1. AGES
Shortcut Methods for Age related Problems
1.
t 1 years earlier the fathers age was x times
that of his son. At present the fathers age is y times that of his son. What
are the present ages of his son and father
Sons age = t1 (x-1) / x-y
2.
The present age of the father is y times the
age of his son. t 2 years hence, the fathers age becomes z times the age of his
son. What are the present ages of the fathesr and son
Sons age = (z-1) t2 / y-z
3.
T1 years earlier the age of the father was x
times the age of his son. t2 years hence, the age of the father becomes z times
the age of his son. the present ages of the son and father
Sons age = (z-1)t2 + t1(x-1)
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(x-z)
4.
n years ago age = Total + Number of Years
(Times-1)
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Times+1
5.
N years hence age = Total - Number of Years
(Times-1)
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Times+1
6.
Present age = x:y, After T years = a:b
Sons age =
y *
T(a-b)/ difference of cross products
Fathers age=
x * T(a-b)/ difference of
cross products
7.
n years before the ages of Father and Son were
in the ratio x:y. In another n years from now the ratio will be a:b. what are
their Present ages
2n(a-b) / Difference of
cross products
8.
If a is n years older than b. After x years y
times of a equal to z times b. then their ages now
a age = zn/ z-y, b age = yn/y-z
Examples
1 The age of the
father 3 years ago was 7 times the age of his son. At present the father's age
is 5 times that of his son. What are the present ages of the father and the son?
Sol: Let the present age of son = x
years
Then the present age of father = 5x yrs
3 years ago, 7(x-3) =5x - 3
2x = 18, x= 9 yrs, so father's age = 45
yrs
2. The ratio of the
father's age to that of son's age is 4:1 the product of their ages is 196.What
will be ratio of their ages after 5 yrs?
Sol: let the ratio of proportionality be x yrs
4x*x = 196 or 4x2 = 196 or x= 7
Thus father's age = 28 yrs, Son's age =
7 yrs
After 5 yrs, father's age = 33 yrs,
son's age =12 yrs
Ratio = 33:12= 11:4
3. The ratio of
Vimals age and Arunas age is 3: 5 and the sum of their ages is 80 years. The
ratio of their ages after 10 years will be
Sol: Let their ages be
3x and 5x years.
Then,3x + 5x = 8O so x = 1O.
Ratio of their ages after 10 years = (3x+ 10) : (5x+ 10)
40 : 60 = 2 : 3.
Then,3x + 5x = 8O so x = 1O.
Ratio of their ages after 10 years = (3x+ 10) : (5x+ 10)
40 : 60 = 2 : 3.
4. One year ago,
Mrs Promila was four times as old as her daughter Swati. Six years hence, Mrs
Promilas age will exceed her daugther’s age by 9 years. The ratio of the
present ages of Promila and her daughter is:
Sol: Let Swatis age 1
year ago = x.
Then, Promilas age 1 year ago = 4x.
(4x+6) - (x+6) = 9 or x = 3.
Present age of Promila = (12 + 1) years = 13 years.
Present age of Swati = (3+ 1) years =4 years.
Ratio of their ages = 13 : 4.
Then, Promilas age 1 year ago = 4x.
(4x+6) - (x+6) = 9 or x = 3.
Present age of Promila = (12 + 1) years = 13 years.
Present age of Swati = (3+ 1) years =4 years.
Ratio of their ages = 13 : 4.
5. The age of
father 10 years ago was thrice the age of his son. Ten years hence, the
father’s age will be twice that of his son. The ratio of their present ages is:
Sol: Let sons age 10
years ago = x.
Then, fathers age 10 years ago = 3x.
2(x+ 10+ 10) = (3x + 10 + 10)so x = 20.
Ratio of their present ages = (3x + 10) : Cx + 10) 70: 30 = 7 :3.
Then, fathers age 10 years ago = 3x.
2(x+ 10+ 10) = (3x + 10 + 10)so x = 20.
