BANK & INSURANCE (AGE BASED PROBLEMS) PART 3

Total Questions: 45

1. Present ages of 'A', 'B' and 'C' are in ratio x:(x + 1):(x − 1), respectively. 5 years ago from now, the sum of ages of 'A' and 'B' was 80% more than the age of 'C', 5 years hence from now. Which of the following cannot be the sum of ages of 'A', 'B' and 'C', 'x' years hence from now? ('x' is an integer that lies between (0 and 100))

I. 90 years
II. 252 years
III. 418.5 years

Correct Answer: (c) All I, II and III
Solution:

Let the present ages of 'A', 'B' and 'C' be 'xy' years, '(x + 1)y' years and '(x − 1)y' years, respectively.

ATQ:

{(xy − 5) + (xy + y − 5)} = {(xy − y) + 5} × 1.8

2xy + y − 10 = 1.8xy − 1.8y + 9

0.2xy + 2.8y = 19

0.2y(x + 14) = 19

y(x + 14) = 95

y = 95/(x + 14)

xy + 14y = 95

Sum of ages of 'A', 'B' and 'C', 'x' years hence from now = (xy + x) + (xy + y + x) + (xy − y + x)

= (3xy + 3x) years

For I:

3xy + 3x = 90

xy + x = 30

(95x/(x + 14)) + x = 30

95x + x² + 14x = 30x + 420

x² + 79x − 420 = 0

x² + 84x − 5x − 420 = 0

x(x + 84) − 5(x + 84) = 0

(x − 5)(x + 84) = 0

x = 5

So, y = 95/(14 + 5) = 5

Age of 'A' = xy = 5 × 5 = 25 years

Age of 'C' = (5 − 1) × 5 = 20 years

Age of 'B' = (5 + 1) × 5 = 30 years

For II:

3xy + 3x = 252

xy + x = 84

(95x/(x + 14)) + x = 84

95x + x(x + 14) = 84(x + 14)

95x + x² + 14x = 84x + 1176

x² + 25x − 1176 = 0

x² + 49x − 24x − 1176 = 0

x(x + 49) − 24(x + 49) = 0

(x − 24)(x + 49) = 0

So, x = 24 or x = −49

So, y = 95/(14 + 24) = 2.5

Age of 'A' = yx = 24 × 2.5 = 60 years

Age of 'C' = ((x − 1) × y) = (24 − 1) × 2.5 = 57.5 years

Age of 'B' = ((x + 1) × y) = (24 + 1) × 2.5 = 62.5 years

For III:

3xy + 3x = 418.5

2xy + 2x = 279

(190x/(x + 14)) + 2x = 279

190x + 2x(x + 14) = 279(x + 14)

190x + 2x² + 28x = 279x + 3906

2x² − 61x − 3906 = 0

2x² − 124x + 63x − 3906 = 0

2x(x − 62) + 63(x − 62) = 0

(x − 62)(2x + 63) = 0

x = 62 or x = −31.5

y = (95/(62 + 14)) = 1.25

Age of 'A' = yx = 62 × 1.25 = 77.5 years

Age of 'C' = ((x − 1) × y) = (62 − 1) × 1.25 = 76.25 years

Age of 'B' = ((x + 1) × y) = (62 + 1) × 1.25 = 78.75 years

Since, we can solve all three equations and gather real values of 'x' and 'y'.
All three cases are possible.

2. Five years ago from now, average age of a family of 9 persons was 'm' years. Because of the birth of a baby who is currently 3 years old, present average age of the family is 2.3 years more than the average age of the family five years ago from now. Find the average age of the family 5 years hence from now.

Correct Answer: (b) 32.3 years
Solution:

According to the question:

9 × (m + 5) + 3 = (9 + 1) × (m + 2.3)

9m + 45 + 3 = 10m + 23

10m − 9m = 48 − 23

m = 25

So, sum of present ages of the whole life = {9 × (25 + 5) + 3}

= 270 + 3 = 273 years

Required average age = (273 + (10 × 5))/10

= (273 + 50)/10

= 323/10 = 32.3 years

3. In a family, the present age of the mother is 30% more than the average of the present age of the son and the father. The age of the father, 4 years hence from now, will be 20% more than that of the mother, 2 years ago from now. 4 years hence from now, the age of the mother will be twice that of the son. How many years hence from now will the son be half as old as the father?

