## Math Puzzle: Tom’s Age

Big Thanks to Bilbao for submitting!

Tom was recently sent to a new school. On the first day of his arrival the teacher asked him his age, and this was his curious reply:

‘Well, you see, it is like this. At the time I was born—I forget the year—my only sister, Ann, happened to be just one-quarter the age of mother, and she is now one-third the age of father.’

‘That’s all very well,’ said the teacher, ‘but what I want is not the age of your sister Ann, but your own age.’

‘I was just coming to that,’ Tom answered; ‘I am just a quarter of mother’s present age, and in four years’ time I shall be a quarter the age of father. Isn’t that funny?’.

This was all the information that the teacher could get out of Tom. From these facts, what was his precise age?

fuzzy| Profile April 2nd, 2010 - 11:28 amTom is 9.

When he was born his sister was 7, mother was 28 (7 x 4).

Now the sister is 16, mother is 36 (9 x 4), father is 48 (16 x 3).

In four years, when Tom is 13, father is 52 (13 x 4).

Mashplum| PUZZLE MASTER | Profile April 2nd, 2010 - 11:31 amTom is 9.6 years old, Ann is 16.8, Mom is 38.4, and Dad is 50.4.

Hex| PUZZLE MASTER | Profile April 2nd, 2010 - 2:05 pmA=F/3

T=M/4

T+4+c=(F+4+d)/4 where a,b,c,d 0 or 1

A-T-a=(M-T-b)/4

We get 3 possible solutions for T.A,M,F:

12,21,48,63

7,13,28,39

11,20,44,60

brianu| Profile April 2nd, 2010 - 2:21 pm1)Ann (A) = .25 Moms initial (Mi) (input in 3)

2)A + Tom (T) = .333 Fathers current (Fc)==> 1.75T = .333Fc ==> 5.25T = Fc (input in 4)

3)T = .25 (Mi + T) == .25Mi + .25T == A + .25T ==> .75T = A (input in 2)

4)T + 4 = .25 (Fc + 4) ==> T + 4 = 1.3125T + 1 ==> T = 9.6 years old

This would be true if Tom means that in four years he will be a quarter of the age his father will be at that time. If he means in four years he will be a quarter of his fathers current age, then he will be 12.8 years old. This is also assuming everyone has the same birthday, or that it is Tom’s birthday at the time in question, otherwise some messed up stuff needs to be factored in.

brianu| Profile April 2nd, 2010 - 2:24 pmI just realized if Tom is 9.6, then it is obviously not his birthday (duh), so I would have to say that everyone needs to have the same birthdays, otherwise someone could transition to a higher age between Toms birthday and the .6 years in question and the numbers would get thrown off. This is also assuming I have done my calculations correctly.

savi| Profile April 2nd, 2010 - 2:28 pm9 years, 7 months, 6 days, 4 hours, 48minutes

Jimmy Anders| PUZZLE MASTER | Profile April 2nd, 2010 - 3:47 pmTom is 9.6 years old.

This makes his mother 38.4 his father 50.4, and his sister 16.8.

Ah, to be 9.6 again…

suineg| PUZZLE MASTER | Profile April 3rd, 2010 - 2:12 pmI think Tom precise age is: 9 years 7 months 6 days 4 hours and 48 minutes “or” 9.6 years.

The equations are:

1)Tom = 0 Sister= 1/4 Mother

2)TomToday = X SisterToday= 1/3 FatherToday

3)TomToday = 1/4 MotherToday

4)TomToday + 4 = 1/4 (FatherToday + 4)

From (2) you get: 3 SisterToday = FatherToday (5)

From (4) you get: TomToday = 1/4 FatherToday – 3 (6)

Now substitute (5) in (6): TomToday = 3/4 SisterToday – 3 (7)

Now TomToday – Tom = X “And” TomToday – Sister = 1/4 ( MotherToday – Mother)= 1/4 X

From this you can infer that when Tom = 0 Sister = 3/4 X (Sister – Tom)

So SisterToday = X + 3/4 X = 7/4 X (8)

Substitute (8) in (7) and get X = 3/4 (7/4 X) – 3

X= 9.6

Cool.

Falwan| Profile April 4th, 2010 - 10:42 amTom’s age now = 9.6 yrs.

Blusummers13| Profile April 4th, 2010 - 1:03 pmWell, I’m not sure if I got the right answer since I was expecting whole numbers, but here goes.

Tom is 9.6 years old.

Variables are all in current age.

