Let the present age of the mother be $M$ and the present age of the daughter be $D$.

We can form two equations based on the given information:

Equation 1:

"The sum of the mother's age and her daughter's age is twice the mother's age 12 years ago."

$M + D = 2(M - 12)$

Simplifying:

$M + D = 2M - 24$ $D = M - 24 \quad \text{(Equation 1)}$

Equation 2:

"Nine years from now, the sum of their ages will be thrice the mother's age 14 years ago."

Nine years from now:

• Mother's age will be $M + 9$
• Daughter's age will be $D + 9$

Fourteen years ago, the mother's age was $M - 14$.

The equation is:

$(M + 9) + (D + 9) = 3(M - 14)$

Simplifying:

$M + 9 + D + 9 = 3M - 42$ $M + D + 18 = 3M - 42$ $M + D = 3M - 60$ $D = 2M - 60 \quad \text{(Equation 2)}$

Solving the system of equations:

From Equation 1, we have $D = M - 24$.

Substitute this into Equation 2:

$M - 24 = 2M - 60$

Solving for $M$:

$M - 2M = -60 + 24$ $-M = -36$ $M = 36$

Now substitute $M = 36$ into Equation 1 to find $D$:

$D = M - 24 = 36 - 24 = 12$

Conclusion:

• The mother is 36 years old.
• The daughter is 12 years old.

Would you like more details or have any questions?

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Here are five related questions to think about:

1. What would their ages be after 10 years from now?
2. How would the problem change if the time spans given were different?
3. Can you model this problem with different age differences between the mother and daughter?
4. How would you approach a similar problem with three people instead of two?
5. What are some common strategies for solving age-related word problems?

Tip: Always define variables carefully to represent unknown quantities clearly before forming equations!