Compound Growth Questions with Answers and Solutions

In the previous chapters, we learned the concepts of Simple Interest and Compound Interest and practiced a variety of questions through our Simple Interest Questions with Answers and Solutions, Compound Interest Questions with Answers and Solutions, and Difference Between SI and CI Questions.

Quick Note: Compound Growth vs Compound Interest

Compound Growth and Compound Interest are closely related because both are based on the principle of compounding. In both cases, each new percentage change is applied to the value obtained after the previous period.

In our Compound Interest Questions with Answers and Solutions, we generally worked with terms such as Principal (P), Rate (R), Time (T), Amount (A), and Compound Interest (CI).

In Compound Growth Questions, we may instead encounter quantities such as population, students, passengers, salary, production, sales, revenue, or subscribers.

When a quantity grows at the same percentage rate every year, the calculation is very similar to Compound Interest. However, Compound Growth questions may also involve different growth rates in different years, repeated decreases, reverse growth, or a combination of increases and decreases.

Therefore, our understanding of Compound Interest provides a strong foundation for Compound Growth, but we should carefully identify the type of growth or decrease given in the question before selecting the appropriate formula.

Quick Note: Compound Growth Formulas

To strengthen our Quantitative Aptitude preparation, we should also explore our:

• Quant Maths Notes
• Quant Maths Formula Section
• Quant Maths Practice Section
• Previous Year Questions (PYQs)
• Math Calculators
• Competitive Exams Maths Syllabus

Now, let’s begin solving Compound Growth Questions with detailed step-by-step solutions.

Question 1: The number of students admitted to PPN College, Kanpur was 2,500 this year. If admissions increase by 8% every year, how many students will there be after 3 years?

Solution

Given

Initial Number of Students = 2,500

Annual Growth Rate (R) = 8%

Time = 3 years

Why Is This Compound Growth?

The important phrase in the question is:

“Admissions increase by 8% every year.”

This means the 8% increase is calculated on the new number of students each year, not repeatedly on the original 2,500 students.

For example:

After 1 year: 8% increase on 2,500
After 2 years: 8% increase on the new value after Year 1
After 3 years: 8% increase on the new value after Year 2

This is why it is called Compound Growth. Each year’s growth becomes part of the value on which the next year’s growth is calculated.

The concept is the same as Compound Interest. Only the terminology changes:

Formula Used

For Compound Growth:

Final Value = Initial Value × (1 + R/100)^n

Compare this with the Compound Interest Amount formula:

A = P × (1 + R/100)^n

They are mathematically the same formula.

Step 1: Calculate the Growth Factor

Growth Factor

= 1 + 8/100

= 1.08

Step 2: Calculate the Number of Students After 3 Years

Final Number of Students

= 2,500 × (1.08)^3

= 2,500 × 1.259712

= 3,149.28

Since the number of students must be a whole number:

= 3,149 students

Answer

The number of students after 3 years will be approximately 3,149.

Exam Tip for SSC, RRB, Defence, Banking, Police Exams

Whenever a question says that a quantity increases by a certain percentage every year, remember:

“Increases by R% every year” → Think Compound Growth

We can also verify the answer year by year:

After 1 year: 2,500 × 1.08 = 2,700

After 2 years: 2,700 × 1.08 = 2,916

After 3 years: 2,916 × 1.08 = 3,149.28

For constant-rate Compound Growth questions like this one, the underlying compounding concept is the same as Compound Interest.

Question 2: The average daily passenger count of the Delhi Metro is 32 lakh. If the number of passengers grows by 5% annually, what will be the average daily passenger count after 4 years?

Solution

Given

Initial Passenger Count = 32 lakh

Annual Growth Rate (R) = 5%

Time (n) = 4 years

Formula Used

Since the passenger count grows by 5% annually, each year’s growth is calculated on the passenger count after the previous year’s growth.

