2. Discounts and Sales

What Is a Discount?

A discount is a reduction in the original price of a product or service.

Stores often advertise discounts using percentages:

10% off

25% off

40% off

50% off

The percentage tells us what fraction of the original price will be removed.

For example:

25% off

means that 25% of the original price is subtracted.

The customer therefore pays:

75% of the original price

because:

100% − 25% = 75%

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Original Price, Discount, and Sale Price

There are three important quantities in a discount problem.

Original price: Price before the discount.

Discount amount: Amount removed from the original price.

Sale price: Price after the discount has been removed.

The basic relationship is:

Sale Price = Original Price − Discount Amount

For example:

Original price = $80

Discount = $20

Sale price:

$80 − $20 = $60

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Calculating a Percentage Discount

To calculate a discount:

Discount Amount = Original Price × Discount Rate

Remember to convert the percentage to a decimal.

For example:

20% = 0.20

Suppose an item costs:

$50

with:

20% off

Calculate:

$50 × 0.20 = $10

Discount:

$10

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Finding the Final Price

Once we know the discount:

Sale Price = Original Price − Discount

For the previous example:

Original price:

$50

Discount:

$10

Therefore:

$50 − $10 = $40

Final price:

$40

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Example: 30% Discount

A jacket originally costs:

$120

It is discounted by:

30%

Step 1: Convert the percentage.

30% = 0.30

Step 2: Calculate the discount.

$120 × 0.30 = $36

Step 3: Subtract the discount.

$120 − $36 = $84

Therefore:

Discount = $36

Final price = $84

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Using Fractions for Familiar Discounts

Some percentages are easy to calculate as fractions.

50% = 1/2

25% = 1/4

75% = 3/4

20% = 1/5

10% = 1/10

For example:

25% off $80

Since:

25% = 1/4

calculate:

$80 ÷ 4 = $20

Discount:

$20

Sale price:

$80 − $20 = $60

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Using Benchmark Percentages

Some discounts can be calculated mentally by combining familiar percentages.

For example:

15% = 10% + 5%

Find 15% of $60.

10%:

$60 ÷ 10 = $6

5%:

$6 ÷ 2 = $3

Therefore:

15% of $60 = $9

Sale price:

$60 − $9 = $51

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Another Mental Strategy

Find:

35% of $200

We can use:

30% + 5%

30%:

$60

5%:

$10

Therefore:

35% = $70

If this is a discount:

$200 − $70 = $130

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Using a Sale-Price Multiplier

There is a faster way to calculate the final price.

If an item is:

20% off

then the customer pays:

100% − 20% = 80%

Convert:

80% = 0.80

Therefore:

Sale Price = Original Price × 0.80

Example:

$150 × 0.80 = $120

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Discount Multipliers

Some useful examples are:

For a 35% discount:

100% − 35% = 65%

Therefore, multiply the original price by:

0.65

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Example Using a Multiplier

Original price:

$240

Discount:

35%

Percentage paid:

65%

Calculate:

$240 × 0.65 = $156

Therefore:

Final price = $156

Savings:

$240 − $156 = $84

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Calculating Savings

The savings from a discount are normally equal to the discount amount.

Suppose:

Original price = $90

Discount = 20%

Calculate:

$90 × 0.20 = $18

Therefore:

Savings = $18

Final price:

$90 − $18 = $72

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Discount Amount vs Final Price

These are easy to confuse.

Suppose:

Original price = $200

Discount = 30%

Discount amount:

$200 × 0.30 = $60

But the final price is:

$200 − $60 = $140

Therefore:

Savings = $60

Price paid = $140

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Comparing Sale Offers

Advertisements may present discounts in different ways.

For example:

Store A: 25% off

Store B: $20 off

Which gives the larger saving?

The answer depends on the original price.

Suppose the item costs:

$100

Store A:

25% of $100 = $25

Store B:

$20

So the savings are:

$25 versus $20

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What If the Original Price Is $60?

Store A:

25% of $60 = $15

Store B:

$20 off

Now the savings are:

$15 versus $20

The comparison has changed.

This demonstrates why a percentage discount cannot always be compared with a fixed discount without knowing the original price.