Correct Answer: (b) 8 years
Solution:

Let the present age of the son = 'y' years

Let the average of present age of the son and the father be 10x years

Then, present age of the mother = 10x × 1.3 = 13x years

Sum of present age of the son and the father

= 10x × 2 = 20x years

So, present age of the father = (20x − y) years

According to the question,

(20x − y + 4) = (13x − 2) × 1.2

20x − y + 4 = 15.6x − 2.4

4.4x − y + 6.4 = 0 [equation I]

Also,

(13x + 4) = (y + 4) × 2

13x + 4 = 2y + 8

2y = (13x − 4)

y = (6.5x − 2)

Substituting this value in equation I,

4.4x − (6.5x − 2) + 6.4 = 0

8.4 − 2.1x = 0

So,

x = 8.4 ÷ 2.1 = 4

So, present age of the mother = 52 years

Present age of the son = (52 + 4) ÷ 2 = 4 = 24 years

And so, present age of the father = 80 − 24 = 56 years

Age difference between the father and the son

= 56 − 24 = 32 years

And so, number of years hence from now, when the son will be half as old as the father

= 32 − 24 = 8 years

4. 2 years ago from now, the average age of 'A' and 'B' was 23 years. Age of 'C', 2 years hence from now, will be 25% less than the age of 'A', 6 years hence from now. The ratio of age of 'B', 7 years ago from now and age of 'C', 1 year ago from now was 7:6, respectively. The average of present age of 'C' and 'D' is 28 years. 'm' years hence from now, the age of 'D' will be 60% more than that of 'A'. Find the value of 'm'.

Correct Answer: (d) 3
Solution:

2 years ago from now, sum of ages of 'A' and 'B' = 23

× 2 = 46 years

So, sum of present ages of 'A' and 'B'

= 46 + 2 + 2

= 50 years

Let the present age of 'A' = 'x' years

Then, present age of 'B' = (50 − x) years

Let present age of 'C' = 'y' years

According to the question,

(y + 2) = (x + 6) × 0.75 = 0.75x + 4.5

y = 0.75x + 2.5 [equation I]

Also we have,

((50 − x) − 7) : (y − 1) = 7 : 6

(43 − x) : (y − 1) = 7 : 6

7y − 7 = 258 − 6x

7y = 258 + 7 − 6x

y = (265 − 6x) ÷ 7 [equation II]

Comparing equations I and II, we have

(0.75x + 2.5) × 7 = 265 − 6x

5.25x + 6x + 17.5 = 265

11.25x = 265 − 17.5 = 247.5

So,

x = 247.5 ÷ 11.25 = 22

So, present ages of 'A' and 'B' are 22 years and 28 years, respectively.

Present age of 'C' = 22 × 0.75 + 2.5 = 19 years

Sum of present ages of 'C' and 'D'

= 28 × 2 = 56 years

So, present age of 'D' = 56 − 19 = 37 years

We have,

(22 + m) × 1.6 = (37 + m)

35.2 + 1.6m = 37 + m

0.6m = 1.8

And,

m = 1.8 ÷ 0.6 = 3

5. Present average age of A, B and C is 58 years. Present average age of A and B is 67 years. If the present age of B is ___% more than the present age of C, then the present age of A is ___ years.

The values given in which of the following options will fill the blanks in the same order in which it is given to make the above statement true:

A. 75, 64
B. 65, 68
C. 60, 70

Correct Answer: (a) All A, B and C
Solution:

Sum of the present ages of A, B and C

= 3 × 58 = 174 years

Sum of the present ages of A and B

= 2 × 67 = 134 years

So, the present age of C = 174 − 134 = 40 years

For option A:

Present age of B = 1.75 × 40 = 70 years

So, the present age of A = 134 − 70 = 64 years

So, option A can be the answer.

For option B:

Present age of B = 1.65 × 40 = 66 years

So, the present age of A = 134 − 66 = 68 years

So, option B can be the answer.

For option C:

Present age of B = 1.60 × 40 = 64 years

So, the present age of A = 134 − 64 = 70 years

So, option C can be the answer.

