SOLUTION: ) Dheeraj is half of the age of his Father. His father is 4 years elder to his mother. 4 years back, Dheeraj was half the age of his mother. What is the present age of Dheeraj?

Question 1132324: ) Dheeraj is half of the age of his Father. His father is 4
years elder to his mother. 4 years back, Dheeraj was half
the age of his mother. What is the present age of Dheeraj?
Found 2 solutions by ikleyn, Theo:
Answer by ikleyn(52908) (Show Source): You can put this solution on YOUR website!
.

Let x be the Dheeraj age. Then the father's age is 2x years, while the mother's age is 2x-4 years. 4 years ago Dheeraj's age was x-4 years, while the mother's age was (2x-4)-4 = 2x - 8 years. The condition says 2*(x-4) = 2x - 8. This equation is, actually, an IDENTITY: it is true for ANY value of x, and, therefore, the value of x CAN NOT be determined uniquely from the equation. ANSWER. The age of Dheeraj can be any positive number, and CAN NOT BE DETERMINED uniquely from the given condition.

Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
this does not appear to have a unique solution.
as far as i can tell, the youngest d can be is 4.
at that time, his father is 8 and his mother is 4.
he is half the age of his father and 4 years ago, he was half the age of his mother.
4 is half of 8.
0 is half of 0.
to be on the safe side and avoiding 0 is half of 0, then the youngest he can be is 5.
the problem is you don't know the age of his father nor do you know the age of his mother nor do you know his age.
you need to know at least one of them to get a unique solution.

for example:

assume his father is 42.
that makes dheeraj 21 and it makes his mother 38.
4 years ago, he was 17 and his mother was 34.
17 is half of 34.

assume his father is 60.
that makes dheeraj 30 and it makes his mother 56.
4 years ago, he was 26 and his mother was 52.
26 is half of 52.

assume his mother is 32.
that means his father is 36.
he is half his father's age, which makes him 18.
4 years ago he was half his mother's age.
4 years ago, he is 14.
4 years ago, his mother was 28.
14 is half of 28.

no matter what age you pick for his father or his mother, he will always be half the age of his father and, 4 years before that, he will always be half the age his mother was then.

algebraically, i get the following:

let d = the current age of dheeraj.
let f = the current age of his father.
let m = the current age of his mother.

he is currrently half the age of his father.

that gets you d = 1/2 * f.

his father is currently 4 years older than his mother.

that gets you f = m + 4

4 years ago, he was half the age his mother was then.

that gets you d - 4 = 1/2 * (m - 4)

you have 3 equations that need to be solved simultaneously.

they are:

d = 1/2 * f
f = m + 4
d - 4 = 1/2 * (m - 4)

from d = 1/2 * f and f = m + 4, you can replace f with m + 4 to get:

d = 1/2 * (m + 4)

from d - 4 = 1/2 * (m - 4), you can add 4 to both sides of this euation to get:

d = 1/2 * (m - 4) + 4

since they're both equal to d, you can set both expressions that are equal to d and make them equal to each other go get:

1/2 * (m + 4) = 1/2 * (m - 4) + 4

simplify to get 1/2 * m + 2 = 1/2 * m - 2 + 4.

this can be further simplified to get 1/2 * m + 2 = 1/2 * m + 2.

this is an identity which means that the value of m can be anything you want it to be.

this means you have an infinite number of solutions.

take another random value for his mother.
assume his mother is 59.
that makes his father 63.
he is half his father's age, so he is 31.5.
4 years ago he was 27.5
4 years ago his mother was 55.
27.5 is half of 55.

the problem statement will always be correct no matter what his mother's age is.

if you can freeze one of them, then you have a unique solution.

otherwise, you have an infinite number of solutions.

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