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Finding the Break-Even Price

Suppose one offer is:

20% off

and another is:

$15 off

At what original price are the savings equal?

Let the original price be:

P

Then:

20% of P = $15

0.20P = 15

Divide:

P = 15 ÷ 0.20

P = $75

At an original price of $75, both offers save:

$15

Below or above this price, the comparison changes.

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Comparing Discounts on Different Original Prices

Suppose:

Store A:

$80 shirt with 25% off

Store B:

$70 shirt with 15% off

Store A:

Discount:

$80 × 0.25 = $20

Final price:

$60

Store B:

Discount:

$70 × 0.15 = $10.50

Final price:

$59.50

A larger percentage discount does not automatically produce a lower final price because the original prices may differ.

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Multiple Discounts

Stores sometimes advertise offers such as:

20% off, then an additional 10% off

It may be tempting to say:

20% + 10% = 30%

But successive discounts are normally calculated one after another.

Suppose the original price is:

$100

First discount:

20% of $100 = $20

New price:

$80

Second discount:

10% of $80 = $8

Final price:

$72

Total savings:

$100 − $72 = $28

Therefore, the total effective discount is:

28%

not 30%.

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Why Successive Discounts Do Not Simply Add

After the first discount, the second percentage is applied to a smaller price.

For:

20% off, then 10% off

the multipliers are:

0.80 × 0.90

= 0.72

The customer pays:

72%

of the original price.

Therefore, the effective discount is:

28%

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Another Multiple-Discount Example

Original price:

$200

Sale:

30% off, then another 20% off

First discount:

$200 × 0.70 = $140

Second discount:

$140 × 0.80 = $112

Final price:

$112

Savings:

$200 − $112 = $88

Effective discount:

$88 / $200 × 100% = 44%

So:

30% off followed by 20% off = 44% effective discount

not 50%.

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Comparing One Discount with Multiple Discounts

Offer A:

40% off

Offer B:

25% off, then another 20% off

Suppose the original price is $100.

Offer A:

$100 × 0.60 = $60

Offer B:

$100 × 0.75 × 0.80 = $60

In this case, both produce the same final price.

The effective discount for Offer B is therefore also:

40%

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Buy One, Get One Free

Another common sale is:

Buy One, Get One Free

Suppose each item normally costs:

$30

Two items normally cost:

$60

Under the offer:

$30

is paid for two items.

Average cost per item:

$30 ÷ 2 = $15

Compared with buying two identical items at the normal price, this is equivalent to a:

50% reduction in the total price of the two-item purchase

provided the customer actually needs two items and both qualify.

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Buy Two, Get One Free

Suppose each item costs:

$12

Normally, three items cost:

3 × $12 = $36

Under:

Buy 2, Get 1 Free

the customer pays:

2 × $12 = $24

Savings:

$36 − $24 = $12

Percentage saving across the three-item bundle:

12 / 36 × 100%

= 33.3%

approximately.

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Percentage Off vs Extra Product

Suppose two packages normally contain 500 g.

Offer A:

20% off the price

Offer B:

20% extra product for the same price

These offers are not mathematically identical.

If the normal price is $10:

Offer A:

500 g for:

$8

Cost per 100 g:

$8 ÷ 5 = $1.60

Offer B:

20% extra of 500 g:

0.20 × 500 = 100 g

New quantity:

600 g

Cost per 100 g:

$10 ÷ 6 ≈ $1.67

Unit rates allow the offers to be compared consistently.

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Unit Price and Sales

A sale price does not automatically mean good value compared with other products.

Suppose:

Brand A:

750 g for $6 after discount.

Brand B:

1 kg for $7.50 at regular price.

Brand A:

$6 ÷ 0.75 = $8/kg

Brand B:

$7.50 ÷ 1 = $7.50/kg

The word sale alone does not tell us which option has the lower unit price.

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What Does "Good Value" Mean?

Evaluating value requires more than identifying the largest discount percentage.

Mathematical information that may matter includes:

• original price
• final price
• discount amount
• quantity received
• unit price
• number of items required
• additional fees
• total cost over time

Other practical considerations can include:

• quality
• durability
• usefulness
• whether the item is actually needed
• whether some of the quantity will be wasted

Mathematics helps provide the evidence needed to evaluate the options.

