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__Quantitative Aptitude Notes: Problems on Ages__

__Quantitative Aptitude Notes: Problems on Ages__

**Introduction**

**1)**If the current age of a person be X, then

- age after
n years = X + n

- age n
years ago = X – n

- n times
the age = nX

- If ages
in the numerical are mentioned in ratio A : B, then A : B will be AX and BX

**2)**If sum of ages of x and y is A and ratio of their ages is p : q respectively, then u can determine age of y by using the formula shown below:

- Age of y
= Ratio of y/Sum of ratios x Sum of Ages

- Age of y
= q/(p + q) x A

**Points to Remember:**

**a)**After reading the question, assume unknown age by some variable x.

**b)**Convert the statements in the question into mathematical equations.

**c)**Calculate the unknown by solving the equations and the obtained value must satisfy the conditions given in the problem.

**Memory Based Example Problems based on Various Types**

**Type 1: Comparing 2 persons**

**1. Rahul is 15 years elder than Rohan. If 5 years ago, Rahul was 3 times as old as Rohan, then find Rahul's present age.**

**Solution:**

Let age of
Rohan be y. Given that,

Rahul is 15
years elder than Rohan = (y + 15). So Rahul’s age 5 years ago = (y + 15 – 5)

Rohan’s age
before 5 years = (y – 5)

5 years
ago, Rahul is 3 times as old as Rohan

(y + 15 –
5) = 3 (y – 5) =>(y + 10) = (3y – 15) =>2y = 25 =>y = 12.5

Rohan’s age
= 12.5 years

Rahul’s age
= (y + 15) = (12.5 + 15) = 27.5 years

**2. Father is aged three times more than his son Sunil. After 8 years, he would be two and a half times of Sunil's age. After further 8 years, how many times would he be of Sunil's age?**

**Solution:**

Assume that
Sunil's present age = x.

Then
father's present age = 3x + x = 4x

After 8
years, fathers age = 2 1/2 times of Sunils' age

=> (4x+8)
= (2) (1/2) (x+8) => 4x + 8 = (5/2) (x + 8) => 8x + 16 = 5x+ 40 => 3x
= 40-16 = 24 => x = 8

After
further 8 years,

Sunil's age
= x + 8+ 8 = 8+8+8 = 24

Father's
age = 4x + 8 + 8 = 4 * 8 + 8 + 8 = 48

Father's
age/Sunil's age = 48/24 = 2

**3. Sharad is 60 years old and Santosh is 80 years old. How many years ago was the ratio of their ages 4: 6?**

**Solution:**

Here, we
have to calculate: How many years ago the ratio of their ages was 4: 6

Let us
assume x years ago

**At present:**Sharad is 60 years and Santosh is 80 years

**x years ago:**Sharad’s age = (60 – x) and Santosh’s age = (80 – x)

Ratio of
their ages x years ago was 4: 6

(60 – x)/
(80 – x) = 4/ 6

6(60 – x) =
4(80 – x)

360 – 6x =
320 – 4x

x = 20

Therefore,
20 years ago, the ratio of their ages was 4: 6

**4. Three years earlier, the father was 7 times as old as his son. Three years hence, the father’s age would be four times of his son. What are the present ages of the father and the son?**

**Solution:**

Let the
present age of son = x yrs and the present age of father = y yrs

3 yrs
earlier, 7(x – 3) = y – 3 or, 7x – y =18………….(i)

3 yes
hence, 4(x+3) = y +3

Or, 4x +12
= y + 3 or, 4x – y = - 9 …………(ii)

Solving (1)
& (2) we get, x = 9 yrs & y =45 yrs

**5. Q is as much younger than R as he is older than T. If the sum of the ages of R and T is 50 years, what is definitely the difference between R and Q’s age?**

**Solution:**

R – Q = Q –
T

(R + T) =
2Q

Now given
that, (R + T) = 50

So, 50 = 2Q
and therefore Q = 25.

Question is
(R – Q) = ?

Here we
know the value (age) of Q (25), but we don’t know the age of R.

Therefore,
(R-Q) cannot be determined.

**6. Father is 3 times more aged than his daughter. If after 5 years, he would be 3 times of daughter’s age, then further after 5 years, how many times he would be of his daughter’s age?**

**Solution:**

Let
daughter’s age be x and father’s age be 3x.