Therefore, we use the Compound Growth formula:

Final Value = Initial Value × (1 + R/100)^n

Step 1: Calculate the Growth Factor

Growth Factor

= 1 + 5/100

= 1.05

Step 2: Calculate the Passenger Count After 4 Years

Final Passenger Count

= 32 × (1.05)^4 lakh

= 32 × 1.21550625 lakh

= 38.8962 lakh

=38.90 lakh

Answer

The average daily passenger count after 4 years will be approximately 38.90 lakh.

Question 3: The average daily passengers using the Kolkata Tram system are 1,20,000. Due to declining popularity, the number of passengers decreases by 10% every year. Find the expected daily passengers after 3 years.

Solution

Given

Initial Number of Passengers = 1,20,000

Annual Decrease Rate (R) = 10%

Time (n) = 3 years

Formula Used

Since the number of passengers decreases by 10% every year, each year’s decrease is calculated on the reduced passenger count of the previous year.

For compound decrease:

Final Value = Initial Value × (1 − R/100)^n

Step 1: Calculate the Decrease Factor

Decrease Factor

= 1 − 10/100

= 1 − 0.10

= 0.90

This means that after each year, 90% of the previous year’s passengers remain.

Step 2: Calculate the Passenger Count After 3 Years

Final Number of Passengers

= 1,20,000 × (0.90)^3

= 1,20,000 × 0.729

= 87,480

Answer

The expected average daily passenger count after 3 years will be 87,480 passengers.

Question 4: An SSC Coaching Institute currently has 2,662 students after experiencing an annual growth of 10% for 2 years. Find the number of students enrolled 2 years ago.

Solution

Given

Final Number of Students = 2,662

Annual Growth Rate (R) = 10%

Time (n) = 2 years

We need to find the Initial Number of Students.

Formula Used

For compound growth:

Final Value = Initial Value × (1 + R/100)^n

Here, the final value is known, so we work backwards to find the initial value.

Therefore:

Initial Value = Final Value ÷ (1 + R/100)^n

Step 1: Calculate the Growth Factor

Growth Factor

= 1 + 10/100

= 1.10

Step 2: Find the Initial Number of Students

Initial Number of Students

= 2,662 ÷ (1.10)^2

= 2,662 ÷ 1.21

= 2,200

Answer

The SSC Coaching Institute had 2,200 students 2 years ago.

Question 5: A TV advertisement campaign reached 8 lakh viewers initially. After 2 years, the viewership increased to 9.68 lakh. Assuming the same percentage growth every year, find the annual growth rate.

Solution

Given

Initial Viewership = 8 lakh

Final Viewership = 9.68 lakh

Time (n) = 2 years

Annual Growth Rate = R%

We need to find R.

Formula Used

For Compound Growth:

Final Value = Initial Value × (1 + R/100)^n

Substituting the given values:

9.68 = 8 × (1 + R/100)^2

Step 1: Divide Both Sides by 8

9.68 ÷ 8 = (1 + R/100)^2

1.21 = (1 + R/100)^2

Step 2: Take the Square Root

1 + R/100 = √1.21

Since:

√1.21 = 1.10

Therefore:

1 + R/100 = 1.10

Step 3: Find the Growth Rate

R/100 = 1.10 − 1

R/100 = 0.10

R = 10%

Answer

The annual growth rate of the TV advertisement’s viewership is 10% per year.

Question 6: The monthly salary of a Junior Accounts Officer (JAO) was ₹45,000. It increased by 8% every year and became approximately ₹56,700. Find the number of years taken for this growth.

Solution

Given

Initial Salary = ₹45,000

Final Salary = ₹56,700

Annual Growth Rate (R) = 8%

Time = n years

We need to find n.

Formula Used

For Compound Growth:

Final Value = Initial Value × (1 + R/100)^n

Substituting the given values:

56,700 = 45,000 × (1 + 8/100)^n

56,700 = 45,000 × (1.08)^n

Step 1: Divide Both Sides by 45,000

56,700 ÷ 45,000 = (1.08)^n

1.26 = (1.08)^n

Now, instead of using logarithms(we will learn in coming chapters), we can check the powers of 1.08, which is more suitable for many competitive exam questions.

