Answer:
Let's express the given information in the form of linear equations.
Let:
- \(m\) be the present age of the mother.
- \(d\) be the present age of the daughter.
According to the first statement, "At present, the age of the mother is three times the age of her daughter," we can write the equation as:
\[m = 3d\] (Equation 1)
According to the second statement, "After 5 years, the sum of their ages will be 54," we can write the equation as:
\[(m + 5) + (d + 5) = 54\] (Equation 2)
Now, we can solve these equations to find the present age of the mother and her daughter.
First, let's solve Equation 1 for \(m\):
\[m = 3d\]
Now, substitute this expression for \(m\) into Equation 2:
\[(3d + 5) + (d + 5) = 54\]
Simplify the equation:
\[3d + d + 10 = 54\]
\[4d + 10 = 54\]
Subtract 10 from both sides:
\[4d = 44\]
Divide both sides by 4:
\[d = 11\]
Now that we have the daughter's present age, we can find the mother's present age using Equation 1:
\[m = 3d\]
\[m = 3(11)\]
\[m = 33\]
So, the present age of the mother is \(33\) years and the present age of the daughter is \(11\) years.
Now, let's find out how many years ago the sum of their ages was \(32\) years.
Let \(x\) be the number of years ago when the sum of their ages was \(32\) years.
So, the equation would be:
\[(m - x) + (d - x) = 32\]
Substitute the present ages we found:
\[(33 - x) + (11 - x) = 32\]
Now, solve for \(x\):
\[33 - x + 11 - x = 32\]
\[44 - 2x = 32\]
\[44 - 32 = 2x\]
\[12 = 2x\]
\[x = 6\]
So, \(6\) years ago, the sum of their ages was \(32\) years.
