Chapter 9

Problems Based on Ages

Rule 1 Theorem: If tyears earlier thefather's age wasx times that of his son. At present the father's age is y times that of his son. Then the present ages of the son and the father are

x-y and y x-y respectively.

Illustrative Examples Ex. 1: The age of the father 3 years ago was 7 times the age

of his son. At present the father's age is five times that of his son. What are the present ages of the father and the son?

3 x ( 7 - l ) n Soln: Son's age = -- - " yrs

and father's age = 9 * 5 = 45 years Ex. 2: Ten years ago A was half of B in age. I f the ratio of

their present ages is 3 :4, what will be the total of their present ages?

Soln: 10 years ago A was ofB'sage.

At present A is of B's age. 4

: . B's age = ^ ^ - = 20 years.

2 ~ 4

A's age = of 20 = 15 years.

.-. Total of their present ages = 20 + 15 = 35 years.

Exercise !. 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 son is .

a) 20 years b) 25 years c) 22 years d) 26 years 2^ A is twice as old as B. 12 years ago, A was five times as

old as B. Find the present age of A. a) 16 years b) 32 years c) 24 years d) 28 years

3. The age of the father 4 years ago was 8 times the age of his son. At present the father's age is 4 times that of his son. Find the present age of son. a) 9 years b) 7 years c) 14 years d) 18 years

4. The age of the father 8 years ago was 5 times the age of his son. At present the father's age is 3 times that of his son. Find the present age of father. a) 48 years b) 36 years c) 46 years d) 58 years

5. The age of the father 6 years ago was 3 times the age of his son. At present the father's age is twice that of his son. Find the present age pf son. a) 8 years b) 16 years c) 14 years d) 12 years

6. 12 years ago, the ratio of the ages of Ram and Rahim is 2 : 3. I f the ratio of their present ages is 5 : 6, what will be the total of their present ages. a) 46 years b) 42 years c) 44 years d) 48 years

7. The age of a man is 4 times that of his son. Five years ago, the man was nine times as old as his son was at that time. The present age of the man is: a) 28 years b) 32 years c) 40 years d) 23 years

8. The age of Arvind's father is 4 times his age. I f 5 years ago, father's age was 7 times of the age of his son at that time, what is Arvind's father's present age? a) 84 years b) 70 years c) 40 years d) 35 years

(SBIPO Exam 1987)

Answers l .c 2. a 3.b 4. a 5.d

2 6. c; Hint: 12 years ago Ram was of Rahim's age

5 At present Ram is of Ram's age.

Now apply the given formula to get the answer. 7. b 8. c; Hint: Here x = 7 and y = 4, Now apply the given formula.

202 PRACTICE BOOK ON QUICKER MATHS

Rule 2 Theorem: If the present age of thefather is x times the age of his son. t years hence, the father's age becomes y times the age of his son. Then the present ages of the father and

his son are x x-y

and ( y - i > x-y

years respectively.

Illustrative Example Ex.: At present the age of the father is five times that of

the age of his son. Three years hence, the father's age would be four times that of his son. Find the present ages of the father and the son.

Soln: Applying the above rule, we have ( 4 - l ) x 3

Son's age : = 9 yrs 5 - 4

and father's age = 9 x 5 = 45 years

Exercise 1. The ratio between the present ages of P and Q is 5 : 8.

After four years, the ratio between their ages will be 2:3. What is Q's age at present.

(BSRB Mumbai PO1998) a) 36 years b) 20 years c) 24 years d) 32 years

2. The age of Mr Ramesh is four times the age of his son. After ten years the age of Mr Ramesh will be only twice the age of his son. Find the present age of Mr Ramesh's son. a) 10 years b) 11 years c) 12 years d) 5 years

(BSRB Bangalore PO 2000) 3. The age of Mrs Anjali is 5 times the age of his son. After

12 years the age of Mrs Anjali will be only twice the age of his son. Find the present age of Mrs Anjali's son. a) 4 years b) 16 years c) 12 years d) 18 years

4. The age of Mrs Rachana is 6 times the age of his son. After 6 years the age of Mrs Rachana wil l be only thrice the age of his son. Find the present age of Mrs Rachana's son. a) 15 years b) 9 years c) 6 years d) 12 years

5. The age of Mr Anuj is 7 times the age of his son. After 7 years the age of Mr Anuj wil l be only 4 times the age of his son. Find the present age of Mr Anuj's son. a) 7 years b) 16 years c) 12 years d) 21 years

6. A man's age is three times that of his son. In 12 years, the father's age will be double the son's age. Man's present age is:

a) 27 years b) 32 years c) 36 years d) 40 years

Answers 1. d; Hint: Consider Q and P as father and son to apply the

given formula. From the question it is clear that Q's age

is - times of P ie x ~ ~ , and after 4 years, Q's age

becomes times of the age of P. Hence t = 4 years and

y - . Now apply the formula and get the answer as 32

years. Hence (d) is the correct answer. Also see Rule-8. 2. d 3. a 4.c 5. a 6.c

Rule 3 Theorem: If /, years earlier the age of the father was x

times the age of his son. t2 years hence, the age of the

father becomes y times the age of his son. Then the present

ages of the son and the father are ( j)+tAx0 a n ( / {x-y)

years respec-^ p i ( , + , ) - | ( , - l ) + ^ - l )

lively.

Note: When /, =t2 =t, then the formulae become the fol-

lowing.

(i) present age of the son =

(i) present age of the father

t{x + y-2) years and

Son's age , \ / \ 2 \ + y ) - ^ K x - y ) years

Illustrative Examples Ex. 1: Three years earlier the father was 7 times as old as his

son. Three years hence the father's age would be four times that of his son. What are the present ages of the father and the son?