Step 2: Check Successive Powers of 1.08

(1.08)¹ = 1.08

(1.08)² = 1.1664

(1.08)³ = 1.259712 ≈ 1.26

Therefore:

n = 3 years

Step 3: Verify the Salary

Salary after 3 years:

= ₹45,000 × (1.08)³

= ₹45,000 × 1.259712

= ₹56,687.04

= ₹56,700

Answer

It took approximately 3 years for the monthly salary to increase from ₹45,000 to ₹56,700.

Exam Tip for SSC/RRB/Defence/Banking/State Police

When the initial value, final value, and growth rate are given and we need to find the time, look for a familiar power of the growth factor.

Here:

56,700 / 45,000 = 1.26

and

(1.08)³ = 1.26

Therefore, we can quickly identify the time as 3 years.

For competitive-exam questions with convenient values, checking successive powers is often easier than using logarithms.

Question 7: The annual salary of a Senior Cashier in a Bank was ₹6,00,000. It increased by 10% in the first year, 8% in the second year, and 12% in the third year. Find the salary after 3 years.

Solution

Given

Initial Annual Salary = ₹6,00,000

Growth Rate in the 1st Year = 10%

Growth Rate in the 2nd Year = 8%

Growth Rate in the 3rd Year = 12%

Time = 3 years

Here, the growth rate is different in each year. Therefore, we cannot use a single growth rate for all three years.

Formula Used

When the growth rates are different for different periods:

Final Value = Initial Value × (1 + R₁/100) × (1 + R₂/100) × (1 + R₃/100)

Therefore:

Final Salary = 6,00,000 × 1.10 × 1.08 × 1.12

Step 1: Calculate Salary After the First Year

Salary after 1st year

= 6,00,000 × 1.10

= ₹6,60,000

Step 2: Calculate Salary After the Second Year

Salary after 2nd year

= 6,60,000 × 1.08

= ₹7,12,800

Step 3: Calculate Salary After the Third Year

Salary after 3rd year

= 7,12,800 × 1.12

= ₹7,98,336

Answer

The annual salary of the Senior Cashier after 3 years will be ₹7,98,336.

Exam Tip

When the percentage growth is different each year, don’t take the average of the given rates and apply the ordinary compound growth formula.

Instead, multiply the individual growth factors:

10% increase → 1.10

8% increase → 1.08

12% increase → 1.12

So:

Final Value = Initial Value × 1.10 × 1.08 × 1.12

Question 8: Two Bank PO Coaching Institutes, A and B, each started with 1,000 students. Institute A increased its enrollment by 12% annually, while Institute B increased its enrollment by 10% annually. Find the difference in the number of students after 3 years.

Solution

Given

Initial Students in Institute A = 1,000

Initial Students in Institute B = 1,000

Annual Growth Rate of Institute A = 12%

Annual Growth Rate of Institute B = 10%

Time = 3 years

We need to calculate the number of students in both institutes after 3 years and then find the difference.

Formula Used

For Compound Growth:

Final Value = Initial Value × (1 + R/100)ⁿ

Step 1: Calculate Students in Institute A

Growth Factor for Institute A:

= 1 + 12/100

= 1.12

Students after 3 years:

= 1,000 × (1.12)³

= 1,000 × 1.404928

= 1,404.928

= 1,405 students

Step 2: Calculate Students in Institute B

Growth Factor for Institute B:

= 1 + 10/100

= 1.10

Students after 3 years:

= 1,000 × (1.10)³

= 1,000 × 1.331

= 1,331 students

Step 3: Find the Difference

Difference

= 1,404.928 − 1,331

= 73.928

Since the number of students must be a whole number:

= 74 students

Answer

After 3 years, Institute A will have approximately 74 more students than Institute B.

Exam Tip

In comparison-based Compound Growth questions, calculate the growth factor for each case separately.

