Guide to Percentages for Digital SAT Math, GCSE, IGCSE, ACT, and High School Mathematics (Part 3)
Successive Percentage Changes, Discounts, Profit and Loss, and Sales Tax
In the previous chapters, you learned how to calculate percentages, percentage increase, percentage decrease, and how to use multipliers to find new values quickly. These ideas form the foundation for solving more realistic percentage problems.
In everyday life, a quantity often changes more than once. A product may be discounted and then taxed. A company's sales may increase one year and decrease the next. A population may grow for several years in succession. Understanding how these repeated changes work is an essential mathematical skill and is frequently tested in Digital SAT Math, PSAT, ACT Math, GCSE Mathematics, IGCSE Mathematics, Cambridge Mathematics, Edexcel Mathematics, AQA Mathematics, and other secondary school mathematics courses.
Rather than memorising separate rules for every situation, this chapter develops each idea from simple mathematical reasoning.
What Are Successive Percentage Changes?
A successive percentage change occurs when a quantity changes more than once.
For example,
a store gives a 20% discount, followed by an additional 10% discount,
a company's revenue increases by 12% one year and 8% the following year,
a town's population grows by 5% each year,
a bank account earns interest every year.
Each new percentage is calculated from the current value, not the original value.
This is the key idea that students often overlook.
Why Can't We Simply Add the Percentages?
Suppose a jacket costs ₹2,000.
The store advertises
20% off, followed by another 10% off.
Many people incorrectly think the total discount is
20% + 10%
= 30%.
This is not correct.
The second discount is calculated after the first discount has already reduced the price.
Therefore, the two percentages apply to different amounts.
Example 1
A jacket costs ₹2,000.
The store offers a 20% discount, followed by another 10% discount.
Find the final selling price.
Step 1
The first discount is 20%.
Multiplier
= 1 − 0.20
= 0.80
Multiply.
₹2,000 × 0.80
= ₹1,600
Step 2
Now apply the second discount.
The second discount is 10%.
Multiplier
= 0.90
Multiply.
₹1,600 × 0.90
= ₹1,440
Answer
The final price is ₹1,440.
Finding the Overall Percentage Decrease
The original price was
₹2,000
The final price is
₹1,440
Decrease
= ₹2,000 − ₹1,440
= ₹560
Percentage decrease
= (560 ÷ 2,000) × 100
= 28%
Notice that
20% + 10%
does not equal the final reduction.
The actual decrease is 28%.
A Faster Method
Instead of calculating each step separately, multiply the multipliers.
20% decrease
→ 0.80
10% decrease
→ 0.90
Combined multiplier
0.80 × 0.90
= 0.72
Now multiply once.
₹2,000 × 0.72
= ₹1,440
This method is especially useful in timed examinations.
Example 2
The population of a town is 48,000.
It grows by 8% in one year and 5% in the following year.
Find the population after two years.
Step 1
Multiplier for an 8% increase
= 1.08
Multiplier for a 5% increase
= 1.05
Step 2
Multiply the multipliers.
1.08 × 1.05
= 1.134
Step 3
Multiply the original population.
48,000 × 1.134
= 54,432
Answer
The population after two years is 54,432.
Discounts
A discount reduces the selling price of an item.
Retail stores commonly advertise discounts during seasonal sales, clearance events, and promotional campaigns.
The mathematical process is identical to percentage decrease.
Example 3
A bicycle costs ₹18,500.
A store offers a 15% discount.
Find the sale price.
Step 1
Multiplier
= 0.85
Step 2
Multiply.
18,500 × 0.85
= 15,725
Answer
The bicycle costs ₹15,725 after the discount.
Sales Tax
Sales tax is added after the original price has been determined.
Unlike a discount, sales tax increases the amount paid.
Example 4
A laptop costs ₹56,000.
A sales tax of 8% is added.
Find the final amount paid.
Step 1
Multiplier
= 1.08
Step 2
Multiply.
56,000 × 1.08
= 60,480
Answer
The customer pays ₹60,480.
Discount Followed by Sales Tax
Many practical problems involve both a discount and a tax.
Apply the changes one after another.
Example 5
A camera costs ₹40,000.
A store gives a 10% discount.
An 8% sales tax is then added.
Find the final price.
Step 1
Discount multiplier
= 0.90
40,000 × 0.90
= 36,000
Step 2
Sales tax multiplier
= 1.08
36,000 × 1.08
= 38,880
Answer
The final amount paid is ₹38,880.
Profit and Loss
Businesses compare the selling price with the cost price.
If the selling price is greater than the cost price, a profit is made.
If the selling price is lower than the cost price, a loss occurs.
Example 6
A shop purchases a calculator for ₹720.
It sells the calculator for ₹864.
Find the profit percentage.
Step 1
Profit
= 864 − 720
= 144
Step 2
Compare with the cost price.
144 ÷ 720
= 0.20
Step 3
Convert to a percentage.
0.20 × 100
= 20%
Answer
The profit is 20%.
Example 7
A retailer buys a chair for ₹4,500.
It is sold for ₹4,050.
Find the loss percentage.
Step 1
Loss
= 4,500 − 4,050
= 450
Step 2
Compare with the cost price.
450 ÷ 4,500
= 0.10
Step 3
Convert to a percentage.
0.10 × 100
= 10%
Answer
The loss is 10%.
Common Mistakes
Mistake 1
Adding successive percentages instead of multiplying the multipliers.
Mistake 2
Calculating profit using the selling price instead of the cost price.
Mistake 3
Applying sales tax before calculating the discount when the question specifies the opposite order.
Mistake 4
Using the wrong multiplier.
Many percentage questions on the Digital SAT and similar examinations combine several ideas in one problem. A question may involve a discount followed by sales tax, or a population that changes over consecutive years. Instead of treating each percentage separately, convert every change into a multiplier and apply the multipliers in the correct order. This approach reduces arithmetic errors and is often the fastest method during timed examinations.
Practice Questions
A jacket costing ₹3,200 receives a 25% discount. Find the sale price.
Increase ₹850 by 12%.
A phone costs ₹30,000. After a 15% discount, an 8% sales tax is added. Find the final price.
A town with 65,000 people grows by 6% and then by 4% the following year. Find the population after two years.
A shop buys a microwave oven for ₹9,600 and sells it for ₹11,040. Find the profit percentage.
A bicycle is purchased for ₹15,000 and sold for ₹13,800. Find the loss percentage.
Answers
₹2,400
₹952
₹27,540
71,656
15%
8%
Chapter Summary
Successive percentage changes require each new percentage to be calculated from the current value rather than the original value. Converting percentage changes into multipliers provides a simple and reliable method for solving problems involving repeated increases, repeated decreases, discounts, sales tax, profit, and loss. These techniques are widely used in finance, commerce, economics, statistics, science, and everyday decision-making, making them essential skills for students preparing for the Digital SAT, ACT, GCSE, IGCSE, and other secondary mathematics examinations.
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