Soln: Quicker Method:

Son's age = 3 (4 -1) + 3(7-1) = 9 + 18 7 - 4

9 yrs

Father's age = | [ 7 + 4 ] + | ( l - 7 > + | ( 4 -1)

99 + 9 9 = 45 years

Ex. 2: One year ago the ratio between Samir's and Ashok's age was 4 : 3 . One year hence the ratio of their ages will be 5 : 4. What is the sum of their present ages in years?

4 Soln: One year ago Samir's age was of Ashok's age.

msij -sq^ni .no.* .moSHBarftasmit 1 esw.-wlgflarftlo V 1 B One year hence Samir's age will be of Ashok's

4 age.

roblems Based on Ages 203

Ashok's age (by formula (3));

A = -1 + 1 1 1 - +

3 4 J_ 12

= 7 years 4 _ 5 3 4

Now, by the first relation:

( S - l ) = 4 7 - 1 3

... S = 8+ 1 = 9years.

.-. Total ages = A + S = 9 + 7=16 years. where A = Ashok's present age and S = Samir's present age

Exercise Ten years ago Mohan was thrice as old as Ram was but 10 years hence, he will be only twice as old as Ram. Find Mohan's present age. a) 72 years b) 70 years c) 30 years d) Cannot be determined

1 A father was 4 times as old as his son 8 years ago. Eight > ears hence, father wil l be twice as old as his son. Find the sum of their present ages, a) 5 6 years b) 5 8 years c) 40 years d) None of these A's mother was four times as old as A, ten years ago. After ten years she wil l be twice as old as A. Then A's present age is . a) 30 years b) 20 years c) 24 years d) 25 years A man says to his son, "seven years ago I was seven times as old as you were and three years hence 1 wil l be three times as old as you will be." There ages are and years. a) 60,12 b)52,12 c)42,12 d)50,15 Sunil was three times as old as Sandeep 6 years back. Sunil will be 5/3 times as old as Sandeep 6 years hence. How old is sandeep today? a) 18 years b) 24 years c) 12 years d) 15 years Ten years ago A was thrice as old as B was but 12 years hence, A will be only twice as old as B. Then the A's present age is years. a) 66 b)32 c)76 d)56 The age of a 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: a)8:5 b)7:3 c)5:2 d)9:5

L 10 years ago Chandravati's mother was 4 times older than her daughter. After 10 years, the mother wil l be twice older than the daughter. The present age of Chandravati is:

(BankPO Exam 1988) a) 5 years b) 10 years c) 20 years d) 30 years

Answers 1. b; Hint: In this problem, you have been asked to find

Mohan's present age. To calculate Mohan's present age by 'Quicker Method' you have to calculate Ram's present age first and then apply this result to the for-mula for Mohan's present age. This involves a complex process. Hence you may solve such problem quickly by "traditional method" as it has been given below. It is up to you which method you are comfortable with. I f you have to find Ram's present age, you can go for 'quicker method': Let Mohan's present age be x years and Ram's present age be y years. Thus, accordingly to the first conditions (x-10) = 3(y-10) o r x - 3 y = -20 Now, Mohan's age after 10 years = (x + 10) years. Ram's age after 10 years = (y + 10) years. .-. (x+10) = 2(y+10)or(x-2y)=10 Solving (i) and (ii) one gets x = 70 and y = 30. .-. Mohan's age = 70 years. Ram's age = 30 years.

2. a 3.b 4.c 5.c 6.c 7. b; Hint: Present ages of father and his son are 70 and 30

years respectively. .-. required ratio = 7:3

8. c

Rule 4 Theorem: If tyears earlier, the father's age was x times that of his son. At present the father's age isy times that of his son. Then the sum total of the age of the father and the son

is x-y years.

Illustrative Example Ex.: Ten years ago A was half of B in age. I f the ratio of

their present ages is 3 :4, what will be the total of their present ages?

Soln: 10 years ago A was -r of B's age

At present A is of B's age

Now, applying the above theorem,

Total of their present ages : V 2

20x7

3 4

= 35

+ 1

years.


Exercise 1. 8 years ago Raman was thrice the age of Chandan. I f the

present age of Raman is twice the age of Chandan, what will be the total of their present ages? a) 48 years b) 42 years c) 36 years d) 54 years

2. 5 years ago Saket was 4 times the age of Alok. I f the present age of Saket is thrice the age of Alok, what will be the total of their present ages? a) 56 years b) 60 years c) 45 years d) 65 years

3. 9 years ago Vimal was 5 times the age of Sudarshan. I f the present age of Vimal is twice the age of Sudarshan, what will be the total of their present ages? a) 66 years b) 54 years c) 36 years d) 46 years

4. 8 years ago Kunal was 8 times the age of Kamal. I f the present age of Kunal is 4 times the age of Kamal, what will be the total of their present ages? a) 54 years b) 60 years c) 65 years d) 70 years

5. 14 years ago Ram was 4 times the age of Pankaj. I f the present age of Ram is twice the age of Pankaj, what will be the total of their present ages? a) 42 years b) 63 years c) 62 years d) 48 years

Answers l.a 2.b 3.c 4.d 5. b

Rule 5 Theorem: If the present age of the father is x times that the age of his son. t years hence, the fathers' age becomes y times the age of his son. Then the sum of the present ages of

father and his son is x-y (x + l ) years.

Illustrative Example Ex.: At present the age of the father is 5 times the age of

his son, three years hence the fathers' age would be four times that of his son. What is the sum of the present ages of father and his son?

Soln: Applying the above formula, we have

the required answer:

years.

