Certainly, let's walk through the given problem and see how we arrive at the answer.
The problem states that Sally is currently 54 years old and her mother is 80 years old. We need to determine how many years ago Sally's mother was three times the age of Sally at that time.
Starting with the present ages:
1. Sally's age now: [tex]\( 54 \)[/tex] years old
2. Mother’s age now: [tex]\( 80 \)[/tex] years old
Let [tex]\( x \)[/tex] be the number of years ago when Sally's mother's age was three times Sally's age.
Years ago, both Sally and her mother were younger by [tex]\( x \)[/tex] years. Therefore:
- The age of Sally [tex]\( x \)[/tex] years ago was [tex]\( 54 - x \)[/tex]
- The age of her mother [tex]\( x \)[/tex] years ago was [tex]\( 80 - x \)[/tex]
We know from the problem that at that point in time, Sally's mother was three times older than Sally. So we set up the equation:
[tex]\[
80 - x = 3 \times (54 - x)
\][/tex]
This equation expresses that Sally's mother's age at that time was three times Sally's age. Let's solve this equation step by step.
1. Start by expanding the right side:
[tex]\[
80 - x = 3 \times (54 - x)
\][/tex]
[tex]\[
80 - x = 3 \times 54 - 3 \times x
\][/tex]
[tex]\[
80 - x = 162 - 3x
\][/tex]
2. Combine like terms by getting all [tex]\( x \)[/tex] terms on one side:
[tex]\[
80 - x + 3x = 162
\][/tex]
[tex]\[
80 + 2x = 162
\][/tex]
3. Isolate [tex]\( 2x \)[/tex] by subtracting 80 from both sides:
[tex]\[
2x = 162 - 80
\][/tex]
[tex]\[
2x = 82
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{82}{2}
\][/tex]
[tex]\[
x = 41
\][/tex]
So, [tex]\( 41 \)[/tex] years ago, Sally's mother was three times Sally's age. To find their ages back then:
- Sally’s age [tex]\( 41 \)[/tex] years ago:
[tex]\[
54 - 41 = 13
\][/tex]
- Mother’s age [tex]\( 41 \)[/tex] years ago:
[tex]\[
80 - 41 = 39
\][/tex]
Therefore, 41 years ago, Sally was 13 years old, and her mother was 39 years old. It was indeed at that time that Sally's mother was three times Sally's age [tex]\( (3 \times 13 = 39) \)[/tex].
So the answer is:
[tex]\[
\text{41 years ago, when Sally was 13 and her mother was 39.}
\][/tex]
