Annie - 30
Betty - 51
Carrie - 55
Darla - 46
Eve - 37

## Reasoning

### Symbolic

Let the ages and names of Annie, Betty, Carrie, Darla and Eve be A, B, C, D and E.

C says to A, that C = A + 10. If C were younger than A, that would be lying, so C must be older than A. (But still lying.)

We have A < C.

C says to A, that B < D. As C > A, C is lying, so B > D.

We have A < C, D < B.

D says to B, that D = E + 9. As D < B, D is telling the truth, so D > E.

We have A < C, E < D < B, D = E + 9.

E says to B, that E = A + 7. As E < B, E is telling the truth, so E > A.

We have A < C, A < E < D < B, D = E + 9, E = A + 7.

Since D = E + 9 and E = A + 7, D = A + 7 + 9 = A + 16.

We have A < C, A < E < D < B, D = E + 9 = A + 16, E = A + 7.

B says to C, that E < C. If B > C then B would be lying, so then E > C, and then A < C < E < D < B. However, C says to D, that C = D ± 6; since C < D, this gives C = D - 6. However, we have E = D - 9, which would make E < C, giving a contradiction. The assumption that B > C is therefore false, so B < C.

We have A < E < D < B < C, D = E + 9 = A + 16, E = A + 7.

A says to B, that B = (17/10)A. As A < B, A is telling the truth.

We have A < E < D < B < C, B = (17/10)A, D = E + 9 = A + 16, E = A + 7.

B says to C, that |C - D| = |D - E| → |C - D| = 9. As B < C, B is telling the truth, so C = D + 9. As D = A + 16, C = A + 16 + 9 → C = A + 25.

We have A < E < D < B < C, B = (17/10)A, C = A + 25, D = A + 16, E = A + 7.

Using D < B < C, we have A + 16 < (17/10)A < A + 25 → 16 < (7/10)A < 25 → 160/7 < A < 250/7 → 22 + 6/7 < A < 35 + 5/7. Since B and A must both be whole numbers, and B = (17/10)A → B - A = (7/10)A, (7/10)A must be a whole number. Hence A must be divisible by 10. The only whole number fitting 22 + 6/7 < A < 35 + 5/7 is A = 30.

We have A = 30, B = (17/10)A, C = A + 25, D = A + 16, E = A + 7.

Hence A = 30, B = 51, C = 55, D = 46, E = 37.

### Verbal

Carrie tells Annie she's older than her by 10 years. If Carrie is younger, she's lying, and that's impossible, so Carrie must be older than Annie, just not by 10 years.

FACT: Carrie is older than Annie (but not by 10 years).

Carrie also lies to (younger) Annie that Betty is younger than Darla.

FACT: Darla is younger than Betty.

Darla tells the truth to (older) Betty that she's 9 years older than Eve.

FACT: Darla is 9 years older than Eve.

Eve tells the truth to (older) Betty that she's 7 years older than Annie.

FACT: Eve is 7 years older than Annie.

Annie tells the truth to (older) Betty that Betty's age is 70% greater than her own. For Betty's age to be a whole number, Annie's age must be a multiple of 10. Since Betty is older than Darla, and Darla is 7 + 9 = 16 years older than Annie, that means Betty has to be more than 16 years older than Annie. The lowest multiple of 7 greater than 16 is 21.

FACT: Annie is at least 30 years old (and definitely a multiple of 10).

At this point, Betty appears to be the oldest, lying lady. Let's assume that, and see if it works.

In that case, Carrie is lying to Darla that the difference in their ages is 6 years, but Betty tells the truth to (older) Carrie that the difference between Carrie's age and Darla's is the same as the difference between Darla's and Eve's, namely, 9 years. Let's test this scenario, assuming Annie's age is 30. Then we get, from youngest to oldest:

TESTING: Annie = 30, Eve = 37, Darla = 46, Betty = 51, Carrie = 55

Checking all statements and the age relations shows that this is an answer. Is this the only answer?

If Annie's age was 40, then Betty's age would be 68, and Carrie's age would be 65, so Carrie would not be the oldest, and that would be a fatal flaw. If Annie is older than 30, Betty is older than Carrie, and Carrie is not the oldest. Hence, it must have been the only answer.