(4~ l )x3 5 - 4 (5 + 1)

9 x 6 = 54

Exercise 1. At present the age of the father is 6 times the age of his

son, 4 years hence the fathers' age would be 5 times that of his son. What is the sum of the present ages of father and his son? a) 116 years b) 112 years c) 114 years d) 111 years

2. At present the age of the father is 4 times the age of his son, 3 years hence the fathers' age would be thrice that of his son. What is the sum of the present ages of father and his son?

a) 20 years b) 25 years c) 30 years d) 60 years 3. At present the age of the father is 7 times the age of his

son, 4 years hence the fathers' age would be 4 times that of his son. What is the sum of the present ages of father and his son? a) 21 years b) 24 years c) 28 years d) 32 years

4. At present the age of the father is 3 times the age of his son, 9 years hence the fathers' age would be twice that of his son. What is the sum of the present ages of father and his son? / a) 36 years b) 38 years c) 32 years d) 46 years

5. At present the age of the father is 7 times the age of his son, 6 years hence the fathers' age would be 5 times that of his son. What is the sum of the present ages of father and his son? a) 80 years b) 64 years c) 96 years d) None of these

Answers l .b 2.c 3.d 4. a 5.C

Rule 6 Theorem: If the sum of the present ages of A and B is x years, 't'years ago, the age of A was 'y' times the age of the B. Then the present ages of A and B are as follows;

xy-t{y-\) x + t{y-l) (i)AgeofB= j (ii) Age of A y + l

Illustrative Example E x : The sum of the ages of a mother and her daughter is

50 years. Also 5 years ago, the mother's age was " times the age of the daughter. What are the presen: ages of the mother and the daughter?

Soln: Detail Method: Let the age of the daughter be x years Then, the age of the mother is (50 -x) years. 5 years ago, 7 ( x - 5 ) = 5 0 - x - 5 or,8x = 50-5 + 35 = 80 ; . x = 1 0 Therefore, daughter's age = 10 years, and mother's age = 40 years. Quicker Method: Daughter's age

Total ages + No. of years ago (Times -11

50 + 5(7-1) 7 + 1

Times +1

years

50x7-5(7 -1 ) Mother's age = 4U

7 + 1 years

Thus, daughter's age = 10 years and mother's aa = 40 years.

Problems Based on Ages 205

Lxercise The sum of the ages of Sweta and her mother is 63 years. Four years back her mother's age was 4 times of Sweta's age at that time. Then the present age of Sweta's mother is years. a) 48 years b) 44 years c) 40 years d) 52 years

1 The sum of the present ages of A and B is 60 years. Also 12 years ago, the ratio of the ages of A and B is 5 :4. Find the present age of A. a) 28 years b) 32 years c) 18 years d) 42 years

I The sum of the ages of Anjali and her mother is 48 years. Six years back her mother's age was twice the age of Anjali. Find the ratio of the present ages of Anjali's mother and Anjali. a)3:5 b)4:5 c)5:4 d)5:3

x The sum of the ages of Vineet and Roshan is 56 years. 4 years back Roshan's age was 3 times the age of Vineet. Find the present age of Vineet. a) 40 years b) 32 years c) 24 years d) 16 years The sum of the ages of P and Q is 42 years. 3 years back the age of P was 5 times the age of the Q. Find the differ-ence between the present ages of P and Q. a) 23 years b) 24 years c) 28 years d) 33 years

i . The sum of the ages of a father and a son is 50 years. Also, 5 years ago, the father's age was 7 times the age of the son. The present ages of the father and the son respectively, are: a) 35 years, 15 years b) 40 years, 10 years c) 38 years, 12 years d) 42 years, 8 years

Aaswers

1 r. Hint: Here A's age is times of the age of B.

: Hint: Age of Anjali = 18 years and the age of her mother = 30 years.

Hence the required ratio = 30: 18 = 5:3. | t i 5.b 6.b

Rule 7 TWorem: If the sum of the present ages of A and B is 'x' J H T 5 . After 'V years, theageofA will be 'y'times that of B. "ten the ages of A and B are given below.

xy + t{y-\) m The age of A = ^ + j and

x-t{y-\) * The age of B= y + ^ years.

strative Example fc. The sum of the ages of a son and father is 56 years.

After 4 years, the age of the father will be three times that of the son. What is the age of the son?

Soln: Detail Method: Let the age of the son be x years. Then , the age of the father is (56 - x) years. After 4 years, 3 (x + 4) = 56 - x + 4 or, 4^ = 56 + 4 -12 = 48 .-. x = 12 years. Thus, son's age = 12 years. Quicker Method: Son's age

Total ages - No. of years after (Times - 1 ) Times +1

56-4(3-1) 48 " = v ' = =12 years. 3 + 1 4

Exercise 1. 4 years hence, mother's age will be twice the age of her

daughter. The sum of their present ages is 46 years. Find the present ages. a) 32 years, 14 years b) 36 years, 10 years c) 38 years, 8 years d) Can't be determined

2. The sum of ages of A and B is 60 years. After 10 years A will be thrice as old as B. Find the difference of their present ages. a) 30 years b) 40 years c) 10 years d) 50 years

3. The sum of the ages of a father and a son is 56 years. After 4 years, the age of the father will be three times that of the son. Then the age of the father is _. a) 35 years b) 40 years c) 42 years d) 44 years

4. The sum of the ages of the father and his son is 41 years. After 17 years the father's age will be twice the age of his son. Then the respective ages of the father and son are

and years. a)32,9 b)34,7 c)33,8 d)31,10

5. The sum of the ages of the father and his son is 66 years. After 3 years, the fathers' age will be thrice the age of his son. Then find the ratio of the present ages of father and his son. a) 25:17 b) 17:6 c)17:5 d) 17:3

6. The sum of the ages of a son and father is 56 years. After four years, the age of the father will be three times that of the son. Their ages respectively are:

(Railway Recruitment Board Exam, 1989) a) 12 years, 44 years b) 16 years, 48 years c) 16 years, 42 years d) 18 years, 36 years

Answers L a 2.b 3.d 4 c 5.c 6.a

Rule 8 Theorem: If the ratio of the ages of A and B at present is a : b. After 'T' years the ratio will become c : d. Then the present ages of A and B are as follows:

Age of A= a x T(c-d) ad -be = ax-

T(c-d) difference of cross products

206

T{c-d)__ Tje-d) Age ofB aj _ difference of cross products

Illustrative Example The ratio of the ages of the father and the son at present is 6: 1. After 5 years the ratio wil l become 7 : 2. What are the present ages of the son and the fa-ther?

