Important Formulas on “Problems on Ages”:
- If the current age is x, then ntimes the age is nx.
- If the current age is x, then age nyears later/hence = x+ n.
- If the current age is x, then age nyears ago = x– n.
- The ages in a ratio a: bwill be ax and bx.
|5. If the current age is x, then||1||of the age is||x||.|
A problem with one variable: How old is Al?
Many single-variable algebra word problems have to do with the relations between different people’s ages. For example:
Al’s father is 45. He is 15 years older than twice Al’s age. How old is Al?
We can begin by assigning a variable to what we’re asked to find. Here this is Al’s age, so let Al’s age = x.
We also know from the information given in the problem that 45 is 15 more than twice Al’s age. How can we translate this from words into mathematical symbols? What is twice Al’s age?
Well, Al’s age is x, so twice Al’s age is 2x, and 15 more than twice Al’s age is 15 + 2x.That equals 45, right? Now we have an equation in terms of one variable that we can solve for x: 45 = 15 + 2x.
|original statement of the problem:||45 = 15 + 2x|
|subtract 15 from each side:||30 = 2x|
|divide both sides by 2:||15 = x|
Since x is Al’s age and x = 15, this means that Al is 15 years old.
It’s always a good idea to check our answer:
|twice Al’s age is 2 x 15:||30|
|15 more than 30 is 15 + 30:||45|
This should be the age of Al’s father, and it is.
|1.||Father is aged three times more than his son Ronit. After 8 years, he would be two and a half times of Ronit’s age. After further 8 years, how many times would he be of Ronit’s age?|
|2.||The sum of ages of 5 children born at the intervals of 3 years each is 50 years. What is the age of the youngest child?|
|3.||A father said to his son, “I was as old as you are at the present at the time of your birth”. If the father’s age is 38 years now, the son’s age five years back was:|
|4.||A is two years older than B who is twice as old as C. If the total of the ages of A, B and C be 27, the how old is B?|
|5.||Present ages of Sameer and Anand are in the ratio of 5 : 4 respectively. Three years hence, the ratio of their ages will become 11 : 9 respectively. What is Anand’s present age in years?|
|6.||A man is 24 years older than his son. In two years, his age will be twice the age of his son. The present age of his son is:|
|7.||Six years ago, the ratio of the ages of Kunal and Sagar was 6 : 5. Four years hence, the ratio of their ages will be 11 : 10. What is Sagar’s age at present?|
|8.||The sum of the present ages of a father and his son is 60 years. Six years ago, father’s age was five times the age of the son. After 6 years, son’s age will be:|
|9.||At present, the ratio between the ages of Arun and Deepak is 4 : 3. After 6 years, Arun’s age will be 26 years. What is the age of Deepak at present ?|
|10.||Sachin is younger than Rahul by 7 years. If their ages are in the respective ratio of 7 : 9, how old is Sachin?|
The present ages of three persons in proportions 4 : 7 : 9. Eight years ago, the sum of their ages was 56. Find their present ages (in years).
|12.||Ayesha’s father was 38 years of age when she was born while her mother was 36 years old when her brother four years younger to her was born. What is the difference between the ages of her parents?|
|13.||A person’s present age is two-fifth of the age of his mother. After 8 years, he will be one-half of the age of his mother. How old is the mother at present?|
|14.||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?|
|15.||The age of father 10 years ago was thrice the age of his son. Ten years hence, father’s age will be twice that of his son. The ratio of their present ages is:|
Answer:1 Option A
Let Ronit’s present age be x years. Then, father’s present age =(x + 3x) years = 4x years.
|(4x + 8) =||5||(x + 8)|
8x + 16 = 5x + 40
3x = 24
x = 8.
|Hence, required ratio =||(4x + 16)||=||48||= 2.|
|(x + 16)||24|
Answer:2 Option A
Let the ages of children be x, (x + 3), (x + 6), (x + 9) and (x + 12) years.
Then, x + (x + 3) + (x + 6) + (x + 9) + (x + 12) = 50
5x = 20
x = 4.
Age of the youngest child = x = 4 years.
Answer:3 Option A
Let the son’s present age be x years. Then, (38 – x) = x
2x = 38.
x = 19.
Son’s age 5 years back (19 – 5) = 14 years.
Answer:4 Option D
Let C’s age be x years. Then, B’s age = 2x years. A’s age = (2x + 2) years.
