The ages of a woman and her daughter add to 45, five years ago, the woman is six times her daughters age. How old was the woman when her daughter was born?
w + d = 45
w - 5 = 6 (d - 5)
solve the system for w and d
the answer is (w - d)
To solve this problem, let's represent the ages of the woman and her daughter with variables. Let's call the woman's current age "W" and the daughter's current age "D."
We are given two pieces of information. First, the sum of their ages is 45: W + D = 45. Second, five years ago, the woman's age was six times her daughter's age: W - 5 = 6(D - 5).
To find how old the woman was when her daughter was born, we need to determine the age difference between them. Let's call this difference "X."
The woman's age when her daughter was born can be expressed as W - X. Similarly, the daughter's age when the daughter was born is 0 (because she was just born).
Since the sum of their ages is 45, we can write (W - X) + 0 = 45.
To solve for X, we need to eliminate the variables in the equation. Rearranging the equation, we have W - X = 45, which can be rewritten as W = X + 45.
Now we have two equations:
1. W - 5 = 6(D - 5)
2. W = X + 45
To continue solving, we need to express W in terms of D in equation 1. From equation 2 (W = X + 45), we can replace W with (X + 45) in equation 1:
(X + 45) - 5 = 6(D - 5)
By simplifying, we get:
X + 40 = 6D - 30
Rearranging the equation, we have:
X = 6D - 30 - 40
X = 6D - 70
Now we can substitute X in equation 2 to find W:
W = (6D - 70) + 45
W = 6D - 25
Since W and D are the woman's and daughter's current ages, respectively, their ages when the daughter was born are:
The woman: W - X = (6D - 25) - (6D - 70), which simplifies to 45.
Therefore, the woman was 45 years old when her daughter was born.