Father : Son Present age = 6 : 1 After 5 years = 7 : 2

5(7-2)

Ex.:

Soln:

Son's age = 1 x 6x2-7x1

= 5 years.

Father's age = 6 x 5(7- -^ = 30 years. 6 x 2 - 7 x 1

Exercise 1. The ratio of present ages of P and Q is 7 : 3. After four

years their ages are in the ratio of 2 : 1. What is the present age of P? (BSRB Chennai PO 2000) a) 24 years b) 28 years c) 32 years d) Data inadequate The present ages of the father and son are in the ratio 6 : 1. After 5 years, the ratio will be 13 : 3. Find the present age of the son. a) 60 years b) 15 years c) 30 years d) 10 years The ages of A and B are in the ratio 3 :5 . After 9 years the ratio of their ages will be 3 :4. The present age of B is

2.

3

a) 9 years b) 15 years c) 20 years d) 16 years 4. The present ages of father and son are in the ratio 5 : 1 .

After 5 years, the ratio will be 11 : 3. Then the present age of the father is years. a) 60 years b) 40 years c) 45 years d) 50 years

5. The ratio of the present ages of P and Q is 8 : 5. After 6 years their ages are in the ratio of 3 : 2. Find the ratio of the sum and difference of the present ages of P and Q. 1)13:3 b)39:19 c)13:2 d) 13:5

6. The ages of A and B are in the ratio 2 : 5 . After 8 years their ages will be in the ratio 1 : 2. The difference of their ages is: a) 20 years b) 24 years c) 26 years d) 29 years

7. The ratio of the ages of father and son at present is 6 : 1. After 5 years, the ratio will become 7 :2. The present age of the son is: a) 10 years b) 9 years c) 6 years d) 5 years

(Bank PO 1991)

Answers l .b 2.d 3.b 4.d 5. a; Hint: Age of P = 48 years and Age of Q = 30 years

.-. Required ratio = 48+ 30:48-30 = 78:18= 13:3 6. b 7.d

Rule 9 Theorem: If the ratio of the ages of A and B at present is a : b. 'T' years earlier, the ratio was c: d. Then the present (i) Age of A

T(c-d) T(c-d) flx-be-ad

(ii) Age ofB

T{c-d)

A X -

m bx bx

difference of cross products

T{c-d) be - ad difference of cross products

Illustrative Example Ex.: The ratio of the ages of the father and the son at

present is 3 :1.4 years earlier, the ratio was 4:1. What are the present ages of the son and the father?

Soln: Father: Son Present age = 3:1 4 years before = 4:1

4(4-1) Son's age : lx-4x1-3x1 = 12 years.

4(4-1) Father's age = 3 * -:~7 4 x 1 - 3 x 1

36 years.

Exercise 1. 6 years ago Jagunnath was twice as old as Badri. I f the I

ratio of their present ages is 9 : 5 respectively, what is the j difference between their present ages in years?

(BSRB Bhopal PO 200* a) 24 b)30 c) 50 d) Cannot be determined

2. 3 years ago Ambuj was thrice as old as Avinash. If tael ratio of their present ages is 8 : 3 respectively, what is ae| difference between their present ages in years? a) 32 years b) 30 years c) 28 years d) 35 years

3. 8 years ago, the ratio of the ages of Nagendra and m was 3 : 2. I f the ratio of their present ages is 7 : 5 resp tively, what is the sum of their present ages? a) 96 years b) 86 years c) 76 years d) 66 years

4. 8 years ago, the ratio of the ages of Rachana and Arch was 4 : 3. I f the ratio of their present ages is 6 : 5 resp tively, what is the ratio of the sum and difference of t present ages? a) 11:2 b) 11:4 c) 11 :1 d) 11:3

5. The ratio of the ages of A and B at present is 4 : years earlier, the ratio was 3 : 2, then find the pres ages of A and B. a) 40 years and 30 years b) 48 years and 36 years c) 64 years and 48 years d) 20 years and 15 years

Answers 1. a; Hint: Here c : d = 2:1 and present ages of Jagunnatr

Badri is 54 and 30 years respectively.


2.b 3. a 4. c; Hint: Their present ages were 24 and 20 years respec-

tively .-. the required ratio = 24 + 20 :24 - 20 = 44:4 = 11:1

5. a

Rule 10 Theorem: If the ratio of the ages of A and B at present is a : b. After 'T'years the ratio will become c: d. Then the sum

'Tied)] of present ages of A and B is ad-be

{a + b) or

T(c-d) difference of cross products


their present ages are 28 and 7 years respectively. Ratio oftheir ages after 5 years will be 28 + 5 : 7 + 5 = 33 : 12= 11:4

Rule 12 Theorem: If the ratio of the ages of A and B at present is a : b. 'T'years earlier, the ratio was c: d, then the sum of the present ages of A and B is

{a + b be-ad

T{c-d) difference of cross products

{a + b)

Illustrative Example Ex.: The ratio of the ages of the father and the son at

present is 3 :1.4 years earlier, the ratio was 4 : 1 . What is the sum of the present ages of the son and the father? Following the above formula, we have Soln:

the required answer: (3 + 1 4(4-1)

4x1-3x1

= 4 x 12 = 48 years.

