Solving the Mother-Daughter Age Problem

This problem involves finding the current age of the daughter based on two conditions relating her age to her mother's age at different times.

Setting Up the Equations

Let $D$ represent the daughter's current age and $M$ represent the mother's current age.

• Condition 1: The mother is currently three times as old as her daughter. $ M = 3D $
• Condition 2: 5 years ago, the mother was four times as old as her daughter. Mother's age 5 years ago: $M - 5$ Daughter's age 5 years ago: $D - 5$ $ M - 5 = 4(D - 5) $

Calculating the Daughter's Current Age

We can solve this system of two equations. Substitute the first equation ($M = 3D$) into the second equation:

1. Substitute $M = 3D$ into $ M - 5 = 4(D - 5) $: $ 3D - 5 = 4(D - 5) $
2. Distribute the 4 on the right side: $ 3D - 5 = 4D - 20 $
3. Rearrange the equation to solve for $D$. Subtract $3D$ from both sides: $ -5 = D - 20 $
4. Add 20 to both sides: $ -5 + 20 = D $ $ 15 = D $

The daughter's current age is 15 years.

Verification

Let's check the conditions:

• Daughter's current age ($D$) = 15 years.
• Mother's current age ($M$) = $3 \times 15 = 45$ years.
• 5 years ago: Daughter was $15 - 5 = 10$ years old. Mother was $45 - 5 = 40$ years old.
• Was the mother four times as old as the daughter 5 years ago? $40 = 4 \times 10$. Yes, the condition holds true.

Therefore, the daughter's current age is 15 years.