6. At present, the average age of 'B' and 'C' together is 3 years less than the average age of 'A', 'B' and 'C' together. Age of 'B', 1 year ago from now, was 75% of the age of 'A', 1 year hence from now. Age of 'A', 3 years hence from now, will be twice the age of 'C', 2 years ago from now. What is the average of the present ages of 'A' and 'C' together?

Correct Answer: (d) 25 years
Solution:

Let the average of present ages of 'A', 'B' and 'C' be 'x' years

Then, average of present ages of 'B' and 'C' = (x − 3) years

Then, sum of present ages of 'A', 'B' and 'C' = x × 3 = 3x years

Sum of present ages of 'B' and 'C' = (x − 3) × 2 = (2x − 6) years

So, present age of 'A' = 3x − (2x − 6) = (x + 6) years

Let the present age of 'B' = 'y' years

Then, present age of 'C' = (2x − 6 − y) years

According to the question,

(y − 1) = (x + 6 + 1) × 0.75

y − 1 = 0.75x + 5.25

y = 0.75x + 6.25  [equation I]

Also, we have,

(x + 6 + 3) = 2 × (2x − 6 − y − 2)

x + 9 = 4x − 16 − 2y

3x − 25 − 2y = 0

y = 1.5x − 12.5  [equation II]

So,

1.5x − 12.5 = 0.75x + 6.25

0.75x = 18.75

So,

x = 25

And so, present age of 'A' = 25 + 6 = 31 years

And y = 1.5 × 25 − 12.5 = 25

So, present age of 'C' = 2x − 6 − y = 19 years

Therefore, required average

= (31 + 19) ÷ 2 = 25 years

7. 'P', 'Q', 'R', 'S' and 'T' are five friends and their average age is ___ years. At present the age 'P' is half of the age of 'R' and the ratio of the present ages of 'Q' to 'S' is 6:13. Eight years hence from now, the ratio of the ages of 'P' to 'T' will become 7:13. If the present average age of 'Q' and 'R' is ___ years, and 'T' is 8 years younger to 'S'. (The age of each person will be in whole years).

The values given in which of the following options will fill the blanks in the same order in which it is given to make the statement true:

I. 40, 36
II. 36, 32
III. 30, 20

Correct Answer: (a) Only II
Solution:

For I:

Let the present age of 'P' = x years

Present age of 'R' = 2x years

Sum of the present ages of 'Q' and 'R'

= 36 × 2 = 72 years

Present age of 'Q' = (72 − 2x) years

Present age of 'S' = [(72 − 2x)/6] × 13 years

After eight years, the age of 'P' = (x + 8) years

After eight years, the age of 'T' = [(x + 8)/7] × 13 years

Present age of 'T' = [(x + 8)/7] × 13 − 8 years

Sum of the ages of 'P', 'S' and 'T' = (40 × 5) − 72

= 128 years

x + {[(72 − 2x)/6] × 13} + {[(x + 8)/7] × 13 − 8} = 128

x + (936 − 26x)/6 + (13x + 104)/7 − 8 = 128

42x + 6552 − 182x + 78x + 624 − 336 = 128 × 42

62x = 1464

x = 1464/62

Therefore, I cannot be true.

For II:

Let the present age of 'P' = x years

Present age of 'R' = 2x years

Sum of the present ages of 'Q' and 'R' = 32 × 2 = 64 years

Present age of 'Q' = (64 − 2x) years

Present age of 'S' = [(64 − 2x)/6] × 13 years

After eight years, the age of 'P' = (x + 8) years

After eight years, the age of 'T' = [(x + 8)/7] × 13 years

Present age of 'T' = [(x + 8)/7] × 13 − 8 years

Sum of the ages of 'P', 'S' and 'T' = (36 × 5) − 64

= 116 years

x + {[(64 − 2x)/6] × 13} + {[(x + 8)/7] × 13 − 8} = 116

x + (832 − 26x)/6 + (13x + 104)/7 − 8 = 116

42x + 5824 − 182x + 78x + 624 − 336 = 116 × 42

62x = 1240

x = 20

Present age of 'T' = [(x + 8)/7] × 13 − 8 = 44 years

Present age of 'S' = [(64 − 2x)/6] × 13 = 52 years

Difference = 52 − 44 = 8 years

Therefore, II can be true.

