Methods for Solving Age Problems Using Simultaneous Equations

Question

Age-based problems are common in competitive exams and require a solid understanding of linear equations. These problems often involve finding the current or past/future ages of individuals based on certain relationships and conditions. Here, we'll break down fundamental concepts and introduce helpful shortcut methods to tackle these questions efficiently.

Key Scenarios in Age Problems

In age-based problems, the following scenarios are frequently encountered:

Age Some Years Ago: Determining someone&rsquo;s age a certain number of years before the present.
	Age in the Future: Calculating an individual&rsquo;s age after a certain number of years.
	Age Relations: Establishing a relationship between the ages of two or more people.

Generally, two of these scenarios are provided, and the task is to find the third. Often, problems may also present a relationship between the ages of two individuals.

Methods of solving age problems

Generally, there are two proven methods of solving Age problems

Simultaneous Equations
	Formula Short-cuts

To solve these problems, we typically frame a simple linear equation based on the information given and solve for the unknowns.

A. Present Ages Based on Past and Current Age

If person A&rsquo;s age t years ago was n1 times the age of person B, and currently A&rsquo;s age is n2 times that of B,

Then,

B's Present Age:
$
y = \frac{(n_1 - 1) t}{n_2 - n_1} \text{ years}
$

A's Present Age:
$
x = \frac{n_2 (n_1 - 1) t}{n_2 - n_1} \text{ years}
$

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Explanation

Let:
- $ x $ = A's present age
- $ y $ = B's present age

Given conditions:
1. $ x - t = n_1(y - t) $  (A's age $ t $ years ago was $ n_1 $ times B's age at that time)
2. $ x = n_2 y $  (A's current age is $ n_2 $ times B's current age)

Solve for $ x $ from the second condition:**
$
x = n_2 y
$

Substitute $ x = n_2 y $ into the first condition:**
$
n_1 y - t = n_1(y - t)
$

$
n_1 y - t = n_2 y - n_1 t
$

$
n_1 y - n_2 y = -t + n_1 t
$

$
(n_1 - n_2) y = (n_1 - 1) t
$

$
y = \frac{(n_1 - 1) t}{n_1 - n_2}
$

Derive A's Present Age:
Substitute $ y $ back into $ x = n_2 y $:
$
x = n_2 \times \frac{(n_1 - 1) t}{n_1 - n_2}
$

B's Present Age:
$
y = \frac{(n_1 - 1) t}{n_1 - n_2} \text{ years}
$

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A's Present Age:
$
x = \frac{n_2 (n_1 - 1) t}{n_2 - n_1} \text{ years}
$

Example A

The age of father is 4 times the age of his son, If 5 years ago father's age was 7 times the age of his son at that time, what is father's present age?

Solution by simultaneous equations

Given:
- $ t = 5 $ (years ago)
- $ n = 7 $ (father's age was 7 times the son's age)
- $ m = 4 $ (father's current age is 4 times the son's age)

Let:
x = father's present age
y = son's present age

Present age: x = 4y

Age condition 5 years ago:
Father's age 5 years ago: x &minus; 5
Son's age 5 years ago: y &minus; 5

Equation from the condition:
x &minus; 5 = 7(y &minus; 5)

Simultaneous Equation Solution:

x = 4y ...........................eq 1
x &minus; 5 = 7(y &minus; 5))............eq 2

Substitute x = 4y  into the second equation:
4y - 5 = 7(y - 5)
4y - 5 = 7y - 35

30 = 3y
y = 10

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But x = 4y
Thus the father's age:
x = 4 * 10 = 40 years

Solving by shortcut method

Given:
- $ t = 5 $ (years ago)
- $ n_1 = 7 $ (father's age was 7 times the son's age 5 years ago)
- $ n_2 = 4 $ (father's current age is 4 times the son's age)

Father's Present Age $ x $:
$
x = \frac{n_2(n_1 - 1)t}{n_1 - n_2}
$

Substitute the given values:
$
x = \frac{4(7 - 1) \times 5}{7 - 4}
$

Calculate:
$
x = \frac{4 \times 6 \times 5}{3}
$

$
x = \frac{120}{3}
$

$
x = 40
$

The father's present age is 40 years.

B. Present Ages Based on Future Age

The present age of A is n1 times the present age of B. If t years hence, the age of A would be n2 times of B,

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Using Simultaneous Equations

Let&rsquo;s solve the given problem using the simultaneous equations method and derive the required expressions.

Given:
- The present age of A is $ n_1 $ times the present age of B.
- After $ t $ years, the age of A will be $ n_2 $ times the age of B.

Set Up Equations

Let:
- $ A $ be the present age of A,
- $ B $ be the present age of B.