Exercise 1. I f the ratio of the ages of A and

years earlier, the ratio was 3 : 1 . present ages of A and B? a) 24 years b) 26 c) 34 years d) 36

2. I f the ratio of the ages of A and years earlier, the ratio was 5 :3 . present ages of A and B? a) 30 years b) 32 years c)35

3. I f the ratio of the ages of A and years earlier, the ratio was 5 : 2. present ages of A and B? a) 37 years b) 31 years c)42

4. I f the ratio of the ages of A and years earlier, the ratio was 6 : 1 . present ages of A and B? a) 36 years b) 63 years c) 64

5. I f the ratio of the ages of A and years earlier, the ratio was 2 : 1 . present ages of A and B? a) 48 years b) 46 years c) 47

Answers l . d 2. a 3.c 4.b 5.a

B at present is 2 : 1.6 What is the sum of the

years years B at present is 3 : 2. 3 What is the sum of the

years d) 40 years B at present is 4 : 3. 2 What is the sum of the



years d) 49 years

Rule 13 Theorem: If a man's age isx% of what it was t^ears ago, buty% of what it will be after t,years. Then his present age

xtx + yt2~> is

I x + y Note: I f t, = t = t, then formula will become V years.

1 2 yx-y)

Illustrative Example Ex: A man's age is 125% of what it was 10 years ago, but

8 3 - % of what it will be after 10 years. What is his

present age?

Soln: Detail method: Let the present age be x years. Then

125%of(x-10) = x;and 8 3 - % 0 f ( x + 10)=x

.-. 125%of(x-10)= 8 3 - % 0 f ( x + 10)

o r > I ( x - 1 0 ) = | ( * + 10)

5 5 50 50 or x x 1" -

' 4 6 6 4 5x 250

.-. x = 50 years

Quicker Method: Applying the above rule, we ha\the required answer

125 x No. of years ago + 83 i x No. of yrs after

125-83

1 .7375 + 250 125x10 + 83^x10 10(

125-83 375-250

10x625 125

= 50 years.

Exercise ) 1. A man's age is 150% of what it was 10 years ago. I

75% of what it will be after 10 years. What is his prey age? a) 25 years b) 30 years c) 35 years d) 40 years

2. A man's age is 135% of what it was 5 years ago, but 8?

of what it wi l l be after 2 years. What is his pres.

years.

age? a) 15 years c) 20 years

b) 25 years d) None of these


3. A man's age is 132% of what it was 3 years ago, but 88% of what it will be after 5 years. What is his present age? a) 11 years b) 17 years c) 29 years d) 19 years



Answers :.b 2.d 3.d 4.c 5.d

Rule 14 The ratio of A's and B's ages is a : b. If the difference be-tween the present ages of A and B 'V years hence is 'x' then

f \

t+x i) the present age of A (younger)

ii) the present age of B (older) =

years

years and

iiii) the sum of the present ages of A and B

1+- years.

Vote: Here A is always younger than B and a: b is younger b _ older

: older. Hence ~ ~ . a younger

Illustrative Example L u The ratio of A's and B's ages is 4 : 5. I f the difference

between the present age ofB and the age of A 5 years hence is 3, then what is the total of present ages of A andB?

*>ln: Detail Method:

A 4 5

B - ( A + 5 ) = 3or,B = A + 8

-A=A+& or, A 1 = 8 4

.-. A = 32 years

5 = 1x32 = 40 years 4

.-. A + B = 40 + 32 = 72 years

Quicker Method: Applying the above theorem, we have

the required answer = 5+3

4

1+1 4

= 8x4x = 72 years. 4

Exercise 1. The ratio of P's and Q's ages is 5 : 7. I f the difference

between the present age of Q and the age of P 6 years hence is 2 then what is the total of present ages of P and Q? (Bank of Baroda PO 1999) a) 52 years b) 48 years c) 56 years d) Data inadequate

2. The ratio of P's and Q's ages is 3 : 5. I f the difference between the present age of Q and the age of P 3 years hence is 3 then what is the total of present ages of P and Q? a) 24 years b) 27 years c) 32 years d) 20 years

3. The ratio of P's and Q's ages is 5 : 8. I f the difference between the present age of Q and the age of P 9 years hence is 6 then what is the total of present ages of P and Q? a) 56 years b) 65 years c) 45 years d) 75 years

4. The ratio of Radha's and Ruchi's ages is 9 : 4. I f the difference between the present age of Radha and the age of Ruchi 5 years hence is 5 then what is the sum of the present ages of Radha and Ruchi? a) 18 years b) 16 years c) 26 years d) 28 years

5. The ratio of the ages of Ram and Shyam is 7 : 4. I f the difference between the present age of Ram and the age Shyam 2 years hence is 4 then what is the sum of the present ages of Ram and Shyam. a) 22 years b) 18 years c) 24 years d) 32 years

Answers l . b 2. a 3.b 4. c; Hint: See Note: Here a: b = 9:4 because Radha is older

than Ruchi. I f you apply the given formula we get Radha's present age = 18 years and Ruchi's present age

= 8 years. 5. a

Rule 15 The difference of present ages of A and Sis'x 'years. If 'V years back their ages were in the ratio a : b, then

(i) the age of A t + - + x years, (ii) the age of


B = t+-- 1

years and (in) the total Age of A and B

t+-- 1 years.

Note: Here A > B ie A is older than B. Hence a > b.

Illustrative Example Ex: The difference of present ages of A and B is 12 years.

If 6 years back their ages were in the ratio 3 : 2, how old are A and B? Detail Method: A-B=12years ....(i)

Soln:

A-6 - => 2A-12 = 3B-18

3 - 6 or,2A-3B = -6 ....(ii) From eqn (i) and eqn (ii), we get A ' a a ^ c 4 2 y&ura a n d D ' s a g e 3 0 y e a r s .

Quicker Method: Applying the above theorem, we have

the age of A - 6 +

the age of B 6 +

12

1-1 2

12

+ 12 =42 years and

1-1 2

30 years.

Exercise

4.

