Problems on Ages with Ages in Geometric Progression
This page discusses the concept of "Problems on Ages with Ages in Geometric Progression". It explains the key concepts involved in solving problems related to this topic, such as geometric progressions, linear equations, and quadratic equations. It also provides a list of formulas that are useful in solving these types of problems.
Introduction to Age Problems with Ages in Geometric Progression
Problems on ages with ages in geometric progression
are a common topic in various competitive exams like SSC, banking, CAT, etc. These types of questions involve the concept of geometric progression, which is a fundamental topic in mathematics. We will discuss the key concepts, formulas, and techniques involved in solving problems on ages with ages in geometric progression.
Formulae Related to Geometric Progression
A list of formulas that can be useful in solving problems on ages with ages in geometric progression
| Formula | Description |
|---|---|
| Sn = a(1 - r^n)/(1 - r) | Sum of a finite geometric progression |
| S∞ = a/(1 - r) | Sum of an infinite geometric progression |
| Pn = a^n*r^(n(n-1)/2) | Product of a finite geometric progression |
| an = a*r^(n-1) | nth term of a geometric progression |
| A = (a + ar + ar^2 + ... + ar^(n-1))/n = a(1 + r + r^2 + ... + r^(n-1))/n | Average of a finite geometric progression |
| 1/(1-x) = 1 + x + x^2 + x^3 + ... | Sum of the infinite series |
| ax + b = c, x = (c - b)/a | Formula for solving linear equations |
| ax^2 + bx + c = 0, x = (-b ± sqrt(b^2 - 4ac))/2a | Formula for solving quadratic equations |
| a^2 - b^2 = (a + b)(a - b) | Difference of squares |
| a^3 - b^3 = (a - b)(a^2 + ab + b^2) | Difference of cubes |
| a^3 + b^3 = (a + b)(a^2 - ab + b^2) | Sum of cubes |
| (a + b)^2 = a^2 + 2ab + b^2 | Square of a binomial |
| (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 | Cube of a binomial |
Key Concepts:
- Finding the Ratio of Ages: To find the ratio of ages when the ages of two or more persons are in GP, we need to set up an equation using the formula for GP.
For example, if the ages of A, B, and C are in GP, and we know that A is 5 years old and C is 45 years old, we can write:
5 : B : 45 = a : ar : ar^2
where a is the age of A, B is the age of B, and r is the common ratio.
To solve for B, we can use the fact that the product of the ages is a constant value, and write:
5 * B * 45 = a^3 * r^3
Simplifying this equation, we get:
B = 15 years
Therefore, the ratio of ages is:
5 : 15 : 45 = 1 : 3 : 9
- Finding the Sum of Ages:
To find the sum of ages when the ages of two or more persons are in GP, we need to use the formula for the sum of a GP.
For example,
if the ages of A, B, and C are in GP, and we know that A is 5 years old and the sum of their ages is 63 years, we can write:
5 + 5r + 5r^2 = 63
Using the formula for the sum of a GP, we can write:
5(1 - r^3)/(1 - r) = 63
Simplifying this equation, we get:
r = 2
Therefore, the ages of A, B, and C are:
5, 10, 20
- Finding the Product of Ages:
To find the product of ages when the ages of two or more persons are in GP, we can use the formula for the product of ages in GP.
For example, if the ages of A, B, and C are in GP, and we know that A is 4 years old and the product of their ages is 256, we can write:
4 * 4r * 4r^2 = 256
Simplifying this equation, we get:
r = 2
Therefore, the ages of A, B, and C are:
4, 8, 16
- Finding the Average of Ages: To find the average of ages when the ages of two or more persons are in GP, we can use the formula for the sum of a GP and the formula for the average.
For example, if the ages of A, B, and C are in GP, and we know that A is 6 years old and the average age is 12 years, we can write:
6 + 6r + 6r^2 = 3 * 12
Using the formula for the sum of a GP, we get:
6(1 - r^3)/(1 - r) = 36
Simplifying this equation, we get:
r = 2
Therefore, the ages of A, B, and C are:
6, 12, 24
And the average age is:
(6 + 12 + 24)/3 = 14
Solved Examples
Question 1 (Easy Level):
The ages of A, B, and C are in GP. The sum of their ages is 56, and A is 4 years old. Find the ages of B and C.
A) 8, 16
B) 16, 32
C) 12, 24
D) 10, 20
Solution:
Let the common ratio be r. Then, the ages of A, B, and C are
4, 4r, and 4r^2. Using the formula for the sum of a GP, we can write:
4(1 - r^3)/(1 - r) = 56
Simplifying this equation, we get: r = 2
Therefore, the ages of A, B, and C are: 4, 8, 16
Hence, option A is the correct answer.
Key Concept used: Using the formula for the sum of a GP to find the value of the common ratio.
Question 2 (Easy Level):
The ages of A, B, and C are in GP. A is 3 years old, and the sum of their ages is 84. Find the product of their ages.
A) 324
B) 81
C) 729
D) 243
Solution:
Let the common ratio be r.
Then, the ages of A, B, and C are 3, 3r, and 3r^2. Using the formula for the sum of a GP, we can write:
3(1 - r^3)/(1 - r) = 84/3
Simplifying this equation, we get: r = 3
Therefore, the ages of A, B, and C are: 3, 9, 27
Hence, the product of their ages is: 3 * 9 * 27 = 729
Therefore, option C is the correct answer.
Key Concept used: Using the formula for the sum of a GP to find the value of the common ratio, and using the product of ages in GP formula.
Question 3 (Medium Level):The ages of A, B, and C are in GP. A is 5 years old, and the product of their ages is 540. Find the ages of B and C.
A) 10, 20
B) 10, 30
C) 12, 36
D) 12, 48
Solution:
Let the common ratio be r. Then, the ages of A, B, and C are 5, 5r, and 5r^2. Using the formula for the product of a GP, we can write:
5 * 5r * 5r^2 = 540
Simplifying this equation, we get: r = 2
Therefore, the ages of A, B, and C are: 5, 10, 20
Hence, option A is the correct answer.
Key Concept used: Using the formula for the product of a GP to find the value of the common ratio.
Question 4 (Medium Level):
The ages of A, B, and C are in GP. The average of their ages is 24, and A is 6 years old. Find the ages of B and C.
A) 12, 36
B) 12, 24
C) 18, 36
D) 18, 54
Solution:
Let the common ratio be r. Then, the ages of A, B, and C are 6, 6r, and 6r^2. The average of their ages is given by:
(6 + 6r + 6r^2)/3 = 24
Simplifying this equation, we get: r = 2
Therefore, the ages of A, B, and C are: 6, 12, 24
Hence, option B is the correct answer.
Key Concept used: Using the formula for the average of a GP to find the value of the common ratio.