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Sale Price vs Value

Suppose:

Option A:

10 pens for $8

Option B:

20 pens for $14

Unit prices:

Option A:

$8 ÷ 10 = $0.80/pen

Option B:

$14 ÷ 20 = $0.70/pen

Option B has the lower unit price.

However, someone who needs only 5 pens would still have to spend:

$14

to receive that unit price.

"Better unit value" and "lower total spending" are not always the same thing.

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Discounts and Budgets

Suppose your budget is:

$100

An item originally costs:

$125

It is:

25% off

Discount:

$125 × 0.25 = $31.25

Final price:

$125 − $31.25 = $93.75

Money remaining:

$100 − $93.75 = $6.25

The sale brings the item within the $100 budget.

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What Percentage Was Saved?

Sometimes the original and sale prices are given instead of the discount percentage.

Suppose:

Original price:

$80

Sale price:

$60

Savings:

$80 − $60 = $20

Discount percentage:

$20 / $80 × 100%

= 25%

Therefore:

the discount was 25%

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Another Reverse Discount Problem

Original price:

$150

Sale price:

$105

Savings:

$150 − $105 = $45

Percentage discount:

45 / 150 × 100%

= 30%

Therefore:

30% off

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Finding the Original Price

Sometimes we know the final price and discount.

Suppose a jacket costs:

$72 after a 20% discount

A 20% discount means the customer pays:

80%

of the original price.

Let the original price be:

P

Then:

0.80P = 72

Therefore:

P = 72 ÷ 0.80

P = $90

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Another Original-Price Example

A device costs:

$195 after a 35% discount

Percentage paid:

65%

Therefore:

0.65P = 195

Calculate:

P = 195 ÷ 0.65

P = $300

Original price:

$300

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Discounts Followed by Additional Charges

A discount may not be the final calculation.

Suppose:

Original price:

$200

Discount:

25%

Discounted price:

$200 × 0.75 = $150

Suppose an illustrative 8% tax is then applied to the discounted price.

Tax:

$150 × 0.08 = $12

Final amount:

$150 + $12 = $162

The order of calculations matters.

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Coupons and Percentage Discounts

Suppose a $100 item has:

20% off

and then a:

$10 coupon

After the percentage discount:

$100 × 0.80 = $80

After the coupon:

$80 − $10 = $70

Final price:

$70

However, real stores may have rules about whether coupons can be combined or the order in which discounts apply.

Always use the conditions stated in the problem or offer.

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Comparing Three Sale Offers

Suppose an item normally costs:

$120

Offer A:

25% off

Final price:

$120 × 0.75 = $90

Offer B:

$35 off

Final price:

$120 − $35 = $85

Offer C:

10% off, then 15% off

First:

$120 × 0.90 = $108

Then:

$108 × 0.85 = $91.80

The final prices are:

A comparison table makes the differences much easier to see.

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Reading Sale Information from Tables

Suppose a store provides:

To interpret the table, we might calculate both the savings and final prices.

Shoes:

Savings = $20

Final = $60

Jacket:

Savings = $36

Final = $84

Backpack:

Savings = $9

Final = $51

Headphones:

Savings = $60

Final = $90

Tables allow several financial options to be compared systematically.

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Interpreting Sale Graphs

A graph might show:

• original prices
• sale prices
• discount percentages
• savings
• prices at different stores

When reading a financial graph:

1. Read the title.

2. Check the axes.

3. Check the units.

4. Identify whether values show prices, percentages, or savings.

5. Compare actual numerical values rather than relying only on visual appearance.

6. Check whether the graph scale could exaggerate differences.

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Sale Advertisements

Advertisements commonly use phrases such as:

UP TO 70% OFF

SAVE $50

FROM $19.99

BUY ONE, GET ONE FREE

20% EXTRA

These statements do not all mean the same thing.

For example:

"Up to 70% off"

means some qualifying items may have a 70% discount. It does not mean every item is discounted by 70%.

Careful interpretation is part of mathematical financial literacy.

