiple Discount Problem
Exercise 5
A store offers a 20% discount on all items. During a special sale, an additional 15% discount is applied to already-discounted prices. What is the final price of a $300 item? What is the overall percentage discount?
Definition:

Sequential Discounts: When multiple discounts are applied one after another to the same item, each discount is applied to the price after the previous discount.

After 1st discount
$300 × 0.80 = $240
After 2nd discount
$240 × 0.85 = $204
Overall discount
($300 - $204)/$300 = 32%
Step 1: Apply first discount

Price after 20% discount = $300 × (1 - 0.20) = $300 × 0.80 = $240

Step 2: Apply second discount to discounted price

Price after additional 15% discount = $240 × (1 - 0.15) = $240 × 0.85 = $204

Step 3: Calculate overall percentage discount

Overall discount = (Original - Final) ÷ Original × 100%

Overall discount = ($300 - $204) ÷ $300 × 100% = 32%

Final price: $204 | Overall discount: 32%
Final answer:

The final price of the item is $204. The overall percentage discount is 32%.

Applied rules:

• Sequential Application: Apply each discount to the current price

• Compound Effect: Each discount reduces the base for the next discount

• Overall Calculation: Use original and final prices for total percentage

Comprehensive Summary: Discounts and Markups
\(\text{Discount}: SP = OP \times (1 - \frac{D}{100}), \quad \text{Markup}: SP = OP \times (1 + \frac{M}{100})\)
Core Discount and Markup Formulas
Core Definitions:

Discount: A reduction in the original price of an item, expressed as a percentage of the original price. The sale price is lower than the original price.

Markup: An increase in the original price of an item, expressed as a percentage of the original price, to determine the selling price. The selling price is higher than the original price.

Cost Price: The original price at which an item is purchased or manufactured.

Selling Price: The price at which an item is sold to customers after applying discounts or markups.

Discount and Markup Problem-Solving Steps:
  1. Identify Original Price: Determine the starting amount (cost price)
  2. Convert Percentage to Decimal: Divide percentage by 100
  3. Determine Operation: Subtract for discounts, add for markups
  4. Calculate Multiplier: (1 - decimal) for discounts, (1 + decimal) for markups
  5. Find New Price: Multiply original by the appropriate multiplier
  6. Calculate Profit/Loss: If needed, find difference between selling and cost prices
Quick Tip: For discounts, multiply by (1 - decimal). For markups, multiply by (1 + decimal).
Memory Aid: Discount = take away (subtract), Markup = add on (add).
Strategy: For sequential operations, apply each to the current price.
Verification: Always check that discount prices are lower and markup prices are higher than original.
Real-Life Applications: Retail pricing, sales promotions, wholesale to retail pricing, profit margins, clearance sales, seasonal discounts.
Common Scenarios: Store sales, manufacturer pricing, restaurant markups, commission calculations, tax calculations.
Key Rules and Properties:

• Discount Rule: Sale Price = Original × (1 - Discount%)

• Markup Rule: Selling Price = Original × (1 + Markup%)

• Sequential Application: Apply each percentage to the current value

• Compound Effect: Multiple changes multiply rather than add percentages

• Verification: Discount prices must be lower, markup prices must be higher

Questions & Answers

Question: I'm confused about when to add or subtract the percentage. How do I know if it's a discount or markup?

Answer: Great question! The key is to understand what each operation does:

Discount:

  • Reduces the original price
  • Always SUBTRACT the percentage
  • Multiply by (1 - decimal)
  • Example: 20% discount → multiply by 0.80

Markup:

  • Increases the original price
  • Always ADD the percentage
  • Multiply by (1 + decimal)
  • Example: 20% markup → multiply by 1.20

Remember: Discount = Take away (subtract), Markup = Add on (add).

A discount makes the price smaller, a markup makes it bigger!

Question: When do I add percentages together versus multiplying the multipliers? I see both approaches.

Answer: This depends on whether you're finding the total percentage change or applying sequential changes:

Adding percentages:

  • Only when finding the total percentage change from original to final value
  • Example: If total went up 15% then 10%, total increase is NOT 25% (it's compound)

Multiplying multipliers:

  • When applying sequential changes to the same base
  • Example: $100 × 1.15 × 1.10 = $126.50
  • Final result: (126.50 - 100)/100 × 100% = 26.5% total increase

Note: 15% + 10% = 25%, but actual total change is 26.5% due to compounding!

Always multiply multipliers for sequential changes, then calculate the total percentage if needed.

Question: I sometimes get confused about whether my answer is correct. How can I verify my discount and markup calculations?

Answer: Here are several verification methods:

Method 1: Reverse calculation

  • For discounts: Take your sale price and increase it by the same percentage
  • Should return close to the original price
  • Example: $80 with 20% discount → $80 × 1.25 = $100 (back to original)

Method 2: Calculate the difference

  • Find the actual discount/markup amount
  • Divide by original amount and multiply by 100
  • Should equal your original percentage

Method 3: Reasonableness check

  • Discount price should definitely be less than original
  • Markup price should definitely be more than original
  • Magnitude should make sense (20% of $100 is $20, not $2)

Always use at least one verification method to ensure accuracy!