Mensa Age Problem

Question

If Tim's age plus Cheryl's age is 80 and Tim's age plus Greg's age is 98 and Cheryl's age plus Greg's age is 94. What are their ages.

Answer 1

Tim is 42, Cheryl is 38 and Greg is 56

This is a system of three equations with three unknowns , so the question is it sovable turns on is this a singular system.

Examine the determinant of the matrix 1 1 0

0 1 1

1 0 1

That is 2 and non-zero so the matrix is not singular.

Then we can use Cramers rule to determine the values for the (T,C,G)^T vector

If you are not clear on determinants and Cramer's rule check out Wikipedia

You did say Mensa!

Not using matrices, one can just write the three equations with variables T,C,G and eliminate varibles between the equations to get an equation dependent on just one variable. Then back-substitute to arrive at the values of the other ages.

Answer 2

There three variables (let them be C, T, and G) and three independent equations, so we can solve the system.

T + C = 80 [eq1]

T + G = 98 [eq2]

C + G = 94 [eq3]

From (eq1), T = 80 - C

From (eq3), G = 94 - C

Now, we can write (eq2) as:

(80-C) + (94-C) = 98

174 -2C = 98

-2C = -76

C = 38

Now, just substitute for C in (eq1) and (eq3):

T + C = 80

T + 38 = 80

T = 42

C + G = 94

38 + G = 94

G = 56

Check (very important):

(eq1): Does 42 + 38 = 80 ?

80 = 80 ?yes

(eq2): Does 42 + 56 = 98 ?

98 = 98 ?yes

(eq3): Does 38 + 56 = 94 ?

94 = 94 ?yes