Ratio of their present ages = (3x + 10) : Cx + 10) 70: 30 = 7 :3.
6. Snehs age
is1/6 th of her fathers age. Snehs fathers age will be twice of Vimals age
after 10 years. If Vimals eighth birthday was celebrated two years before, then
what is Snehs present age?.
Sol: Vimals age after 10
years = (8 + 2 + 10) years = 20 years.
Snehs father’s age after 10 years = 40 years.
Snehs father’s present age = 30 years.
Snehs age= 1/6 * 30 years = 5 years.
Snehs father’s age after 10 years = 40 years.
Snehs father’s present age = 30 years.
Snehs age= 1/6 * 30 years = 5 years.
7. Ten years
ago, Chandrawati s mother was four times older than her daughter. After ten
years, the mother will be twice older than daughter. The present age of
Chandrawati is
Sol: Let Chandrawatis
age 10 years ago is x years.
Her mother’s age 10 years ago 4x.
(4x+10)+10=2(x+10+10) or 2x=20 or x=10.
Present age of Chandrawati (x + 10) = 20 years.
Her mother’s age 10 years ago 4x.
(4x+10)+10=2(x+10+10) or 2x=20 or x=10.
Present age of Chandrawati (x + 10) = 20 years.
8. Ten years
ago A was half of B in age. If the ratio of their present ages is 3: 4, what
will be the total of their present ages
Sol: Let As age 10 years
ago = x.
Then, Bs age 10 years ago = 2x.
(x + 10) / (2x+ lO) = 3/4 hence x = 5. Total of their present ages =(x + 10 + 2x + 10)
(3x + 20) = 35 years.
Then, Bs age 10 years ago = 2x.
(x + 10) / (2x+ lO) = 3/4 hence x = 5. Total of their present ages =(x + 10 + 2x + 10)
(3x + 20) = 35 years.
9. Kamla got
married 6 years ago. Today her age is 1.25 times her age at the time of
marriage. Her son’s age is 1/10 times her age. Her sons age is:
Sol: Let Kamlas age 6
years ago = x years. Kamlas present age = (x + 6) years.
x+6= (5/4)*x or x=24.
Kamla’s present age= 30 years.
Sons present age = (1/10)*30 years = 3 years.
x+6= (5/4)*x or x=24.
Kamla’s present age= 30 years.
Sons present age = (1/10)*30 years = 3 years.
10.Pushpa is
twice as old as Rita was 2 years ago. If the difference of their ages be 2
years, how old is Pushpa today
Sol: Let Ritas age 2
years ago = x years.
Pushpas age now = 2x years.
2x - (x + 2) = 2 so x = 4.
Pushpas age now = 8 years.
Pushpas age now = 2x years.
2x - (x + 2) = 2 so x = 4.
Pushpas age now = 8 years.
11. Sushil
was thrice as old as Snehal 6 years back. Sushil will be times as old as Snehal
6 years hence. How old is Snehal today?
Sol: Let Snehals age 6
years back = x.
Then, Sushils age 6 years back = 3x.
(5/3) * (X + 6 + 6) = (3X + 6 + 6) hence 5(x+ 12) = 3(3x+ 12) so x=6.
Snehal today = (x+ 6) years = 12 years.
Then, Sushils age 6 years back = 3x.
(5/3) * (X + 6 + 6) = (3X + 6 + 6) hence 5(x+ 12) = 3(3x+ 12) so x=6.
Snehal today = (x+ 6) years = 12 years.
12. The sum of the ages
of a father and son is 45 years. Five years ago, the product of their ages was
four times the father’s age at that time. The present age of father and son,
respectively are:
Sol: Let sons age = x
years. Then fathers age = (45 - x)years.
(x—5)(45—x—5) = 4(45- x - 5) hence (x—5) = 4 so x = 9
their ages are 36 years and 9 years.