For III:

Let the present age of 'P' = x years

Present age of 'R' = 2x years

Sum of the present ages of 'Q' and 'R' = 20 × 2 = 40 years

Present age of 'Q' = (40 − 2x) years

Present age of 'S' = [(40 − 2x)/6] × 13 years

After eight years, the age of 'P' = (x + 8) years

After eight years, the age of 'T' = [(x + 8)/7] × 13 years

Present age of 'T' = [(x + 8)/7] × 13 − 8 years

Sum of the ages of 'P', 'S' and 'T' = (30 × 5) − 40

= 110 years

x + {[(40 − 2x)/6] × 13} + {[(x + 8)/7] × 13 − 8} = 110

x + (520 − 26x)/6 + (13x + 104)/7 − 8 = 110

42x + 3640 − 182x + 78x + 624 − 336 = 110 × 42

−62x = 692

x = −692/62

Age cannot be negative.

Therefore, III is false.

8. Directions (8-10): Answer the questions based on the information given below.

In a family, there are five members, Ajeet, Sakshi, Diksha, Priya and Eshant. The ratio of the age of Sakshi 4 years hence from now to age of Eshant 6 years hence from now will be 6:5. Average present age of Sakshi, Diksha and Eshant is 'A' years. 2 years ago from now, age of Ajeet was (A + 8) years. Diksha is 'B' years elder to Eshant and present average age of Priya and Sakshi is (C + 6) years.

Additional Information:

I. The value of 'B' is equal to the larger root of the quadratic equation (x + 1)² − 6x − 33 = 4(x + 24).

II. 3C − B + 4 = 2A − 16

III. The value of 'C' is "[√(x + 9)] + x − 17" more that of 'B'.

What is the present age of Priya?

Correct Answer: (d) 20 years
Solution:

Given,

(x + 1)² − 6x − 33 = 4(x + 24)

x² + 1 + 2x − 6x − 33 = 4x + 96

x² + 2x − 6x − 4x + 1 − 33 − 96 = 0

x² − 8x − 128 = 0

x² − 16x + 8x − 128 = 0

x(x − 16) + 8(x − 16) = 0

(x − 16)(x + 8) = 0

x = 16, −8

Larger root of the equation = 16

So, value of 'B' = 16

Given, the value of C is [(√(x + 9)) + x − 17] more than that of 'B'

And value of x = 16 and the value of B = 16

So,

C − [(√(x + 9)) + x − 17] = B

C − (5 − 1) = B

C − 4 = B

Or,

C = 20

Given,

3C − B + 4 = 2A − 16

Therefore,

3 × 20 − 16 + 4 = 2A − 16

Or,

2A = 60 + 4

Or,

A = 64/2 = 32

2 years ago from now, the age of Ajeet = (A + 8)

= 32 + 8 = 40 years

Present age of Ajeet = 40 + 2 = 42 years

Present age of Priya and Sakshi

= 20 + 6 = 26 years

Sum of present ages of Priya and Sakshi

= 26 × 2 = 52 years

Let present age of Eshant = ‘m’ years

So, present age of Diksha = (m + 16) years

Age of Eshant 6 years hence from now will be = ‘m + 6’ years

Age of Sakshi 4 years hence from now will be
= [(m + 6)/5] × 6 = ((6m + 36)/5) years

Present age of Sakshi = [((6m + 36)/5) − 4] years

Sum of present ages of Sakshi, Diksha and Eshant

= 32 × 3 = 96 years

So, [(6m + 36)/5] − 4 + m + 16 + m = 96

6m + 36 − 20 + 5m + 80 + 5m = 96 × 5

16m + 96 = 96 × 5

16m = 96 × 4

m = 24

So, present age of Eshant = 24 years

Present age of Diksha = 24 + 16 = 40 years

Present age of Sakshi = [(6 × 24 + 36)/5] − 4 = 32 years

Present age of Priya = 52 − 32 = 20 years

Present age of Priya = 20 years

9. Eight years hence from now, what will be the ratio of the ages of Diksha and Sakshi, respectively?

Correct Answer: (c) 6:5
Solution:

(x + 1)² – 6x – 33 = 4(x + 24)

x² + 1 + 2x – 6x – 33 = 4x + 96

x² + 2x – 6x – 4x + 1 – 33 – 96 = 0

x² – 8x – 128 = 0

x² – 16x + 8x – 128 = 0

x(x – 16) + 8(x – 16) = 0

(x – 16)(x + 8) = 0

x = 16, –8

Larger root of the equation = 16

So, value of ‘B’ = 16

Given, the value of C is [(√(x + 9)) + x – 17] more than that of ‘B’.