Father’s
age is 3 times more aged than his daughter, therefore father’s present age = x
+ 3x = 4x

After 5
years, father’s age is 3 times more than his daughter age.

(4x + 5) =
3 (x + 5) =>(4x+5)=3 (x+5) =>(4x + 5) = 3 (x + 5) => x = 10

After 5
years it was (4x + 5), then after further 5 years, father’s age = (4x +10) and
daughter’s age = (x + 10)

[(4x +
10)/(x + 10)] = ?

Substitute
the value of x, we get 2.5

After
further 5 years, father will be 2.5 times of daughter’s age

**7. 5 years ago, sister’s age was 5 times the age of her brother and the sum of present ages of sister and brother is 34 years. What will be the age of her brother after 6 years?**

**Solution:**

Let present
age of brother be x and sister’s age be 34 – x

**Past Age (5 yrs Ago) Present Age Future Age (After 6 yrs)**

**Brother**(x – 5) X (x + 6) = ?

**Sister**(34 – x) – 5 (30 – x)

We are
given, 5 years ago sister’s age was 5 times the age of her brother.

Therefore,

(34 – x) –
5 = 5 (x – 5) =>34 – x – 5 = 5x – 25 =>5x + x = 34 – 5 +25 =>6x = 54
=>x = 9

Future age
(after 6 yrs) = (x + 6) = (9 + 6) = 15 years

**Type 2: Comparing 3 persons**

**1. Age of mother 10 years ago was 3 times the age of her son. After 10 years, mother’s age will be twice that of his son. Find the ratio of their present ages.**

**Solution:**

We are
given that, age of mother 10 years ago was 3 times the age of her son

So, let age
of son be x and as mother’s age is 3 times the age of her son, let it be 3x,
three years ago.

At present:
Mother’s age will be (3x + 10) and son’s age will be (x + 10)

After 10
years: Mother’s age will be (3x + 10) +10 and son’s age will be (x + 10) + 10

Mother’s
age is twice that of son

(3x + 10)
+10 = 2 [(x + 10) + 10]

(3x + 20) =
2[x + 20]

Solving the
equation, we get x = 20

We are asked
to find the present ratio.

(3x + 10) :
(x + 10) = 70 : 30 = 7 : 3

**2. The total age of A and B is 12 years more than the total age of B and C. C is how many years younger than A ?**

**Solution:**

(A+B) -
(B+C) = 12

=> A - C
= 12.

=> C is
younger than A by 12 years.

**3. If two times of the daughter’s age in years is included to the mother’s age, the total is 70 and if two times of the mother’s age is included to the daughter’s age, the total is 95. So the Mother’s age is,**

**Solution:**

Let daughter’s
age = A and mother’s age = B

Given: 2A+B
= 70 and A+2B = 95

Solving B,
we will get B = 40.

**4. If 25 is the sum of ages of X, Y and Z and if X is 17 years younger than Z who is thrice as old as Y. Then how old is Z?**

**Solution:**

Let Y's age
be a.

Then Z's
age is 3a.

And X's age
= 3a-17.

Now, a + 3a
+ 3a-17 = 25.

7a - 17 =
25. => 7a = 25 + 17 = 42

a = 6.
Therefore, Z's age = 3(6) = 18.

**Type 3: Based on Ratio**

**1. Eight years ago, Poorvi’s age was equal to the sum of the present ages of her one son and one daughter. Five years hence, the respective ratio between the ages of her daughter and her son that time will be 7:6. If Poorvi’s husband is 7 years elder to her and his present age is three times the present age of their son, what is the present age of the daughter?**

**Solution:**

Let present
age of her son = x

Present age
of Poorvi's husband = 3x

Present age
of Poorvi = 3x - 7

Eight years
ago, Poorvi’s age = 3x−15

present age
of her daughter => (3x−15) – x = (2x−15)

after 55
years, age of her daughter =(2x−10)

age of her
son =(x+5)

(2x−10):
(x+5)=7: 6 => 12x − 60= 7x + 35 => 5x = 95 => x=19

Present age
of her daughter = (2x − 15) = 23

**2. The ratio of Rohan’s age 4 years ago and Rahul’s age after 4 years is 1: 1. If at present, the ratio of their ages is 5: 3, then find the ratio between Rohan’s age 4 years hence and Rahul’s age 4 years ago.**

**Solution:**

At present:
Ratio of their ages = 5 : 3. Therefore, 5 : 3 will be 5x and 3x.