Here:

Institute A → (1.12)³ = 1.404928

Institute B → (1.10)³ = 1.331

Since both started with the same number of students, we can also solve it directly:

Difference = 1,000 × [(1.12)³ − (1.10)³]

= 1,000 × (1.404928 − 1.331)

= 73.928 = 74 students

Question 9: The number of online banking users of a bank increased by 20% in the first year. In the following year, it decreased by 10%. If the bank initially had 5,00,000 users, find the number of users after 2 years.

Solution

Given

Initial Number of Users = 5,00,000

Increase in the 1st Year = 20%

Decrease in the 2nd Year = 10%

This question involves growth followed by a decrease, so we apply the two percentage changes successively.

Formula Used

When a quantity first increases and then decreases:

Final Value = Initial Value × (1 + Increase%/100) × (1 − Decrease%/100)

Therefore:

Final Users = 5,00,000 × (1 + 20/100) × (1 − 10/100)

= 5,00,000 × 1.20 × 0.90

Step 1: Calculate Users After the First Year

Users after 20% increase:

= 5,00,000 × 1.20

= 6,00,000

Step 2: Calculate Users After the Second Year

Now the number decreases by 10%.

Users after 2 years:

= 6,00,000 × 0.90

= 5,40,000

Answer

The bank will have 5,40,000 online banking users after 2 years.

Exam Tip: 

Don’t Cancel 20% Increase with 10% Decrease

The percentages are applied to different values.

The 20% increase is calculated on 5,00,000, while the 10% decrease is calculated on the new value of 6,00,000.

We can also find the overall percentage change using successive percentage change:

Net % Change = x + y + (xy/100)

Taking increase as +20% and decrease as −10%:

= 20 − 10 + [20 × (−10)]/100

= 10 − 2

= 8% increase

Therefore:

5,00,000 × 108/100 = 5,40,000

So the final number of users is again 5,40,000.

Question 10:  A company had an annual revenue of ₹80 crore. Its revenue increased by 15% in the first year, 10% in the second year, and then decreased by 5% in the third year. Find the company’s revenue at the end of the third year.

Solution

Given

Initial Annual Revenue = ₹80 crore

Increase in 1st Year = 15%

Increase in 2nd Year = 10%

Decrease in 3rd Year = 5%

We need to find the revenue at the end of the third year.

Formula Used

Since the percentage change is different each year, we apply each growth or decrease factor successively:

Final Value = Initial Value × (1 + R₁/100) × (1 + R₂/100) × (1 − R₃/100)

Therefore:

Final Revenue = 80 × 1.15 × 1.10 × 0.95

Step 1: Revenue After the First Year

Revenue after 15% increase:

= 80 × 1.15

= ₹92 crore

Step 2: Revenue After the Second Year

Revenue after another 10% increase:

= 92 × 1.10

= ₹101.20 crore

Step 3: Revenue After the Third Year

In the third year, revenue decreases by 5%.

Revenue:

= 101.20 × 0.95

= ₹96.14 crore

Answer

The company’s annual revenue at the end of the third year will be ₹96.14 crore.

Exam Tip

This question combines compound growth and compound decrease.

Remember:

15% increase → multiply by 1.15

10% increase → multiply by 1.10

5% decrease → multiply by 0.95

So we can solve it directly:

₹80 × 1.15 × 1.10 × 0.95 = ₹96.14 crore

Do not simply calculate:

15% + 10% − 5% = 20%

because each percentage change is applied to the new value obtained after the previous change.

Practice Questions On Compound Growth

Question 1: Kota Book Centre stocked 50,000 competitive exam books at the start of the year. Due to peak exam season, sales grew by 20% in the first year. However, due to the shift toward digital e-books, sales decreased by 10% in the second year. Find the total number of books sold at the end of the second year.