And value of x = 16 and the value of B = 16

So, C – [(√(x + 9)) + x – 17] = B

C – (5 – 1) = B

C – 4 = B

C = 20

Given, 3C – B + 4 = 2A – 16

Therefore, 3 × 20 – 16 + 4 = 2A – 16

2A = 60 + 4

A = 64/2 = 32

2 years ago from now, the age of Ajeet = (A + 8) = 32 + 8 = 40 years

Present age of Ajeet = 40 + 2 = 42 years

Present average age of Priya and Sakshi = 20 + 6 = 26 years

Sum of present ages of Priya and Sakshi = 26 × 2 = 52 years

Let present age of Eshant = ‘m’ years

So, present age of Diksha = (m + 16) years

Age of Eshant 6 years hence from now will be = ‘m + 6’ years

Age of Sakshi 4 years hence from now will be = [(m + 6)/5] × 6 = ((6m + 36)/5) years

Present age of Sakshi = [((6m + 36)/5) – 4] years

Sum of present ages of Sakshi, Diksha and Eshant = 32 × 3 = 96 years

So, [(6m + 36)/5] – 4 + m + 16 + m = 96

6m + 36 – 20 + 5m + 80 + 5m = 96 × 5

16m + 96 = 96 × 5

16m = 96 × 4

m = 24

So, present age of Eshant = 24 years

Present age of Diksha = 24 + 16 = 40 years

Present age of Sakshi = [(6 × 24 + 36)/5] – 4 = 32 years

Present age of Priya = 52 – 32 = 20 years

Required ratio = (40+8) : (32+8) = 48:40 = 6:5

10. If the average present age of Eshant, Ajeet and Ritika is 34 years, then find the ratio of the present age of Ritika and Sakshi, respectively.

Correct Answer: (a) 9:8
Solution:

Given,

(x + 1)² − 6x − 33 = 4(x + 24)

x² + 1 + 2x − 6x − 33 = 4x + 96

x² + 2x − 6x − 4x + 1 − 33 − 96 = 0

x² − 8x − 128 = 0

x² − 16x + 8x − 128 = 0

x(x − 16) + 8(x − 16) = 0

(x − 16)(x + 8) = 0

x = 16, −8

Larger root of the equation = 16

So, value of ‘B’ = 16

Given, the value of C is [√(x + 9)] + x − 17 more than that of ‘B’.

And value of x = 16 and the value of B = 16

So, C − [√(x + 9) + x − 17] = B

C − (5 − 1) = B

C − 4 = B

C = 20

Given, 3C − B + 4 = 2A − 16

Therefore, 3 × 20 − 16 + 4 = 2A − 16

2A = 60 + 4

A = 64/2 = 32

2 years ago from now, the age of Ajeet = (A + 8)

= 32 + 8 = 40 years

Present age of Ajeet = 40 + 2 = 42 years

Present average age of Priya and Sakshi = 20 + 6 = 26 years

Sum of present ages of Priya and Sakshi = 26 × 2 = 52 years

Let present age of Eshant = ‘m’ years

So, present age of Diksha = (m + 16) years

Age of Eshant 6 years hence from now will be = ‘m + 6’ years

Age of Sakshi 4 years hence from now will be
= [(m + 6)/5] × 6 = ((6m + 36)/5) years

Present age of Sakshi = [((6m + 36)/5) − 4] years

Sum of present ages of Sakshi, Diksha and Eshant = 32 × 3 = 96 years

So, [(6m + 36)/5] − 4 + m + 16 + m = 96

6m + 36 − 20 + 5m + 80 + 5m = 96 × 5

16m + 96 = 96 × 5

16m = 96 × 4

m = 24

So, present age of Eshant = 24 years

Present age of Diksha = 24 + 16 = 40 years

Present age of Sakshi = [(6 × 24 + 36)/5] − 4 = 32 years

Present age of Priya = 52 − 32 = 20 years

Sum of present ages of Eshant, Ajeet and Ritika
= 34 × 3 = 102 years

Present age of Ritika = 102 − 42 − 24 = 36 years

Required ratio = 36 : 32 = 9 : 8