Growth and Progression Questions - CCAT Test Prep

This type of question tests your understanding of Exponential Growth and how numbers increase repeatedly over a period of time. In this example, the number of guests triples every half hour for the next two hours. To solve it, you need to calculate how the initial number of guests grows step by step through multiple intervals. By breaking down the intervals, you can determine the final number of guests present at the party.

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How to Calculate Exponential Growth

Question 1 - Exponential Growth

Only two guests have arrived at a party by 8:00 PM, but the number of guests present triples every half hour for the next two hours. How many guests are present at the party by 10:00 PM?

  • A) 64
  • B) 80
  • C) 120
  • D) 136
  • E) 162

Step-by-Step Solution:

Start by noting the given information:

  • The initial number of guests at 8:00 PM is 2.
  • The number of guests triples every half hour.
  • You need to determine the total number of guests at 10:00 PM, which is 2 hours later.

Two hours contain four half-hour intervals. Let’s calculate the number of guests at each interval:

  1. Initial Guests at 8:00 PM:

Initial guests = 2
  1. After the First Half Hour (8:30 PM):

Guests = 2 × 3 = 6
  1. After the Second Half Hour (9:00 PM):

Guests = 6 × 3 = 18
  1. After the Third Half Hour (9:30 PM):

Guests = 18 × 3 = 54
  1. After the Fourth Half Hour (10:00 PM):

Guests = 54 × 3 = 162

Answer: 162

Question 2 - Geometric Progression

If a plant grows to double its size every hour, starting at 5 cm at 2:00 PM, how tall will it be at 6:00 PM?

  • A) 10 cm
  • B) 15 cm
  • C) 20 cm
  • D) 40 cm
  • E) 80 cm

Step-by-Step Solution:

The growth pattern is doubling each hour, starting with 5 cm at 2:00 PM.

  1. Initial Height at 2:00 PM:

Initial height = 5 cm
  1. After One Hour (3:00 PM):

Height = 5 × 2 = 10 cm
  1. After Another Hour (4:00 PM):

Height = 10 × 2 = 20 cm
  1. After Another Hour (5:00 PM):

Height = 20 × 2 = 40 cm
  1. After Another Hour (6:00 PM):

Height = 40 × 2 = 80 cm

Answer: 80 cm

Question 3 - Population Growth

A population of bacteria doubles every 30 minutes. If the initial population is 100 at 9:00 AM, what will be the population at 11:00 AM?

  • A) 800
  • B) 1600
  • C) 3200
  • D) 4000
  • E) 6400

Step-by-Step Solution:

  1. Initial Population at 9:00 AM:

Initial population = 100
  1. After Each 30-Minute Interval:
    • After 9:30 AM:
    
Population = 100 × 2 = 200
    • After 10:00 AM:
    
Population = 200 × 2 = 400
    • After 10:30 AM:
    
Population = 400 × 2 = 800
    • After 11:00 AM:
    
Population = 800 × 2 = 1600

Answer: 1600

How to Solve These Questions in Your Head

  1. Identify the Growth Rate: Determine by how much the value changes each interval (e.g., doubling, tripling).
  2. Count the Number of Intervals: Understand how many intervals occur in the given timeframe.
  3. Apply Repeated Multiplication: Multiply the value by the growth rate repeatedly for each interval.
  4. Calculate the Final Value: The final result will be the value after applying the growth rate for the total number of intervals.

This approach helps you solve questions involving growth rates and progression by breaking down the problem into manageable intervals and applying multiplication in each step.

