SOLUTION: Hannah, Mark and Tom have a combined age of 68. Hannah is twice as old as the difference between Mark and Tom's age. Mark and Tom's combined age is 16 times Hannah's age. Mark is

Question 287596: Hannah, Mark and Tom have a combined age of 68. Hannah is twice as old as the difference between Mark and Tom's age. Mark and Tom's combined age is 16 times
Hannah's age. Mark is older than Tom. How old is Tom?
(a) 31 (b) 32 (c) 33 (d) 34 (e) None of the above

Found 2 solutions by ptaylor, Deina:
Answer by ptaylor(2198) (Show Source): You can put this solution on YOUR website!
Let x=Hannah's age
y=Mark's age
z=Tom's age
x+y+z=68--------------------eq1
y-z=difference between Mark and Tom's age
x=2(y-z)--------------------------eq2
y+z=16x---------------------------------eq3
From eq 1, y+z=68-x----substitute this for y+z in eq3
68-x=16x add x to each side
17x=68
x=4--------------------------Hannah's age
Substitute x=4 into eq2
4=2y-2z or
y-z=2----eq2a
substitute x=4 into eq3
y+z=64-----eq3a
add eq 2a and eq 3a
2y=66
y=33---------------------- Mark's age
substitute y=33 into eq3a
33+z=64
z=31-----------------------------Tom's age
CK
Difference between Mark and Tom's age =33-31=2; Hannah is twice that old OK
Mark and Tom's combined age 33+31=64 is 16 times Hannah's age OK
Hope this helps--ptaylor
Answer by Deina(147) (Show Source): You can put this solution on YOUR website!
First, let's put this in a form where we can see what we're looking for:

Hannah, Mark and Tom have a combined age of 68. So:
H + M + T = 68

Hannah is twice as old as the difference between Mark and Tom's age.
H = 2(|M - T|)

Mark and Tom's combined age is 16 times Hannah's age.
M + T = 16(H)

Mark is older than Tom.

Now we have a series of equations that we can solve:
H + M + T = 68
H = 2(M - T)
M + T = 16(H)

The 1st thing that I see is the similarity between the 1st & 3rd equations:

so
Subtract H from both sides, giving us:

and from the 3rd equation we know that:
so
Add H to each side and:

Divide both sides by 17 and we find that:
H = 4 So Hannah is 4 years old.

So
From the 4th clue we know that Mark is older than Tom, and from the 2nd equation we know that:
so we now know that the difference between Mark & Tom's age is 2 years

So taking our , we will substitute Tom's age + 2 for Marks age (and reverse their order, just to clarify things):
= Subtract 2 from each side, then divide each side:

so Tom is 31 years old, making Mark = 33

Substitute those numbers back into the equations to check, reread the problem to make sure you're answering the right question and bob's your uncle!

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