(x—5)(45—x—5) = 4(45- x - 5) hence (x—5) = 4 so x = 9
their ages are 36 years and 9 years.
13. Two years
ago a man was 6 times as old as his son. After 18 years, he will be twice as
old as his son. Their present ages (in years) are
Sol: Let sons age 2
years ago be x.
Mans age 2 years ago = 6x.
2(x + 2 + 18) = (6x + 2 + 18) hece 4x=20 so x = 5.
Their present ages are (6x +2) and (x +2) i.e. 32 years and 7 years.
Mans age 2 years ago = 6x.
2(x + 2 + 18) = (6x + 2 + 18) hece 4x=20 so x = 5.
Their present ages are (6x +2) and (x +2) i.e. 32 years and 7 years.
14. A father
is twice as old as his son. 20 years ago, the age of the father was 12 times
the age of the son. The present age of the father (in years) is:
Sol: Let sons age = x.
Then, fathers age = 2x.
12(x—2O) = (2x - 20) so x = 22
Fathers present age = 44 years.
12(x—2O) = (2x - 20) so x = 22
Fathers present age = 44 years.
15. The ages of
two persons differ by 20 years. If 5 years ago, the elder one be 5 times as old
as the younger one, their present ages (in years) are respectively:
Sol: Let their ages be x
and (x + 20) years.
5 (x - 5) = (x + 20 - 5) or 4x = 40 or x = 10.
Their present ages are 30 years and 10 years.
5 (x - 5) = (x + 20 - 5) or 4x = 40 or x = 10.
Their present ages are 30 years and 10 years.
16. The ages of
Ram and Mukta are in the ratio of 3: 5. After 9 years, the ratio of their ages
will becomes 3:4. The present age of Mukta (in years) is.
Sol: Let Rams age = 3x
& Muktas age = 5x.
3x + 9 / 5x + 9 = 3/ 4 so 4*(3x+ 9) = 3 (5x+ 9) hence x= 3.
Muktas age = 15 years.
3x + 9 / 5x + 9 = 3/ 4 so 4*(3x+ 9) = 3 (5x+ 9) hence x= 3.
Muktas age = 15 years.
17. If 6 years are subtracted from the present age of Gulzar and the
remainder is divided by 18, then the present age of his grandson Anup is
obtained. If Anup is 2 years younger to Mahesh whose age is 5 years, then what
is the present age of Guizar
Sol: Anups age
=(5—2)years = 3 years.
Let Guizar’s age x years
Then x - 16/18 = 3 so x - 6 = 54 so x 60
Let Guizar’s age x years
Then x - 16/18 = 3 so x - 6 = 54 so x 60
18. The average age of an
adult class is 40 years. Twelve new students with an average age of 32 years
join the class, thereby decreasing the average of the class by 4 years. The
original strength of the class was
Sol: Let, original
strength = x.
40x + 12*32 / (x + 12) = 36 or x = 12
40x + 12*32 / (x + 12) = 36 or x = 12
19. Sachin
was twice as old as Ajay 10 years back. How old is Ajay today if Sachin will be
40 years old 10 years hence
Sol: Sachins age today =
30 years.
Sachins age 10 years back = 20 years.
Ajays age 10 years back = 10 years.
Ajays age today =20 years.
Sachins age 10 years back = 20 years.
Ajays age 10 years back = 10 years.
Ajays age today =20 years.
20. Five years
ago, the total of the ages of father and his son was 40 years. The ratio of
their present ages is 4 : 1. What is the present age of the father?
Sol: Let sons age = x.
Then, fathers age = 4x.
(x - 5)+(4x - 5) = 40 ⇒ x = 10
Present age of father = 40 years.
(x - 5)+(4x - 5) = 40 ⇒ x = 10
Present age of father = 40 years.
21.The ratio of the ages of Meena and Meera is 4 : 3. The sum of
their ages is 2 years. The ratio Of their ages after 8 years will be
Sol: Let Meenas age = 4x
& Meeras age = 3x. Then, 4x+3x=28 hence x=4.