From the first condition:
$
A = n_1 B \tag{1}
$

From the second condition, after $ t $ years:
$
A + t = n_2 (B + t) \tag{2}
$

Substitute and Simplify

Substitute $ A = n_1 B $ from equation (1) into equation (2):
$
n_1 B + t = n_2 (B + t)
$

Expand and simplify:
$
n_1 B + t = n_2 B + n_2 t
$
$
n_1 B - n_2 B = n_2 t - t
$
$
(n_1 - n_2) B = t (n_2 - 1)
$

Solve for $ B $:
$
B = \frac{t (n_2 - 1)}{n_1 - n_2} \tag{3}
$

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Solve for $ A $

Now, substitute $ B $ from equation (3) into equation (1):
$
A = n_1 B = n_1 \cdot \frac{t (n_2 - 1)}{n_1 - n_2}
$

Thus, the present age of A is:
$
A = \frac{n_2 t (n_2 - 1)}{n_1 - n_2} \tag{4}
$

Final Expressions

- The present age of A:
  $
  A = \frac{n_2 t (n_2 - 1)}{n_1 - n_2}
  $

- The present age of B:
  $
  B = \frac{t (n_2 - 1)}{n_1 - n_2}
  $

Example B

The age of Mr. Bed is 4 times the age of his son. After 10 years, the age of Mr. Bed will be only twice the age of his son. Find the present age of Mr. Bed's son.

Given:
- The present age of Mr. Bed $ A = 4 B $,
- After 10 years, Mr. Bed&rsquo;s age will be twice the age of his son: $ A + 10 = 2(B + 10) $.

Set Up Equations

From the first condition:
$
A = 4 B \tag{1}
$

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From the second condition:
$
A + 10 = 2 (B + 10) \tag{2}
$

Substitute and Solve

Substitute $ A = 4 B $ into equation (2):
$
4 B + 10 = 2 (B + 10)
$

Simplify:
$
4 B + 10 = 2 B + 20
$
$
4 B - 2 B = 20 - 10
$
$
2 B = 10
$
$
B = 5
$

So, Mr. Bed&rsquo;s son is currently 5 years old.

Find Mr. Bed's Age

Using $ A = 4 B $:
$
A = 4 \times 5 = 20
$

Thus, Mr. Bed&rsquo;s current age is 20 years.

C. Deriving Ages of A and B based on past and future time

The age of A, t1 years ago was n1 time the age of B. If t2 years later A's age would be n2 times that of B, then

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Simultaneous Equations

Given:
- The age of A $ t_1 $ years ago was $ n_1 $ times the age of B.
- After $ t_2 $ years, A's age will be $ n_2 $ times the age of B.

Let:
- $ A $ be the present age of A,
- $ B $ be the present age of B.

Set Up Equations Based on the Given Conditions

From the first condition, $ t_1 $ years ago:
$
A - t_1 = n_1 (B - t_1) \tag{1}
$

From the second condition, after $ t_2 $ years:
$
A + t_2 = n_2 (B + t_2) \tag{2}
$

Solve the Equations Simultaneously

Simplifying Equation (1):
$
A - t_1 = n_1 B - n_1 t_1
$
$
A = n_1 B - n_1 t_1 + t_1
$
$
A = n_1 B + t_1 (1 - n_1) \tag{3}
$

Simplifying Equation (2):
$
A + t_2 = n_2 B + n_2 t_2
$
$
A = n_2 B + n_2 t_2 - t_2
$
$
A = n_2 B + t_2 (n_2 - 1) \tag{4}
$

Solve for $ A $ and $ B $

Shortcut Formula Method

- $ A = \frac{t_2 (n_2 - 1) + t_1 (n_1 - 1)}{n_1 - n_2} $
- $ B = \frac{t_2 (n_2 - 1) - t_1 (n_1 - 1)}{n_1 - n_2} $

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Example C

10 years ago, Mike's mother was 4 times older than him. After 10 years, his mother will be twice as old as Mike. Find Mike's present age.

Given:
- 10 years ago, Mike's mother was 4 times older than him: $ A - 10 = 4 (B - 10) $,
- After 10 years, his mother will be twice as old as Mike: $ A + 10 = 2 (B + 10) $.

Set Up Equations

From the first condition:
$
A - 10 = 4 (B - 10) \tag{1}
$

From the second condition:
$
A + 10 = 2 (B + 10) \tag{2}
$

Solve Simultaneously

Simplify Equation (1):
$
A - 10 = 4 B - 40
$
$
A = 4 B - 30 \tag{3}
$

Simplify Equation (2):
$
A + 10 = 2 B + 20
$
$
A = 2 B + 10 \tag{4}
$

Solve the Two Equations

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Equating (3) and (4):
$
4 B - 30 = 2 B + 10
$
$
4 B - 2 B = 10 + 30
$
$
2 B = 40
$
$
B = 20
$

So, Mike's present age is 20 years.

Mike's Mother's Age

Substitute $ B = 20 $ into equation (4):
$
A = 2 \times 20 + 10 = 50
$

So, Mike's mother is currently 50 years old.

Shortcut Method

Subtituting the givine data into

- $ A = \frac{t_2 (n_2 - 1) + t_1 (n_1 - 1)}{n_1 - n_2} $
- $ B = \frac{t_2 (n_2 - 1) - t_1 (n_1 - 1)}{n_1 - n_2} $

Subtituting the conditions give, we have that

- Mike&rsquo;s present age = 20 years
- Mike&rsquo;s mother&rsquo;s present age = 50

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