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Percentage Discount vs Dollar Savings

A large percentage discount does not necessarily mean a large amount of money saved.

Item A:

$20 with 50% off

Savings:

$10

Item B:

$500 with 10% off

Savings:

$50

Item A has the larger percentage discount.

Item B produces the larger dollar saving.

Which measure matters depends on the question being asked.

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Original Price Matters

Compare:

50% off $20

and:

20% off $100

First:

0.50 × $20 = $10 saved

Second:

0.20 × $100 = $20 saved

The smaller percentage produces the larger dollar saving because it applies to a larger original price.

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Estimating Sale Prices

Estimation can help check calculations.

Suppose:

32% off $198

Estimate:

About 30% of $200:

≈ $60

So the sale price should be roughly:

$140

Exact discount:

$198 × 0.32 = $63.36

Exact sale price:

$198 − $63.36 = $134.64

The exact answer is reasonably close to the estimate.

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Checking Whether an Answer Is Reasonable

If an item is:

20% off

the final price should be:

• lower than the original price
• greater than 50% of the original price
• exactly 80% of the original price

For a $100 item:

A final price of:

$80

makes sense.

A final price of:

$20

would mean the discount amount has been confused with the sale price.

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Worked Example 1: Basic Discount

Original price:

$70

Discount:

20%

Savings:

$70 × 0.20 = $14

Final price:

$70 − $14 = $56

Answer:

Savings = $14

Final price = $56

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Worked Example 2: 35% Off

Original price:

$160

Discount:

35%

Savings:

$160 × 0.35 = $56

Final price:

$160 − $56 = $104

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Worked Example 3: Using a Multiplier

Original price:

$280

Discount:

15%

Percentage paid:

85%

Calculate:

$280 × 0.85 = $238

Final price:

$238

Savings:

$42

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Worked Example 4: Comparing Offers

A $200 item has two possible offers.

Offer A:

30% off

Savings:

$60

Final:

$140

Offer B:

$50 off

Savings:

$50

Final:

$150

The offers produce final prices of:

$140 and $150

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Worked Example 5: Successive Discounts

Original price:

$250

Discounts:

20% off, then 15% off

First:

$250 × 0.80 = $200

Second:

$200 × 0.85 = $170

Final price:

$170

Savings:

$250 − $170 = $80

Effective discount:

80 / 250 × 100% = 32%

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Worked Example 6: Find the Discount Percentage

Original price:

$240

Sale price:

$180

Savings:

$240 − $180 = $60

Percentage discount:

60 / 240 × 100%

= 25%

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Worked Example 7: Find the Original Price

Sale price:

$126

Discount:

30%

Percentage paid:

70%

Therefore:

0.70P = 126

P = 126 ÷ 0.70

P = $180

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Worked Example 8: Buy Two, Get One Free

Each shirt costs:

$24

Normally three shirts cost:

3 × $24 = $72

Under the offer:

2 × $24 = $48

Savings:

$24

Percentage saving:

24 / 72 × 100%

≈ 33.3%

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Worked Example 9: Unit-Price Comparison

Offer A:

600 g for $4.80

Offer B:

900 g for $6.75

Offer A:

$4.80 ÷ 6 = $0.80 per 100 g

Offer B:

$6.75 ÷ 9 = $0.75 per 100 g

The unit prices provide a common basis for comparing the offers.

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Worked Example 10: Budget and Sale

Budget:

$150

Original price:

$180

Discount:

25%

Final price:

$180 × 0.75 = $135

Money remaining:

$150 − $135 = $15

The discounted item fits within the stated budget.

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Evaluating a Sale

When deciding whether a sale represents useful value, ask:

What is the original price?

What percentage is actually discounted?

How many dollars will I save?

What is the final price?

How does the unit price compare with alternatives?

Are there additional charges?

Do I need to buy more items to receive the discount?

Will I actually use the extra quantity?

Is another store's regular price lower?

Does the purchase fit the available budget?

A large discount percentage by itself does not answer all of these questions.

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A Reliable Discount Strategy

Step 1: Identify the original price.

Step 2: Identify the discount percentage or offer.

Step 3: Convert the percentage to a decimal or fraction.

