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Chapter 04 Problems on Ages It is easy to solve the problems relating to ages. The statements given in the problems are made into mathematical expressions and solved. Practicing how such statements are made into mathematical expressions, will help in solving the problems. For example read the question below Let a person’s age as X. Ten years before his/her age would be X -10 and Ten years hence / after his / her age would be X + 10. Two or more person’s ages are related and expressed mathematically as below Let A’s age is twice as that of B’s age then Let A’s age as X & B’s age as Y Mathematically the statement may be expressed as X=2Y Eg.1. Kannan’s age is twice as that of Kumar’s age ten years ago. If Kumar’s age is 30 what is Kannan’s age? Solution: Let Kannan’s age = X Kumar’s age = 50 years Kumar’s and Kannan’s ages have been connected / related in the question viz Kannan’s age is twice as that of Kumar’s age ten years ago. Kumar’s age before ten years = 30-10= 20 years Kannan’s age = Twice of kumar’s age before ten years = 2×20 = 40 years Eg. 2. Ramu’s age is 2 years more than Kumar’s age. After ten Years the ratio of their ages will be 10 : 9. What is kumar’s age? Let Kumar’s Age = X ∴ Ramu’s age = X + 2 After ten years. Kumar’s age = X + 10 Ramu’s age = X + 2 + 10 = X + 12 Now equate the ratio ie., X + 12 : X + 10 = 10 : 9
10X-9X=108-100 Years Kumar’s age = 8 Years Eg. 3. Now Arun’s father age is twice as that of Arun. Ten years before their ages were in the ratio of 3 : 1. Find the current age of Arun. Solution: Let Arun’s age be X Arun’s father age 2x 10 year before. Arun’s father age 2 x - Arun’s age x – 10 As per the statement 2x – 10 / x – 10 = 3/ Arun’s Current age = 20. Eg. 4. The average age of Bhavan and Vimal is 20 years. 10 years before their average age was in the ratio of 3:7.Find bhava’s age. Solution : Let Bhavan’s age be x Vimal’s age be y x + y= 40 ---------------eqn 1 10 year before Bhavan’s age X– 10 Vimal’s age Y – 10 7x – 70 = 3y – 30 7x – 3y = 40 ------------- eqn 2 x + y = 40 ----------------- Eqn 1 X 3 3x + 3y = 120 ---------------- 3 7x – 3y = 40 ----------------- 2
Vijay’s age = 5x= 5×6 = 30 years Eg. 8. Four men joined with an industry in which 20 men with the average age of 35 years were working and thereby the average age of the new group increased by two years. If three of them would have joined, the average age of the group would be increased by only one year. Find the age of the fourth men who had newly joined. Solution: The total age of the group initially = 20×35 = 700 years Total age of the new group with 24 men = 24×37= 888 Total age of the group with 23 men = 23×36= The age of the fourth men newly joined = 888 – 828 = 50 years Eg.9. In a group of 5 men, two men of ages 40 and 45 were replaced by two women thereby the average age of the group decreased by one year. What is the average age of the women? Solution: Let the average age of 5 men =x Total age of five men = 5x Total age of three men when two were left = 5x – (40+45)= 5x- After replacing two women the average age of the group = x- Total age of the current group = 5(x-1)= 5x- The total age of the two women = 5x-5 – (5x-85) =85-5=80 years The average age of the two women = years Eg.10. The present age of the father is 5 years more than the thrice of his son’s age 5 years ago. The total age of the father and the son is 50 years. What will be the ratio of their ages after 5 years? Solution: Let the son’s age = x 5 years ago the son’s age = x- Father’s age = 3(x-5) + 5 The total age= 50
Father’s age + Son’s age = [3(x-5) + 5]+x [3(x-5) + 5]+x = 50 3x-15+5+x= Son’s age = 15 years Father’s age = 3(x-5) + 5= 3(15-5) + 5= 3×10+ = 35 years After 5 years Fathers age: Son’s age = 40:20 = 2: Exercise 3
- 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? a) 12 b) 24 c) 8 d ) none of these.
- The sum of the ages of a son and his father is 56 years. 4 years ago their ages were is the ratio of 1:5. Find the age of the father in years after 4 years a) 42 b) 48 c) 36 d) 40.