Answer: 54,000 books

Question 2: The daily footfall of passengers at Kanpur Central Railway Station was recorded at 1,50,000. If the footfall grows consistently at an annual rate of 8% due to new train connections, what will be the estimated daily passenger footfall after 2 years?

Answer: 1,74,960 passengers

Quick Revision: Common Growth Factors and Successive Powers

In SSC, RRB, Banking, Defence, Police, and other competitive exams, becoming familiar with the successive powers of common growth factors can help us solve some Compound Growth and Compound Interest questions quickly without using logarithms.

What We Learned in This Section

In this section, we learned how to:

• Solve constant compound growth and decrease questions
• Use growth and decrease factors
• Find the initial value, final value, growth rate, and time
• Handle different growth rates in different years
• Solve successive increase and decrease questions
• Compare two compound growth cases
• Use quick shortcuts and familiar growth-factor powers to solve SSC, RRB, Banking, Defence, Police, and other competitive exam questions faster

Related Topics

To strengthen these concepts further, we should also practise:

• Percentage – increase, decrease, and successive percentage
• Simple Interest & Compound Interest – compounding concepts
• Ratio and Proportion – comparison and proportional change
• Profit, Loss & Discount – percentage-based increases and decreases
• Depreciation – repeated percentage decrease over time

FAQ

1. What is the formula for Constant Compound Growth?

When a quantity increases at the same percentage rate in each growth period, we use:

Final Value = Initial Value × (1 + R/100)ⁿ

where:

R = Growth Rate per period
n = Number of growth periods

2. What is the formula for Compound Decrease?

When a quantity decreases at the same percentage rate in each period, we use:

Final Value = Initial Value × (1 − R/100)ⁿ

where:

R = Decrease Rate per period
n = Number of decrease periods

3. How do we find the Initial Value in a Compound Growth question?

If the final value, growth rate, and number of growth periods are known:

Initial Value = Final Value ÷ (1 + R/100)ⁿ

This is obtained by rearranging the Compound Growth formula.

4. What is the formula when the Growth Rate is different in different periods?

When a quantity grows at different percentage rates in successive periods, we use:

Final Value = Initial Value × (1 + R₁/100) × (1 + R₂/100) × (1 + R₃/100) × …

where R₁, R₂, R₃, … are the growth rates for the respective periods.

5. What is the formula when the Decrease Rate is different in different periods?

When a quantity decreases at different percentage rates in successive periods, we use:

Final Value = Initial Value × (1 − R₁/100) × (1 − R₂/100) × (1 − R₃/100) × …

where R₁, R₂, R₃, … are the decrease rates for the respective periods.

6. What is the formula when a value first increases and then decreases?

If a quantity first increases by a certain percentage and then decreases by another percentage:

Final Value = Initial Value × (1 + Increase%/100) × (1 − Decrease%/100)

The increase and decrease are successive percentage changes, so both percentages are not simply added or subtracted.

7. What is the Growth Factor in Compound Growth?

For a growth rate of R% per period, the Growth Factor is:

Growth Factor = 1 + R/100

For example, for an 8% increase:

Growth Factor = 1 + 8/100 = 1.08

This means the new value becomes 1.08 times the previous value after one growth period.

8. What is the Decrease Factor in Compound Growth?

For a decrease rate of R% per period, the Decrease Factor is:

Decrease Factor = 1 − R/100

For example, for a 10% decrease:

Decrease Factor = 1 − 10/100 = 0.90

This means 90% of the previous value remains after one decrease period

Important Resources Related to Compounding Growth 

1. Simple Interest & Compound Interest
2. Simple Interest Questions with Answers and Solutions
3. Simple Interest & Compound Interest Formula with Examples and Explanations
4. Compound Interest Questions with Answers and Solutions (15 Solved)
5. Difference Between Simple Interest and Compound Interest Questions with Answers and Solutions
6. Depreciation Questions with Answers and Solutions
7. Simple Interest & Compound Interest Practice Questions
8. Simple Interest & Compound Interest PYQs SSC, RRB, Banking & Defence Exams With Solution.