Rahul is younger than Radha by 10 years. I f 5 years back their ages were in the ratio 1 : 2, how old is Radha? a) 20 years b) 15 years c) 25 years d) Data inadequate The ages of Ram and Mohan differs by 16 years. Six years ago, Mohan's age was thrice as that of Ram's. Find the sum of their ages. a) 30 years b) 27 years c) 24 years d) 25 years The difference between the ages of two persons is 8 years. 15 years ago, the elder one was twice as old as the younger one. Then the present age of the elder person is years. a)23 ' b)31 c)28 d)24 The difference between the ages of two persons is 12 years. 6 years ago, the ratio of the elder and younger

5.

one is 5 : 2. Then the present age of younger one is years.

a) 16 b)28 c)14 d)24 The difference of present ages of A and B is 10 years. I f 4 years back their ages were in the ratio of 7 : 5. Find the sum of their present ages.

a) 39 years b) 58 years c) 78 years d) 68 years

Answers 1. c; Hint: Here Radha - Rahul = 10 years,

ie Radha - Rahul = A - B and Radha : Rahul = 2 : 1 . Now by applying the given for-

mula, we have the age of Radha = 5 + 10

2 - l 1

+ 10

= 25 years. 2. a; Hint: Here A = Mohan, B = Ram and a: b = 3 : 1

Now apply the given rule. 3. b; Hint: Here a: b = 2 : 1 , and apply the given Rule -15 (i) 4. c 5.d

Miscellaneous 1. I f the ages of P and R are added to twice the age of Q, the

total becomes 59. I f the ages of Q and R are added I thrice the age of P, the total becomes 68. And i f the agj of P is added to thrice the age of Q and thrice the age I R, the total becomes 108. What is the age of P?

|SBI Bank PO, 1999 a) 15 yrs b) 19 yrs c) 17 yrs d) 12 yrs e) None of these

2. The product of the ages of Harish and Seema is 240. If twice the age of Seema is more than Harish's age b> -years, what is Seema's age in years?

[SBI Bank PO, 199*t a) 12 years b) 20 years c) 10 years d) 14 years e) Data inadequate

3. Jayesh is twice as old as Vijay and half as old as Surest. I f the sum of Suresh's and Vijay's ages is 85 years, what is Jayesh's age in years?

[BSRB Bhopal PO, 2000 a) 34 b)36 c)68 d) Can not be determined e) None of these

4. Present age of Rahul is 8 years less than Ritu's preseJ age. I f 3 years ago Ritu's age was x, which of the follow-ing represents Rahul's present age?

[BSRB Delhi PO,2 a)x + 3 b ) x - 5 c ) x - 3 + 8 d) x + 3 + 8 e) None of these

5. The ratio of the present ages of a son and his father is II : 5 and that of his mother and father is 4 : 5. After 2 yeaJ the ratio of the age of the son to that of his mothd


becomes 3:10. What is the present age of the father? a) 30 years b) 28 years c) 3 7 years d) Data inadequate e) None of these

1

6. 20 years ago my age was of what it is now. What is my

present age? a) 30 years b) 25 years c) 35 years d) 40 years

7. 15 years hence, A will be twice as old as B, but five years ago A was 4 times as old as B. Find the difference of their present ages. a) 15 years b) 45 years c) 30 years d) 25 years

8. A says to B " I am twice as old as you were when I was as old as you are". The sum of their ages is 63 years. Find the difference of their ages. a) 27 years b) 12 years c) 9 years d) 6 years

9. A is as much younger than B as he is older than C. I f the sum of B's and C's ages is 40 years. Find the age of A. a) 20 years b) 25 yerars c) 30 years d) 27 years

10. A is twice as old as B was two years ago. I f the differ-ence in their ages be 2 years, find A's age. a) 14 years b) 18 years c) 8 years d) 12 years

11. In ten years, A will be twice as old as B was 10 years ago. If A is now 9 years older than B. Find the present age of B. a) 39 years b) 40 years c) 36 years d) 49 years

12. Five years ago, the total of the ages of father and son was 60 years. The ratio of their present ages is 4 : 1. Then the present age of the father is . a) 48 years b) 51 years c) 56 years d) 61 years

13. Two years ago, a mother was four times as old as her daughter. 8 years hence, mother's age will exceed her daughter's age by 12 years. The ratio of the present ages of mother and daughter is . a ) 3 : l b ) 4 : l c )3 :2 d) 5: 1

14. A is 3 years younger to B. C is two years older than A. Then B's relation to C is . a) two years older b) one year younger c) one year older d) two years younger

15. I f C's age is twice the average age of A, B and C. A's age is one half the average of A, B and C. I f B is 5 years old, the aveage age of A, B and C is . a) 10 years b) 15 years c) 12 years d) 9 years

16. A father's age is three times the sum of the ages of his two children, but 20 years hence his age will be equal to the sum of their ages. Then the father's age is . a) 30 years b) 40 years c) 35 years d) 45 years

17. A father's age is four times as much as the sum of the ages of his three children but 6 years hence his age will be only double the sum of their ages. Then the age of the father is . a) 30 years b) 40 years c) 60 years d) 45 years

18. The respective ages of a father and his son are 41 and 16 years. In how many years wil l the father be twice as old

as his son? a) 19 years b) 9 years c) 10 years d) 15 years

19. The total ages of A, B and C at present is 90 years. Ten years ago the ratio of their ages was 1 : 2 : 3. Then the present age of B is . a) 30 years b) 20 years c) 40 years d) None of these

20. 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, then the present ages of the father and son respectively are and years. a)39,6 b)35,10 c)36,9 d)40,10

21. The ratio of the father's and son's age is 7 :4. The prod-uct of their ages is 1008. The ratio of their ages after 6 years hence will be . a)5:3 b)8:5 c )7 :4 d)5:8

22. Ratio of Sujeet's age to Sameer's age is 4 :3. Sujeet will be 26 years old after 6 years. Then the present age of Sameer is . a) 21 years b) 15 years c) 24 years d) 18 years