Step 4: Calculate the discount amount.

Step 5: Subtract the discount from the original price.

Step 6: Include any additional calculations stated in the problem.

Step 7: If comparing offers, calculate each final price on the same basis.

Step 8: Compare savings and unit prices when relevant.

Step 9: Estimate to check your calculation.

Step 10: Interpret the numbers in the context of the purchase.

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Common Mistakes

Mistake 1: Treating the discount as the final price

20% off $100 means:

$20 saved

not:

$20 final price

The final price is:

$80

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Mistake 2: Forgetting to convert the percentage

Incorrect:

$80 × 25

Correct:

$80 × 0.25

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Mistake 3: Subtracting the percentage number directly

Incorrect:

$100 − 20 = $80

This happens to produce the correct number only because the original price is $100.

For $200:

20% off is:

$40

not $20.

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Mistake 4: Adding successive discounts

20% off followed by 10% off is:

28% effective discount

not 30%.

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Mistake 5: Assuming the largest percentage means the lowest price

Original prices may be different.

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Mistake 6: Assuming a sale automatically means good value

Another product may have a lower regular price or lower unit price.

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Mistake 7: Ignoring conditions

Some discounts require:

• buying multiple items
• spending a minimum amount
• membership
• using a coupon
• purchasing during a particular period

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Mistake 8: Ignoring how much is actually needed

A lower unit price may require purchasing much more than necessary.

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Did You Know?

Discount mathematics combines several mathematical ideas.

When comparing sales, we may use:

• percentages
• decimals
• fractions
• ratios
• unit rates
• proportions
• estimation
• tables
• graphs
• algebra

This is why shopping provides a useful real-world application of mathematics.

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Key Terms

• Original price: Price before a discount.
• Discount: Reduction in price.
• Discount rate: Percentage by which the original price is reduced.
• Discount amount: Amount of money removed from the original price.
• Sale price: Price after the discount.
• Final price: Amount paid after the calculations specified in the problem.
• Savings: Difference between the original price and the discounted price.
• Multiplier: Decimal used to calculate a percentage of an amount directly.
• Successive discount: One discount applied after another.
• Effective discount: Overall percentage reduction produced by one or more discounts.
• Unit price: Cost per one item or standard unit of quantity.
• Coupon: Offer that reduces a price when its conditions are satisfied.
• Break-even point: Value at which two financial options produce the same result.
• Value: Relationship between what is received and what it costs.

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Key Equations

Discount amount:

Discount = Original Price × Discount Rate

Sale price:

Sale Price = Original Price − Discount

Percentage paid:

Percentage Paid = 100% − Discount Percentage

Multiplier method:

Sale Price = Original Price × Percentage Paid as a Decimal

Savings:

Savings = Original Price − Sale Price

Discount percentage:

Discount % = Savings / Original Price × 100%

Original price:

Original Price = Sale Price / Percentage Paid as a Decimal

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Key Takeaways

• A discount reduces the original price of a product or service.
• Percentage discounts are calculated from the original price unless the offer specifies otherwise.
• The discount amount is the amount saved, not usually the final price.
• The sale price equals the original price minus the discount.
• A multiplier can calculate a sale price directly.
• A 20% discount means paying 80% of the original price.
• Savings can be calculated in dollars or as a percentage.
• Fixed-dollar discounts and percentage discounts should be compared using actual savings or final prices.
• Different original prices can make percentage comparisons misleading.
• Successive discounts are applied one after another and normally should not simply be added.
• Unit prices are useful when sale packages contain different quantities.
• "Buy one, get one free" and similar offers should be analyzed using the total cost and quantity received.
• The largest advertised discount does not automatically produce the lowest final price.
• A lower unit price does not automatically mean lower total spending.
• Additional fees or charges may affect the final amount.
• Estimation is useful for checking discount calculations.
• Sale advertisements should be interpreted carefully, especially phrases such as "up to", "from", and "extra."
• Evaluating value requires comparing the numerical evidence with the quantity, cost, conditions, and purpose of the purchase.
• The most useful question is not simply "How big is the discount?" but:

"What will I actually pay, what will I receive, and how does that compare with the alternatives?"