T=Tom, S=Sister, M=Mother, F=Father

1) S-T=(M-T)/4

2) S=F/3

3) T=M/4

4) T+4=(F+4)/4

Four equations, four unknowns. Solve and I get T=9.6

Bobo The Bear| PUZZLE MASTER | Profile April 4th, 2010 - 4:38 pmAt first, I kept getting the following:

Tom = 9.6 yrs old

Ann = 16.8 yrs old

Mother = 38.4 yrs old

Father = 50.4 yrs old

I was concerned about the non-integer results (most people don’t report their ages in decimals). Then I considered that the second paragraph might be taken to mean “… just one-quarter the age that my mother is now, …” instead of “…just one-quarter the age that my mother was at that time, …”

Working it out again, I got:

Tom = 6 yrs old

Ann = 12 yrs old

Mother = 24 yrs old

Father = 36 yrs old

The math works, but, unless these children were adopted (or some other similar arrangement), I became concerned about the mother’s age when Ann was born. Made me a little queasy, actually.

So rereading the question and seeing that the puzzle asks “what was his PRECISE age?”, I’m happier with saying that Tom is 9.6 years old.

Shawn| PUZZLE GRANDMASTER | Profile April 5th, 2010 - 10:39 am9.6 years

You have to take all the information and put it into the same time period. I chose year zero for Tommy, the year he was born.

A=Ann’s age at year 0

M=Mom’s age at year 0

D=Dad’s age at year 0

T=Tommy’s age today, used to reference the passage of time

The problem gives us:

Ann is 1/4 her mother’s age at T=0

A = 1/4*M

Presently, Ann’s age (A+T) is 1/3 her father’s age (D+T)

A+T = 1/3*(D+T)

Presently, Tommy’s age (T) is 1/4 his mother’s age (M+T)

T = 1/4*(M+T)

In 4 years, Tommy’s age (T+4) will be 1/4 that of his father’s age (D+T+4)

T+4 = 1/4*(D+T+4)

Solve these last 2 equations for M & D respectively, and substitute into the first 2 equations will result in 2 equations in terms of A & T:

A = 3/4*T

A = 1/3*(T+3T+12)

Solving for T = 9.6 years

A = 7.2 years

M = 28.8 years

D = 40.8 years

joe| Profile April 6th, 2010 - 7:54 amI ran through some simultaneous equations and got that he was 9.6 years old at the moment.

Adam| Profile April 6th, 2010 - 11:01 amTom is 9.6 years old.

His sister is 16.8 years old.

His mother is 38.4 years old.

His father is 50.4 years old.

NitzOO| Profile April 6th, 2010 - 11:26 am9.6 years.

bilbao| Profile April 7th, 2010 - 12:21 am9.6 years old is the solution I had for this puzzle

Hex| PUZZLE MASTER | Profile April 7th, 2010 - 5:02 amMy equations above yield Bilbao’s solution if non integer ages were allowed.

Decimal ages? woaaaahh

What are we going to invent next?

RK| Founder | Profile April 7th, 2010 - 7:12 amThanks again Bilbao for the puzzle

bilbao| Profile April 7th, 2010 - 7:29 amHi Hex, to tell you the truth, the decimal ages issue is something I don´t like either.

(I know about another puzzle where you must deal with ages counted in days!!!!).

On the other hand, something must be invented to make these Age Puzzles more original…

Hex| PUZZLE MASTER | Profile April 7th, 2010 - 4:33 pmHi Bilbao, keep the puzzles flowing.

Any new goat puzzles?

Sam| Profile April 20th, 2010 - 8:30 pmLet t= Tom’s current age

a= Ann’s current age

m= Mother’s current age

f= Father’s current

From the conversation between Tom and the teacher, we find that

4(a-t)=m-t

3a=f

4t=m and

4(t+4)=f+4

Simple algebra reduces these four equations to t=9.6, so Tom is 9.6 years old.

AJ| Profile August 27th, 2010 - 4:41 amconsider the current Ages of the boy,mother,father,sister as Bc,Mc,Fc,&Sc respectively.

statment#1:At the time I was born—I forget the year—my only sister, Ann, happened to be just one-quarter the age of mother,

Therefore,

Sc-Bc=1/4(Mc-Bc)

statment#2:she is now one-third the age of father

Therefore,

Sc=1/3(fc)

statment#3:‘I am just a quarter of mother’s present age…..

Therefore,

Bc=1/4(mc)

statment#4:……and in four years’ time I shall be a quarter the age of father. Isn’t that funny?’

Therefore,

Bc+4=1/4(fc+4)

Solve these 4 equations in ‘substution method’ in Algebra & you get the answer as 9.6 for

the Boy’s age.

Algebra was tough…but it is easy once you get o know it..Have fun!!!

AJ| Profile August 27th, 2010 - 4:55 amLet me just tell you how I wrote Sc-Bc=1/4(Mc-Bc)

from the fact,

“At the time I was born—I forget the year—my only sister, Ann, happened to be just one-quarter the age of mother”

say, current year is 2010 & boys age is 15..so at the time the boy as born, the time is 2010 – 15. Say his moms current age is 45, back then it would have been 30 which is the difference between her age & boys age,

that’s why in the equation, you see mothers & sisters age Sc-Bc & Mc-Bc as a difference of the boys age….when he was born. Again back then Ann happened to be a quarter of her mother’s age, so Sc-Bc=1/4(Mc-Bc).

This is how you write equations from facts.