20 Practice Questions - Exponential Growth

  1. A bacteria colony triples every 15 minutes. If the initial population is 4, how many bacteria will be present after 1 hour?
  • A) 108
  • B) 162
  • C) 192
  • D) 216
  • E) 252
  1. A group of ants doubles every 10 minutes. Starting from 8 ants, how many ants will there be after 40 minutes?
  • A) 64
  • B) 80
  • C) 96
  • D) 128
  • E) 256
  1. A plant grows by multiplying its height by 4 every hour. Starting at 1 cm at 12:00 PM, what will be the height at 3:00 PM?
  • A) 64 cm
  • B) 48 cm
  • C) 16 cm
  • D) 32 cm
  • E) 128 cm
  1. A virus population doubles every hour. Starting with 100 viruses at 9:00 AM, how many viruses will there be at 12:00 PM?
  • A) 400
  • B) 600
  • C) 800
  • D) 1000
  • E) 1200
  1. The number of bees in a hive triples every 20 minutes. Starting from 6 bees, how many will be present after 1 hour?
  • A) 54
  • B) 72
  • C) 108
  • D) 162
  • E) 180
  1. A population of cells quadruples every 30 minutes. If the initial population is 3, what is the population after 2 hours?
  • A) 48
  • B) 96
  • C) 192
  • D) 384
  • E) 768
  1. A drop of water doubles in volume every 10 minutes. Starting with 1 mL, what will be the volume after 50 minutes?
  • A) 16 mL
  • B) 24 mL
  • C) 32 mL
  • D) 48 mL
  • E) 64 mL
  1. A rabbit population increases fivefold every hour. Starting with 2 rabbits at 1:00 PM, how many will there be at 4:00 PM?
  • A) 50
  • B) 100
  • C) 125
  • D) 250
  • E) 500
  1. The number of fish in a pond doubles every 15 minutes. Starting from 10 fish, how many will be present after 1 hour and 15 minutes?
  • A) 80
  • B) 160
  • C) 320
  • D) 640
  • E) 1280
  1. A virus population triples every hour. Starting with 5 viruses at 10:00 AM, how many will there be at 1:00 PM?
  • A) 45
  • B) 90
  • C) 135
  • D) 180
  • E) 225
  1. A plant doubles its leaves every hour, starting with 4 leaves at 2:00 PM. How many leaves will it have by 6:00 PM?
  • A) 32
  • B) 48
  • C) 64
  • D) 128
  • E) 256
  1. The population of frogs doubles every 45 minutes. Starting from 8 frogs, what is the population after 3 hours?
  • A) 64
  • B) 96
  • C) 128
  • D) 256
  • E) 512
  1. A certain mold grows to triple its size every 20 minutes. Starting at 1 cm, what will its size be after 1 hour?
  • A) 9 cm
  • B) 18 cm
  • C) 27 cm
  • D) 36 cm
  • E) 45 cm
  1. The number of birds in a park doubles every 30 minutes. Starting with 3 birds, how many will there be after 2 hours?
  • A) 24
  • B) 48
  • C) 96
  • D) 192
  • E) 384
  1. A company's profit triples every month. If the profit in January is $100, how much will it be in April?
  • A) $300
  • B) $900
  • C) $2700
  • D) $8100
  • E) $24300
  1. A tree grows by multiplying its branches by 2 every month. Starting with 4 branches, how many will there be after 5 months?
  • A) 64
  • B) 72
  • C) 96
  • D) 128
  • E) 256
  1. A population of cells quadruples every hour. Starting with 10 cells, how many cells will there be after 3 hours?
  • A) 160
  • B) 240
  • C) 320
  • D) 400
  • E) 640
  1. A group of fireflies doubles every 15 minutes. Starting with 5 fireflies, how many will be present after 1 hour and 15 minutes?
  • A) 80
  • B) 90
  • C) 100
  • D) 160
  • E) 320
  1. The number of bacteria in a dish doubles every 20 minutes. Starting with 6 bacteria, how many will there be after 2 hours?
  • A) 96
  • B) 192
  • C) 384
  • D) 768
  • E) 1536
  1. A balloon doubles in volume every 10 minutes. Starting with 2 cubic cm, what will be the volume after 1 hour?
  • A) 64 cubic cm
  • B) 128 cubic cm
  • C) 256 cubic cm
  • D) 512 cubic cm
  • E) 1024 cubic cm

Answers to Practice Questions:

  1. D) 216
  2. D) 128
  3. A) 64 cm
  4. C) 800
  5. D) 162
  6. D) 384
  7. C) 32 mL
  8. D) 250
  9. C) 320
  10. C) 135
  11. C) 64
  12. C) 128
  13. C) 27 cm
  14. D) 192
  15. C) $2700
  16. A) 64
  17. E) 640
  18. D) 160
  19. E) 1536
  20. C) 256 cubic cm
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