Age Shocks

Let the present age (age in the year 2002) of grandpa be $x$ years. So, Jack's present age is $\frac{x}{2}$ years. The year of birth of grandpa is $x$ years before 2002 and that of Jack is $\frac{x}{2}$ years before 2002.

So, grandpa's year of birth = $2002-x$ and Jack's year of birth = $2002- \frac{x}{2}$

It is given that the sum of the years in which they were born is $3854$ . Hence, $(2002-x) + (2002- \frac{x}{2}) = 3854$ Solving this we get grandpa's current age. $(2002-x) + (2002- \frac{x}{2}) = 3854$ $or, 4004 - \frac{3x}{2} = 3854$ $or, \frac{3x}{2} = 150$ $or, x=100$

So, Jack's current age = $\frac{x}{2}= \frac{100}{2} = 50$ . Therefore, Jack's age at the end of the year 2003 is $50 + 1= \boxed{51}$

J : Jack's Age

2J : Jack's Grandpa's Age

(2002 - J) + (2002 - 2J) = 3854

4004 - 3J = 3854

150 = 3J

J = 50 (in 2002)

so, in 2003 Jack's age will be 51

The statement "At the end of the year 2002" means that Jack is at his current age in 2002, regardless when is his birthday or his grandpa's birthday. So, we can safely assume that Jack's birthday in 2002 is J and his grandpa's birthday in 2002 is G.

"Jack was half old as his grandpa" simply means that $J=\frac {1}{2} G$

"The sum of the years in which they were born is 3854" This means that if Jack is born in the year $x$ and his grandpa in the year $y$ , then $x+y=3854$

This leads to the following equations:

The year in which Jack was born, $x=2002 - J$

The year in which Jack's grandpa was born, $y=2002-G$

Hence,

$(2002-J) + (2002-G)=3854$

$4004 - J - G=3854$

$J+G=4004-3854$

$J+G=150$

But, $J=\frac{1}{2} G$ $=>$ $G=2J$

$J+2J=150$

$3J=150$

$J=50$

By the end of the year 2003, his age is 51.

the year which Jack is A and his grandpa is B We ll get this : " A+B = 3854 and 2(2002 - A ) = 2002 - B" Can u so do with that ? " it is easy

Assume that the age of Jack in year $2002=J$ , then the age of grandpa in year $2002=2J$ .

Jack was born in year $2002-J$ .

Grandpa was born in year $2002-2J$ .

Sum of years in which they're born $=2002-J+2002-2J=4004-3J=3854$

$J=\frac{4004-3854}{3}=\frac{150}{3}=50$

Jack's age in year $2002$ is 50, but $50$ is not the answer! It is because the problem is the age of Jack in $year$ $2003$ , so we need to add $1$ into $50$ .

Therefore, our answer is $50+1=\boxed{51}$ .

let he was born x years ago form 2002.

that is, he was born in (2002-x)

so his grandpa was born 2x years ago from 2002

that is, grandpa was born in (2002-2x)

now, according to question,

(2002-2x) +(2002-x) =3854

or, 4004-3x=3854

or,3x= 150

=> x= 50

so his age in 2002 was 50

and so , he was 51 in 2003.

Assume that Jack is J years old, and his grandpa is G years old. At the first statement, we can conclude that J=0.5G. At the second statement, we need to find another equation. To find the years they were born, we need to take 2002 minus their ages to get the years. From that, we can conclude the second equation: ( 2002 - J) + (2002 - G) = 3854. From these two equation, we can find that J=50 and G=100. The problem asks age of Jack one year later, so the answer is 51.

In 2002's end, Let Jack's age be x. His grandpa's age will be 2x. Subtracting their ages from 2002 would give us their birth years whose sum is 3854. So.....

(2002-x)+(2002-2x)=3854 4004-3854=3x 3x=150 x =50

So Jack is 50 years old at the end of 2002. At the end of 2003 he will of course be 51

Let's assume:

Jack's age $= x$ years

Grandpa's age $= 2x$ years

Sum of birth years:

$(2002-2x)+(2002-x)=3854$

$4004-3x=3854$

$-3x=3854-4004$

$-3x=-150$

$x=\frac{-150}{-3}$

$x=50$

So, Jack's current age is $50$ years. Therefore, Jack's age at the end of year $2003$ will be: $50+1=\boxed{51}$

Let Jack's age = x

grandpa's age = 2x

so, (2002-x) + (2002-2x) = 3854

x = 50

Jack's age at the end of 2002 = 50

So at the end of 2003 his age is 51

(2002), Let grandpa age =2x then Jack age = x, So grandpa birth year=2002 - 2x and Jack Birth year = 2002 - x i.e. (2002 -2x) + (2002 -x) = 3854, or 3x = 4004 - 3854= 150 or, x=150/3=50 (jack age) So Age of Jack at the end of year 2003 is 50+1=51

Let Jack's age at the end of 2002 be equal to x

Then his grandpa's age at the end of 2002 would be 2x (given).