(2x + 2) + 2x + x = 27
5x = 25
x = 5.
Hence, B’s age = 2x = 10 years.
Answer:5 Option A
Let the present ages of Sameer and Anand be 5x years and 4x years respectively.
|Then,||5x + 3||=||11|
|4x + 3||9|
9(5x + 3) = 11(4x + 3)
45x + 27 = 44x + 33
45x – 44x = 33 – 27
x = 6.
Anand’s present age = 4x = 24 years.
Answer:6 Option D
Let the son’s present age be x years. Then, man’s present age = (x + 24) years.
(x + 24) + 2 = 2(x + 2)
x + 26 = 2x + 4
x = 22.
Answer:7 Option A
Let the ages of Kunal and Sagar 6 years ago be 6x and 5x years respectively.
|Then,||(6x + 6) + 4||=||11|
|(5x + 6) + 4||10|
10(6x + 10) = 11(5x + 10)
5x = 10
x = 2.
Sagar’s present age = (5x + 6) = 16 years.
Answer:8 Option D
Let the present ages of son and father be x and (60 –x) years respectively.
Then, (60 – x) – 6 = 5(x – 6)
54 – x = 5x – 30
6x = 84
x = 14.
Son’s age after 6 years = (x+ 6) = 20 years..
Answer:9 Option B
Let the present ages of Arun and Deepak be 4x years and 3x years respectively. Then,
4x + 6 = 26 4x = 20
x = 5.
Deepak’s age = 3x = 15 years.
Answer:10 Option D
Let Rahul’s age be x years.
Then, Sachin’s age = (x – 7) years.
|x – 7||=||7|
9x – 63 = 7x
2x = 63
x = 31.5
Hence, Sachin’s age =(x – 7) = 24.5 years.
Answer:11 Option B
Let their present ages be 4x, 7x and 9x years respectively.
Then, (4x – 8) + (7x – 8) + (9x – 8) = 56
20x = 80
x = 4.
Their present ages are 4x = 16 years, 7x = 28 years and 9x = 36 years respectively.
Answer:12 Option C
Mother’s age when Ayesha’s brother was born = 36 years.
Father’s age when Ayesha’s brother was born = (38 + 4) years = 42 years.
Required difference = (42 – 36) years = 6 years.
Answer:13 Option C
Let the mother’s present age be x years.
|Then, the person’s present age =||2||x||years.|
|2||x + 8||=||1||(x + 8)|
2(2x + 40) = 5(x + 8)
x = 40.
Answer:14 Option D
1. The difference of age b/w R and Q = The difference of age b/w Q and T.
2. Sum of age of R and T is 50 i.e. (R + T) = 50.
Question: R – Q = ?.
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.
Answer:15 Option B
Let the ages of father and son 10 years ago be 3x and x years respectively.
Then, (3x + 10) + 10 = 2[(x + 10) + 10]
3x + 20 = 2x + 40
x = 20.
Required ratio = (3x + 10) : (x + 10) = 70 : 30 = 7 : 3.
Answer:16 Option E
|I. S = 5D D =||S||….(i)|
- S – 5 = 25 (D – 5) S = 25D – 120 ….(ii)
|Using (i) in (ii), we get S =||25 x||S||– 120|
4S = 120.
S = 30.
Thus, I and II both together give the answer. So, correct answer is (E).
Answer:17 Option E
- Retirement age is 60 years.
- There are 50 employees in the department.
Average age of 50 employees = 30 years.
Total age of 50 employees = (50 x 30) years = 1500 years.
Number of employees next year = 40.
Total age of 40 employees next year (1500 + 40 – 60 x 10) = 940.
|Average age next year =||940||years = 23||1||years.|
Thus, I and II together give the answer. So, correct answer is (E).
Answer:18 Option C
Let Divya’s present age be D years and Shruti’s present age b S years
Then, D = 2 x S D – 2S = 0 ….(i)
|I.||D + 5||=||9||….(ii)|
|S + 5||5|
|II.||D – 10||=||3||….(iii)|
|S – 10||1|
From (ii), we get : 5D + 25 = 9S + 45 5D – 9S = 20 ….(iv)
From (iii), we get : D – 10 = 3S – 30 D – 3S = -20 ….(v)
Thus, from (i) and (ii), we get the answer.
Also, from (i) and (iii), we get the answer.
I alone as well as II alone give the answer. Hence, the correct answer is (C).