- The ages of three members in a group 4 years ago was at the ratio of 2 : 3 : 5. Now the age of the elderly person is 64. What is the present age of the younger one? a) 24 years b) 32 years c) 60 years d) 28 years
- The average age of three members was calculated as 38. It was found later one of the members was read as 61 instead of 67. What is the correct average age in years? a) 36 b) 38 c) 40 d) 42
- A man spent 1/8th^ of his life in childhood, 1/9th^ in youth and some more years as a bachelor and 6 years after his marriage he built a house at half his final age. If the life period of the man is 72 years , how many years did he spend as bachelor? a) 14 Years b) 11 Years c) 13 Years d) 12 Years
- The ratio of Rita's age to her mother's age is 3:8. The difference of their ages is 35 years. The ratio of their ages after 4 years will be: a) 7:12 b) 5:12 c) 38:43 d) 42:
- The present age of Ashok and Shiva is in the ratio of 3 : 5. Shiva is eight year older than Ashok. What will be the ratio of their ages after four years?
- After 5 years Sampath age will be 5 times as that of his age 15 years before. What is Sampath age now? a) 20 Years b)25 Years c) 30 Years d) 32 Years
- The ratio of ages between A & B is 1:2. The ratio of the ages between B & C is 3:4. What will be the age of A after 8 years if the age of C is 48 at that time? a) 6 b) 32 c) 24 d) 18
- The ratio of the ages of a daughter and her mother is 1:4. However the ratio between the ages of the daughter and father is 1:5. In 5 years time the age of the daughter will become 12. What will be the age of the mother after 5 years? a) 40 years b) 33 years b) 28 years d) 35 years
- The ratio of the ages of A and B is 2:3. After 5 years their ages will be in the ratio of 3:4. What will be the ratio of their ages after 10 years? a) 2 :3 b) 3:5 c) 3:4 d) 4:
- The average age of four brothers born at the intervals 3 years each is 58 years. What is the ratio of the ages between the youngest and the eldest? a) 1 : 4 b) 1: 9 c) 10 : 19 d) 4:
- A father said to his younger son, “I was as old as your elder brother’s present age at the time of your birth”. The father age is 40 years at present and the age difference between the brothers is 4 years. Find the age of the younger brother. a) 16 Years b) 20 Years c) 12 Years d) data in adequate
- The ratio of the ages of a gentleman and his wife is 5:4. Six years hence, the ratio of their ages will be 6 : 5. At the time of their marriage the ratio of their ages was 9:7. How many years ago did they get married together? a) 3 Years b) 2 Years c) 1 Year d) 4 Years
- A mother and her daughter are such that (a) 8 years ago mother’s age was twice as that of her daughter’s age (b) 8 years hence Mother’s age would be 1½ times as that of her daughter’s age. What are their current ages in years? a) 20, 40 b) 32, 48 c) 16, 36 d) 24, 40
- The sum of twice the age of Kamala and thrice the age of Arun is 72. Arun is elder and the age difference between them is 4. What is the age of Kamala? a) 12 years b) 14 years c) 16 years d) None of the above
- A Father is aged three times more than his son Varun. After 8 years he would be two and a half times of Varun’s age. After further 8 years how many times would be the father age to that of his son varun’s age? a) 2.5 b) 2 c)1.5 d) 3
- A father’s age is thrice as that of the sum of his two sons’ age. But twenty years hence his age will be equal to the sum of their sons’ age. Then the father’s age is a) 30 Years b)42 Years c) 35 Years d) 45 Years
- The average age of 10 members of a committee is the same as it was 4 years ago, because an old member has been replaced by a young member. Find how much younger is the new member? a) 30 Years b) 40 Years c) 35 Years d) 60 Years
- A wizard named Nepo says "I am only three times my son's age. My father is 40 years more than twice my age. Together the three of us are a mere 1240 years old." How old is Nepo? a)140 Years b) 150 Years c) 320 Years d) 360 Years
Key answers
1-a 2-c 3-d 4-c 5-c 6ba
7-a 8-b 9-d 10-a 11-a 12-c
13-d 14-b 15-b 16–a 17-a 18-a
19-a 20-b 21-d 22-c 23-a 24-a
25- d 26-a 27- b 28-a 29-b 30-d
Explanations
1 Option a Solution : Hence C is 12 years younger than A
