Christine is 7 years younger than twice as old as Karen. Karen is 6 years younger than Christine was when Christine was 3 times as old as Karen. How old Are they now?

Question

Christine is 7 years younger than twice as old as Karen. Karen is 6 years younger than Christine was when Christine was 3 times as old as Karen. How old Are they now?

Answer 1

C and K are their past ages, when Christine was three times as old as Karen.

So C = 3K

Since then, N years have passed, so their current ages are C+N and K+N.

Currently, Karen is 6 years younger than Christine's past age back then.

K+n = C-6

Finally Christine is CURRENTLY 7 years younger than twice as old as Karen.

So C+n = 2(K+n)-7

The 3 by 3 system is:

C = 3k

K + n = C - 6

C + n = 2(k+n) - 7

Substituting C=3k

k + n = 3k - 6

n = 2k - 6

Substituting these into the third equation:

C + n = 2(k+n) - 7

3k + (2k-6) = 2(k+ 2k-6) - 7

5k - 6 = 2(3k-6) - 7

5k - 6 = 6k - 12 - 7

5k - 6 = 6k - 19

-6 = k - 19

13 = k

Karen was 13. Christine was 39.

n = 2k-6 = 2(13) - 6 = 26 - 6 = 20.

So 20 years have passed.

Karen is now 33 and Christine is 59.

Christine is in fact 7 years younger than twice Karen's age.

Thank you for the challenging problem!