23. I f 6 years are subtracted from the present age of Randheer and the remainder is divided by 18, then the present age of his grandson Anup is obtained. I f Anup is 2 years younger to Mahesh whose age is 5 years, then what is the age of Randheer? | Bank PO 19931 a) 96 years b) 84 years c) 48 years d) 60 years

24. The ratio of Vimal's age and Arun's age is 3 : 5 and sum of their ages is 80 years. The ratio of their ages after 10 years will be . [Bank PO 1990J a)2:3 b) 1:2 c)3:2 d)3:5

25. Shyam is 3 times as old as his son. After 10 years, the sum of their ages will be 76 years. The respective ages of the father and the son are and years. a)42,14 b)39,13 c) 45,15 d) None of these

26. A is 20 years older than B. He is also 6 times as old as B. Then the respective ages of A and B are and

years. a) 24,4 b)42,7 c)30,5 d) None of these

27. The ages of A, B and C together total 185 years. B is twice as old as A and C is 17 years older than A. Then the respective ages of A, B and C are a) 40,86 and 59 years b) 42, 84 and 59 years c) 40,80 and 65 years d) None of these

28. The ratio of Vimal's age and Arun's age is 3 : 5 and the sum of their ages is 80 years. Find the ratio of their ages, (i) after 10 years and (ii) 10 years ago a ) 2 : 3 , 2 : l b ) 2 : 3 , l : 2 c ) 3 : 2 , l : 2 d) 3 :2,2 : 1

29. In 10 years, A will be twice as old as B was lOyearsago. I f A is now 9 years older than B, the present age ofB is: a) 29 years b) 39 years c) 19 years d) 49 years

| L I C Exam 1989|


30. The difference between the ages of two persons is 10 years. 15 years ago, the elder one was twice as old as the younger one. The present age of the elder person is: a) 35 years b) 25 years c) 45 years d) 55 years

31. One year ago a father was four times as old as his son. In 6 years time his age exceeds twice his son's age by 9 years. Ratio of their ages is: a) 13:4 b)12:5 c) 11:3 d)9:2

32. Ratio of Ashok's. age to Pradeep's age is equal to 4 : 3. Ashok will be 26 years old after 6 years. How old is Pradeep now?

[Railway Recruitment Board Exam 1989|

) v e a r s D ) 2 1 years c) 12 years d) 15 years

33. Deepak is 4 times as old as his son. Four years hence the sum of their ages will be 43 years. How old is Deepak's son now? a) 5 years b) 7 years c) 8 years d) 10 years

34. The ratio of Mona's age to the age of her mother is 3 : 11. The difference of their ages is 24 years. The ratio of their ages after 3 years will be: a) 1:3 b)2:3 c )3 :5 d)None ofthese

i 1 35. Kamla got married 6 years ago. Today her age is 1 times

her age at the time of marriage. His son's age is (1/10) times her age. The age of her son is: a) 2 years b) 3 years c) 4 years d) 5 years

[Bank PO Exam 1988] 36. Sachin was twice as old as Ajay 10 years back. How old

is Ajay today i f Sachin will be 40 years old 10 years hence? [Bank PO Exam 1991] a) 20 years b) 10 years c) 30 years d) 15 years

37. 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 the father is:

[Hotel Management, 1991] a) 3 9 years b) 3 6 years c) 25 years d) None of these

Answers 1. d; P + R + 2Q = 59;Q + R + 3P = 68

andP + 3(Q + R)=108 Solving the above two equations, we get P = 12 yrs.

2. a; Let the ages of Harish and Seema be x and y respec-tively. According to the question, x.y=240....(i) 2y-x=4....(ii) Solving equations (i) and (ii), we get y = 12 years

3. a; J :V :S = 2:1 :4 2

.-. Jayesh'sage = T j ^ j -

4. b; Let the Rahul's present age is 'A' years. Then Ritu's present age is A + 8 Now, according to the question, A + 8- 3= x .-.A = x-5years Hence, (b) is the correct answer

S I _ M 4 5 e - - = - r > F = 5S =

' F 5 F 5 M =-F

S+2 => 105 + 20 = 3*/+ 6

-x85 = 34 years

M+2 '0

= 3 x - x 5 S + 6 = 125' + 6 5

. \2S=14 => S = 7 years .-. F = 5S = 35 years 6. a; Let my present age be x years.

V (* - 2 0 ) = ! or (3x - 60) = x or, 2x = 60

.-. x = My present age = 30 years. 7. c; Let A's age = x, and B's age = y years.

As per the first condition : (x + 15) = 2(y +15) orx-2y=15... .(i) As per the second condition : (x - 5) = 4 (y - 5) or x - 4y = -15 ....(ii) Solving (i) and (i i), one gets, x = 45, y = 15. .-. A's age = 45 years, B's age = 15 years

8. c; Let A's age be x years and B's age be y years. .-. x + y = 63 ....(i) (x - y) years ago, A was of y years age. Now according to the question, x = 2 [y - (x -y ) ]o r ,x = 2(y-x) or, 3x-4y = 0 (ii) Solving equ (i) and equ (ii), we get x = 36, y = 27 .-. A's age = 36 years, B's age = 27 years. .-. Difference of their ages = 36 - 27 = 9 years.

9. a; Given that (B - A) = (A - C) or (B + C) = 2A ...(f) and B + C=40....(ii) .-. A = 20 years.

10. c; Let B's age 2 years ago be x years. .-. A's present age = 2x years Also 2x - (x + 2) = 2 or, x = 4 .-. A's age = 2 x 4 = 8 years

11. a; Let B's age be x years, then A's age be (x + 9) years. As per the given condition (x + 9+10)=2(x-10)or,x = 39 .-. The present age ofB = 39 years

12. c; Let the present age of the son be x and that of the father be 4x years. .-. (x-5) + (4x-5) =60 or5x = 70 .-. x = 14 years .-. Father's present age = 56 years.