The year when Jack was born was - The present year minus Jack's age. That gives us (2002 - x )

The year when Jack's grandpa was born was - The present year minus grandpa's age. That gives us (2002 - 2x )

It is also given that the sum of the years in which they were born is 3854.

That means (2002 - x ) + (2002 - 2x ) = 3854

                    = 2002 - _x_ + 2002 -  *2x* = 3854

                    = 4004 -3 _x_ = 3854

                    = -3 _x_  = 3854 - 4004

                    = -3 _x_  = -150

Negative sign and negative sign will cancel out each other,

That will give us,

                    3 _x_ = 150

                   _x_ = 150 divided by 3

                   _x_ = 50

    50 is the age of Jack at the end of 2002. So, at the end of 2003, it will be 50 + 1 = 51

                                                                  ...................................... SOLVED..........................................

Let J = Jacks birth year

   G = His grandpa's birth year

E1 J + G = 3854

E2 2(2002 - J) = 2002 - G

Simplify E2:

   4002 - 2J = 2002 - G

   G = 2J - 2002

Merge with E1: J + G = 3854

  J + 2J - 2002 = 3854

  3J = 5856

    J = 1952 <----Jack was born in the year 1952

Thus,

   Jacks age is 50 in year 2002

   And **51** in the 2003

Let, age of Jack at the end of year 2002 is X. Then I can write, (2002-X)+(2002-2*X)=3854

=>X=50

so at the end of 2003 the age of Jack is 50+1=51.

Consider the age of Jack at the end of year 2002 be J, and that of grandpa be G.

So, from the first sentence, we have:

J = $\frac{1}{2} \times G$

The year in which Jack was born is $2002 - J$ , and the year in which grandpa was born is $2002 - G$ .

Sum of these years is 3854.

Therefore, $2002 - J + 2002 - G = 3854$

$4004 - J - G = 3854$

$J + G = 150$

Substitute J = $\frac{1}{2} \times G$ into the above equation, we get $G = 100$ .

Therefore, J = $\frac{1}{2} \times 100 = 50$

This was the age of Jack at the end of year 2002.

So, his age at the end of year 2003 would be $50 + 1 = \boxed{51}$ years.

That's the answer!

We know if Year born ( Y) of Jack and Grandpa are (J) and (G). If YJ + YG = 3854. If YJ = 2002 - J and YG = 2002 - 2J because G = 2J in this year.Then , YJ + YG = 3854 ---> 2002 - J + 2002 - 2J = 3854 ----> 4004 - 3J = 3854 ---> 3J = 150 ---> J = 150/3 =50. So, Age of Jack at the end of year 2003 is Age of Jack at the end of year 2002 + 1 = 50 + 1 = 51. Answer : 51

Let x be age of jack in 2002 and y be the age of grandpa in 2002 then 2x=y....now 2002-x +2002-y=3854 Thus we get x=50...der fore age in 2003 will be 51

Taking grandpa's age as 'x' and jack's age as 'x/2' equation formed is "(2002-(x/2))+(2002-x)=3854, i.e., x=100, jacks age at the end of 2002=50, hence age at the end of 2003 is 51

let the year of birth of Jack and Grandpa be x and y respectively

then in the year 2002 Jack's age was $(2002-x)$ and that of granda was $(2002-y)$

it is mentioned that in the year 2002, Jacks age was half as his grandpa's so $2002-x =$ $\frac{1}{2}$ $* 2002 - y$ ....(i)

and the sum of their years of birth is 3854

that is $x + y = 3854$ ......(ii)

equating equations i and ii we get x = 1952 & y =1902

so in the year 2002, Jack was 50 years old and grandpa completed a century

*and in the year 2003 Jack was 51 years old *

AT YEAR 2002

age of grandpa = p,

age of jack = j,

then year at which grandpa was born = 2002 - p,

year at which jack was born = 2002 - j,

according to given data,

(2002 - j) + (2002 - p) = 3874.

=> p + j = 150.

given jack was half old as his grandpa,

=> p = 100, j = 50,

therefore at 2003 age of jack = j+1 = 51

Let age of G, Father be x years in 2002 so he was born on (2002 – x)

then age of Jack = 2x, and he was born on (2002 – 2x)

Sum = (2002 – x) + (2002 – 2x) = 3854 x = 50 age of Jack on 2002 so on 2003 his age was 50 + 1 = 51