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# Children at Large

##### Age 11 to 14 Challenge Level:

We received many correct solutions, especially from students at Weyford Junior School (who found it useful to start off by trying "what if..." situations), Ardingly College Junior School and the Overseas Family School in Singapore. Well done to you all.

Sara from Pleckgate School used a trial and improvement method to arrive at her solution:

Tom was twice as old as Sally 1 year ago so that means that Sally can't be very old. My first prediction was that Tom would be 8 and Sally would be 4 for that clue so that would mean that Tom is now 9 and Sally 5. Kate being twice as old as Sally would mean she would be 10 and Ben being 2 years younger than Tom would mean he would be 7. But this prediction was wrong because the combined ages of the girls and the combined ages of the boys aren't the same.

So, I decided to try Sally as 3 and Tom as 6 to make Tom twice as old last year, so now they would be 4 and 7. As Sally is now 4 that makes Kate 8, and as Tom is 7 that makes Ben 5. Adding up the boys totals and the girls totals would make:
7 + 5 = 12 for Boys
8 + 4 = 12 for Girls

Rushad, Oliver and Rayan from Tanglin Trust School in Singapore reasoned as follows:

We found out that Tom's age had to be an odd number, because if, for example, we take away 1 from 10 we get nine, and halve it to get Sally's age we don't end up with a whole number.

So we first started with Tom's age as 9, and worked the rest out, but that made a difference of 1 in their combined ages.

Then we tried 11, and worked the rest out. It made a difference of 2.

So we knew that if we keep increasing Tom's age the age difference would increase too.

So we knew Tom's age was 7, and worked the rest out: Tom-7, Ben-5, Sally-4, Kate-8!!!!

Tiffany Harte & Agathe Lapointe from Stamford High School used algebraic notation to help them reason as follows:

We decided that Sally was x years old. Therefore Kate would be 2x.

Last year Sally was age x-1, and Tom was twice that age, making Tom 2(x-1), which is equal to 2x-2.

This year Tom must be 2x-1 years old (a year older).

Ben must be 2x-3, as he is two years younger than Tom.

We could then form an equation using the information about their combined ages:

Kate's age + Sally's age = Tom's age + Ben's age
2x + x = 2x-1 + 2x-3
3x = 4x - 4
3x + 4 = 4x
4 = x

Therefore Sally is 4, Kate is 8, Tom is 7 and Ben is 5.

Andrei Lazanu from School No. 205 in Bucharest also used algebraic notation to arrive at the solution:

I noted with their initials the ages of the four children in the family (Kate - K, Sally - S, Tom - T, Ben - B).

I wrote the following equations:

T = B + 2 (1)
K + S = T + B (2)
K = 2S (3)
T - 1 = 2(S - 1) (4)

Equation (4) could be written as:

T - 1 = 2S - 2 ? T = 2S - 1 (5)

I substitute K with 2S, from equation (3), into equation (2):

3S = T + B (6)

Now, I substitute T from equation (1) into equation (6):

3S = 2B + 2 (7)

I put T from equation (5) into equation (6):

3S = 2S - 1 + B
therefore
B = S + 1 (8)

I put B from equation (8) in equation (7):

3S = 2(S + 1) + 2
3S = 2S + 4

and I finally find

S = 4

And I calculate B using equation (8):

B = 5

From equation (1) I calculate T:

T = 7

And, from equation (3),

K = 8.

I verify my results in the initial equations and they are correct. So, the final answer is:

Kate: - age 8
Sally: - age 4
Tom: - age 7
Ben: - age 5.