Meenas age = 16 years and Meeras age = 12 years.
Ratio of their ages after 8 years = (16 + 8): (12 + 8) = 24 : 20 = 6 : 5.
Meenas age = 16 years and Meeras age = 12 years.
Ratio of their ages after 8 years = (16 + 8): (12 + 8) = 24 : 20 = 6 : 5.
22. Ratio of
Ashoks age to Sandeeps age is 4 : 3. Ashok will be 26 years old after 6 years.
How old is Sandeep now?
Sol: Let their ages be
4x and 3x years.
Now 4x + 6 = 26 hence x = 5
. so Age of Sandeep = 3x = 15 years.
Now 4x + 6 = 26 hence x = 5
. so Age of Sandeep = 3x = 15 years.
23. In 10 years, A
will be twice as old5as B was 10 years ago. If A is now 9 years older than B,
the present age of B is:
Sol: Let Bs age = x
years. Then, As age = (x+ 9) years.
(x+9+10)=2(x—10) hence x=39.
Present age of B = 39 years.
(x+9+10)=2(x—10) hence x=39.
Present age of B = 39 years.
24. The average age of 12 students is 20 years1f the age of one
more student is included, the average decreases by 1. What is the age of the
new student?
Sol: New average = (20 -
1) = 19.
Age of new student = (13*19 - 12*20) years = 7 years
Age of new student = (13*19 - 12*20) years = 7 years
25. Average age of A and B is 24 years and average age of B, C and D
is 22 years. The sum of the ages of A, B, C and D is:
Sol: A+B = (24X2) = 48
and B + C + D = (22*3) = 66
With this data, we cannot find A + B + C + D.
So, the data is inadequate,
With this data, we cannot find A + B + C + D.
So, the data is inadequate,
26. The ratio of the
ages of Swati and Varun is 2: 5. After 8 years, their ages will be in the ratio
of 1: 2. The difference in their present ages (in years) is:
Sol: Let Swati age = 2x &
Varuns age = 5x.
2x + 8 / 5x + 8 = 1/2 hence 2(2x+8) = (5x+8) so x=8.
Swatis age = 16 years & Varun’s age =40 years.
Difference of their ages = 24 years.
2x + 8 / 5x + 8 = 1/2 hence 2(2x+8) = (5x+8) so x=8.
Swatis age = 16 years & Varun’s age =40 years.
Difference of their ages = 24 years.
27. The Average age of a
class of 22 subjects in 21 years. The average increased by 1 when the teacher’s
age also included. What is the age of the teacher?
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28. Ten years ago, Kumar was thrice as old as Selva was but 10
years hence, he will be only twice as old. Find Kumar’s present age
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Sol: Let Kumar’s present age be X years and Selva’s
present age be Y years.
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Then, according to the first condition,
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X - 10 = 3(y - 10)
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Now. Kumar's age after 10 years
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Selva's age after 10 years = (Y + 10)
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Solving (1) and (2), we get
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29. A father is four times as old as his son today. After 20
years, he would be just twice as old. At the time of birth of his son, how much
old must the father be?
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6 years hence. How old is Snehal today?
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Let Snehal's age 6 years back = X Then,
Sushil's age 6 years back = 32.
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Then
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5
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(X + 6 + 6) = (3X + 6 + 6)
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3
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5(X + 12) = 3 (3X + 12)
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4X = 24
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X = 6
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Snehal's age today = (X + 6) years = 12 years.
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30. The ages of Kamaraj and Deenan differ by 16 years, Six years
ago, Mohan’s age was thrice as that of Kamaraj’s find their present ages.
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Sol:
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Let Kamaraj's age = X years
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So, Mohan's age = (X + 16) years
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Also, 3(X - 6) = X + 16 - 6 or, X = 14
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and, Mohan's age = 14 + 16 = 30 years.
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