Let the Rita’s age and her mother age be 3X and 8X The difference of their ages 8X-3X = 35 ∴ X= 7 The ratio of their ages after 4 years = 3X+4 : 8X+ = 25 : 60 = 5:
- Option a Solution: The ratio of the ages of Ashok and shiva =3: ∴ The age of Ashok =3X The age of Shiva = 5X 5X- 3X= 8 (since Shiva is eight years elder) X= After 4 years Ashok’s age = 3X+4= 3(4)+4= Shiva’s age = 5X+4= 5(4)+4= Age ratio after 4 years = 16:24=2:
- Option b Solution: Rahul’s Age= Karuna’s age = Assume X years after their age ratio = 3: On cross multiplication 4(40+X) = 3(56+X) 160+ 4X = 168+3X X= 168-160=8 Years
- Option d Solution: Let Kumar’s age = x
Kumar’s age after 5 years = x+ After 5 years Mohan’s age = 2x x= 15 Mohan’s age (^) = 2x-5= 30-5= 25 years
- Option a Solution: Let Rajen’s age be 8X and his sister age be 5 X at the time of his marriage Rajan’s sister age at the time of marriage = 5X=25 ∴X=5 & 8X= After 5 years of Rajan’s marriage his sister’s present age = 25+5 = After 5 years of Rajan’s marriage his present age = 40 + 5 = 45 Sum of their current ages = 45 + 30 = 75 years
- Option a Solution: Let A’s age and B’s age as X and Y respectively at present After 10 years A’s age will be 4 times as that of B’s age 10 years ago Sum of their ages at present Solving eqn 1 and eqn 2 Y = 20 After 10 years B’s age = 20+10 = 30 years
- Option c Solution: Let Mahesh, Rajesh and Sharath age be X,Y and Z respectively Y=2X , Z= 2Y=4X All of their ages increased by 50%
After 10 years Dinesh age will be 41 x= 41-10= 31 Substituting x value in eqn 1 Mother’s age = 63 years
- Option b Solution: The sum of the ages of A, B and C is 116 A+B+C =116 eqn The total age of B and C is 82 eqn From eqns 1 & A is 8 years younger than C B’s age after 5 years years
- Option a Solution: Sonia’s present age = X & Deepak’s present age = Y X=5Y ------Eqn 1 Five years ago Sonia’s age was twenty five times Deepak’s age at that time
X-5 =25 (Y-5)
X-5 =25 Y-125 X+120 = 25 Y
∴ X+ 120 =25 Y-----Eqn Eqn 2 Eqn 1 ∴ X= 30 years
- Option a Solution: Let age’s of 3 persons be x, 2x, 3x After 5 years Difference between the eldest & youngest years
- Option a Solution: Let Sampath’s age = x After 5 years his age= X+ 15 years before his age = X- X-15= 5 (X+5) X-15= 5 X+ 5X- 4X= 25-5= 20 ∴ X = 5 Sampath’s age 4X= 20 years
Let the age of the youngest be x years The age of the brothers are x, x+3 , x+ 6 and x+ 9 respectively from the youngest to the eldest brother. ∴ The total age = x+x+3 + x+ 6+ x+ 9 4 x + 18 = 58 x = The ratio of the ages of the youngest brother to the eldest brother = The required ratio = 10 : 19
- Option a Solution: Let the elder brother’s present age be x years Before x years, the fathers age = The present age of the elder brother 40- x = x 2 x = 40 x = 20 Age of the younger brother = x – 4 = 20 – 4 = 16 years
- Option a Solution: Let the ages of gentleman and his wife be 5 X and 4 X Six years hence 25 X – 24 X = 36- 30 = 6 X = 6 The age of the gentleman = 5 X = 5×6= 30 The age of his wife = 4 X = 4×6= 24 Let before Y years they got married and the ratio was 9: 216 – 9 Y = 7 Y - 210 Y= 3Years
- Option d
Solution: Let 8 years ago the daughter’s age be X Then the age of the mother = 2X At present Mothers age = 2X+8 & Daughter’s Age = X+ After 8 years Mothers age = 2X+8+8 = 2X+16 & Daughter’s Age = X+8+8 = X+ = 4X+ 32 = 3X+48 ∴ X = 16 Current age of the daughter = X=8 = 16+8= 24 Current age of the mother = 2X=8 = 2(16)+8=
- Option a Solution: Let Kamal’s age be x and Arun’s age be y The sum of twice the age of Kamala and thrice the age of Arun is 72. Age difference between them is 4 By solving eqn 1 and eqn 2 Substituting the value of y in eqn The age of Kamala = 12 years
After 4 years the new total would be = 10X+(4×10) = 10X+ (All 10 members age will be increased by 4 years) Since the average is same as that of the average before 4 years The new member is (10X+40) -10X = 40 years younger
- Option d Solution: Let the ages of Nepo, his son and his father be N, S and F respectively. From the statement of Nepo N+S+F= F= 2N+40 & S= Eqn 1 is written in terms of S 3S+S+2(3S)+40 = ∴ 10S+40 = 1240 S =1240-40= S= Nepo’s age = 3S= 3 ×120 =360 years ******