13. a; Let the mother's age 2 years ago be 4x and daughter's age 2 years ago be x. .-. (4x + 8)-(x + 8)=12


or,3x= 12orx = 4 .-. Mother's present age = 4x + 2 = 18 years and daughter's present age = x + 2 = 6 years .-.requiredratio = 3 :1

14. c; Let the age of A be (x - 3) years .-. B's age = x years .-. C'sage = (x-3) + 2 = x - 1 .-. B's age - C's age = x - (x -1) = 1 year .-. B is one year older than C.

15. a; Let the average age of A, B and C be x years .-. total age of A, B and C = 3 x x = 3x years Now, according to the question,

3x-^2x + ^j = 5 . x =iOyears

16. a; Let the present, age of father be x years and the present age of son be y years. .-. x = 3y ....(i) Also,(x + 20) = (y + 20 + 20) ....(ii) [20 will be added twice as for 2 children] Solving (i) and (ii), we get x = 30 years.

17. c; Let father's age be x years and the sum of ages of chil-dren be y years. .-. x = 4y....(i) Also(x + 6) = 2(y + 6 + 6 + 6) ....(ii) [6 is added thrice for three children] Solving (i) and (ii) x = 60 years and y = 15 years.

18. b; Suppose x years hence the father will be twice as old as his son. x + 41=2(x+16) or,x = 41 -32 = 9years

19. a; Let the respective ages of A, B and C ten years ago be x, 2x and 3x years. .-. (x+10) + (2x+10) + (3x+10) = 90 or,x=10 .-. B's present age = 2x + 10 = 30 years

20. c; Let son's age be x years or Father's age = (45 - x) years. .-. (x-5)(45-x-5) = 4(45-x-5) or, x = 9 years .-. The son's age = 9 years Father's age = 45 - 9 = 36 years

21. b;Let father's and son's age be 7x and 4x respectively.

.-. 28x 2 = 1008 or x 2 = 36 orx = 6

.-. Father's age = 7x = 42 years Son's age = 4x = 24 years Father's age after 6 years hence = 48 years Son's age 6 years hence = 30 years Ratio = 48:30or8:5

22. b; Let the respective ages of Sujeet and Sameer be 4x and 3x years.

.-. 4x + 6=26orx = 5

.-. Sameer's present age = 3 * 5 = 15 years 23. d;Anup's age = (5 - 2) years = 3 years

Let Randheer's age be x years.

. fdL . 3 18 or, x = 54 + 6 = 60 years

24. a; Let their ages be 3x and 5x years. .-. 3x + 5x = 80orx= 10 .-. Vimal's age 10 years hence = (3x + 10) = 40 years Aruna's age 10 years hence = (5x + 10) = 60 years Ratio = 40:60 or2:3

25. a; Let son's present age be x years. .*, Father's present age = 3x years. Son's age 10 years hence = x + 10 Father's age 10 years hence = 3x + 10 As per the condition, (x + 10) + (3x + 10) = 76 or4x = 56 .-. x = 14 .-. Son's present age = 14 years Father's present age = 42 years

26. a; Let the age of B be x years According to the question, x + 20 = 6x .-. x = 4 years .-. A' age = 4 x 6 = 24 years and B's age = 4 years

27. b;Let A's age be x years B's age be 2x years C's age = (x + 17) years According to the question, x + 2x + (x+17)=185 .-. 4x= 185-17= 168 .-. x = 42, .-. A's age = 42 years B's age = 84 years C's age = 42 + 17 = 59 years

28. b;Let the ages be 3x and 5x years. .-. 3x + 5x = 80 .-. x = 10 (i) Ratio of their ages after 10 years = 3x+10:5x+10 or 40:60 = 2:3 (ii) Ratio of their ages before 10 years = 3x -10: 5x -10 o r 2 0 : 4 0 = l : 2

29. b; Let the present ages ofB and A be x yrs & (x + 9) yrs or,(x + 9+ 10)=2(x- 10)orx=39.

30. a; Let the present age of the elder person be x years. Then, the present age of another person = (x -10) years. (x-15) = 2(x-10-15)orx = 35 .-. The present age of the elder person is 35 years.

31. c; Let the present ages of father & son be x & y respec-tively. Then,(x- l ) = 4 ( y - l ) o r 4 y - x = 3 ....(i) And,(x + 6)-2(y + 6) = 9or-2y + x = 15....(ii) Solving (i) & (ii) we get, x = 33 and y = 9. .-. Ratio of their ages = 33 :9 or, 11 : 3.

32. d; Ashok's present age = (26 - 6) years = 20 years.

214

.-. Pradeep's present age = [ ^ x 2 0 J years = 15 years.

33. b;Let the son's age be x years. Then,(x + 4) + (4x + 4) = 43 or5x = 35 orx = 7

34. a; Let the ages of Mona and her mother be 3x and 1 lx years respectively. Then,(llx-3x) = 24orx = 3. So, their present ages are 9 years and 33 years. Ratio of their ages after 3 years = 12 :36 or 1 : 3.

35. b;Let son's age be x. Then, Kamla's age = lOx years. Kamla's age at the time of marriage = (1 Ox - 6) years.

PRACTICE BOOK ON QUICKER MATHS

.-. 1 Ox = -(l Ox - b) or 40x = 50x - 30 or x = 3 4

36. a; Let Ajay's age 10 years back be x years. Then, Sachin's age 10 years back = 2x years. .-. 2x + 20 = 40orx=10. Ajay's present age = (10 + 10) years = 20 years.

37. b;Let father's present age = x years. Then, son's present age = (45 - x) years. (x-5)(45-x-5) = 4(x-5)

or * 2 -4 i ; c + 180 = 0 or(x-36)(x-5) = 0 .-. x = 36 years.

Chapter 9

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