Rohan’s age
4 years ago = 5x – 4

Rahul’s age
after 4 years = 3x + 4

-> Ratio
of Rohan’s age 4 years ago and Rahul’s age after 4 years is 1 : 1

Therefore,
[(5x – 4)/(3x + 4)] = 1 => x = 4

-> We
are asked to find the ratio between Rohan’s age 4 years hence and Rahul’s age 4
years ago.

Rohan’sage
: (5x + 4)

Rahul’s
age: (3x – 4)

Ratio of
Rahul’s age and Rohan’s age [(5x + 4)/(3x – 4)] = 24/8 = 3 : 1

**3. Present ages of Kiran and Syam are in the ratio of 5:4 respectively. Three years hence, the ratio of their ages will become 11:9 respectively. What is Syam's present age in years?**

**Solution:**

Let the
present ages of Kiran and Syam be 5

*x*years and 4*x*years respectively.
Then, (5x +
3)/(4x + 3) = 11/9

=> 9(5

*x*+ 3) = 11(4*x*+ 3) => 45*x*+ 27 = 44*x*+ 33 => 45*x*- 44*x*= 33 – 27 =>*x*= 6.
Syam’s
present age = 4

*x*= 24 years.**4. The present ages of A, B and C are in proportions 4:7:9. Eight years ago, the sum of their ages was 56. What are their present ages (in years)?**

**Solution:**

Eight years
ago, their ages were (4x-8), (7x-8) and (9x-8)

(4x-8) +
(7x-8) + (9x-8) = 56

20x - 24 =
56

20x = 80

x = 4

Their
present ages are (4*4, 7*4 , 9*4) = 16, 28, 36

**5. Present ages of Harish and his daughter Lalima are in the ratio of 9:2 respectively. Harish’s wife Richa is 3 years younger than him and two years hence the ratio between ages of Richa and Lalima will be 7:2 respectively. What is Lalima’s present age in years?**

**Solution:**

Let present
age of Harish =9x

Present age
of Lalima =2x

Present age
of Richa =9x−3

After two
years, age of Richa =9x−3+2 = 9x−1

After two
years, age of Lalima =2x+2

9x−1:
2x+2=7: 2

18x−2=14x+14

4x=16 =>
x=4

Lalima’s
present age = 2x = 8

**6. If 13:11 is the ratio of present age of Jothi and Viji respectively and 15:9 is the ratio between Jothi's age 4 years hence and Viji's age 4 years ago. Then what will be the ratio of Jothi's age 4 years ago and Viji's age 4 years hence?**

**Solution :**

Let the
present age of Jothi and Viji be 13X and 11X respectively.

Given,
Jothi's age 4 years hence and Viji's age 4 years ago in the ratio 15:9.

That is,
13X + 4 / 11X - 4 = 15/9

9(13X + 4)
= 15(11X - 4)

117X + 36 =
165X - 60

48X = 96

X = 2.

Now,
required ratio is (13X – 4)/(11X+4) = 13(2)-4 / 11(2)+4 = 22/26 = 11/13.

Hence the
answer is 11:13

**Type 4: Miscellaneous**

**1. A father said his son, " I was as old as you are at present at the time of your birth. " If the father age is 38 now, the son age 5 years back was:**

**Solution:**

Let the
son's present age be x years .Then, (38 - x) = x x= 19.

Son's age 5
years back = (19 - 5) = 14 years

**2. The ages of two persons differ by 16 years. 6 years ago, the elder one was 3 times as old as the younger one. What are their present ages of the elder person?**

**Solution:**

Let's take
the present age of the elder person = x

and the
present age of the younger person = x – 16

(x – 6) = 3
(x – 16 – 6)

=> x – 6
= 3x – 66

=> 2x =
60

=> x =
60/2 = 30

**3. In a family, a couple has a son and daughter. The age of the father is three times that of his daughter and the age of the son is half of his mother. The wife is nine years younger to her husband and the brother is seven years older than his sister. What is the age of the mother?**

**Solution:**

Let the
daughter's age be x years.

Then,
father's age = (3x) years.

Mother's
age = (3x - 9) years; Son's age = (x + 7) years.

So, x + 7 =
(3x-9)/2 => 2x + 14 = 3x - 9 => x = 23.

Therefore
Mother's age = (3X - 9) = (69 - 9) years = 60 years.