Question
A mother is now two and a half times as old as her daughter. Four years ago the ratio of their ages was 3:1. Find the present age of the mother
Answers
GPT-4o mini
Let's denote the present age of the daughter as \( d \) years. Then, the present age of the mother is \( 2.5d \) years.
According to the problem, four years ago, the daughter's age was \( d - 4 \) years, and the mother's age was \( 2.5d - 4 \) years. At that time, the ratio of their ages was 3:1, which gives us the equation:
\[
\frac{2.5d - 4}{d - 4} = 3
\]
To eliminate the fraction, we can cross-multiply:
\[
2.5d - 4 = 3(d - 4)
\]
Expanding the right side:
\[
2.5d - 4 = 3d - 12
\]
Now, let's rearrange the equation to isolate \( d \):
\[
2.5d - 3d = -12 + 4
\]
\[
-0.5d = -8
\]
Dividing both sides by -0.5:
\[
d = 16
\]
Now that we have the daughter's age, we can find the mother's age:
\[
\text{Mother's age} = 2.5d = 2.5 \times 16 = 40
\]
Thus, the present age of the mother is \( \boxed{40} \) years.
According to the problem, four years ago, the daughter's age was \( d - 4 \) years, and the mother's age was \( 2.5d - 4 \) years. At that time, the ratio of their ages was 3:1, which gives us the equation:
\[
\frac{2.5d - 4}{d - 4} = 3
\]
To eliminate the fraction, we can cross-multiply:
\[
2.5d - 4 = 3(d - 4)
\]
Expanding the right side:
\[
2.5d - 4 = 3d - 12
\]
Now, let's rearrange the equation to isolate \( d \):
\[
2.5d - 3d = -12 + 4
\]
\[
-0.5d = -8
\]
Dividing both sides by -0.5:
\[
d = 16
\]
Now that we have the daughter's age, we can find the mother's age:
\[
\text{Mother's age} = 2.5d = 2.5 \times 16 = 40
\]
Thus, the present age of the mother is \( \boxed{40} \) years.