Step-by-step explanation:
The present age of mother is 40 years and the daughter's age is 20 years.
Answer:
Mother present age: 40 years
Mother present age: 40 yearsDaughter present age: 20 years
Step-by-step explanation:
Let's denote the present age of the mother as [tex]\sf x [/tex] years and the present age of the daughter as [tex]\sf y [/tex] years.
According to the problem, the sum of their present ages is 60 years.
So, we have the equation:
[tex]\sf x + y = 60 [/tex]
After 10 years, the mother's age will be [tex]\sf x + 10 [/tex] years, and the daughter's age will be [tex]\sf y + 10 [/tex] years.
According to the problem, the product of their ages after 10 years will be 1500.
So, we have the equation:
[tex]\sf (x + 10)(y + 10) = 1500 [/tex]
Now, let's solve these equations simultaneously.
Expanding the second equation, we get:
[tex]\sf x(y + 10) +10(y+10)= 1500 [/tex]
[tex]\sf xy + 10x + 10y + 100 = 1500 [/tex]
Simplify this equation:
[tex]\sf xy + 10x + 10y = 1400 [/tex]
Now, we can use the first equation to express one variable in terms of the other.
Let's express [tex]\sf y [/tex] in terms of [tex]\sf x [/tex] from the first equation:
[tex]\sf y = 60 - x [/tex]
Substitute this expression for [tex]\sf y [/tex] into the second equation:
[tex]\sf x(60 - x) + 10x + 10(60 - x) = 1400 [/tex]
Now, solve for [tex]\sf x [/tex]:
[tex]\sf 60x - x^2 + 10x + 600 - 10x = 1400 [/tex]
[tex]\sf -x^2 + 60x + 600 = 1400 [/tex]
[tex]\sf -x^2 + 60x - 800 = 0 [/tex]
To simplify further, let's divide the entire equation by -1:
[tex]\sf x^2 - 60x + 800 = 0 [/tex]
Now, we need to solve this quadratic equation. Let's use the quadratic formula:
[tex]\sf x = \dfrac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} [/tex]
where [tex]\sf a = 1 [/tex], [tex]\sf b = -60 [/tex], and [tex]\sf c = 800 [/tex].
Plugging in the values:
[tex]\sf x = \dfrac{{-(-60) \pm \sqrt{{(-60)^2 - 4 \cdot 1 \cdot 800}}}}{{2 \cdot 1}} [/tex]
[tex]\sf x = \dfrac{{60 \pm \sqrt{{3600 - 3200}}}}{2} [/tex]
[tex]\sf x = \dfrac{{60 \pm \sqrt{{400}}}}{2} [/tex]
[tex]\sf x = \dfrac{{60 \pm 20}}{2} [/tex]
Now, we have two possible values for [tex]\sf x [/tex]:
[tex]\sf x_1 = \dfrac{{60 + 20}}{2} = \dfrac{80}{2} = 40 [/tex]
[tex]\sf x_2 = \dfrac{{60 - 20}}{2} = \dfrac{40}{2} = 20 [/tex]
Since [tex]\sf x [/tex] represents the mother's age, and it cannot be less than the daughter's age, we discard the solution [tex]\sf x = 20 [/tex].
So, the present age of the mother, [tex]\sf x [/tex], is 40 years.
Now, we can find the present age of the daughter, [tex]\sf y [/tex], using the first equation:
[tex]\sf y = 60 - x [/tex]
[tex]\sf y = 60 - 40 [/tex]
[tex]\sf y = 20 [/tex]
So, the present age of the mother is 40 years, and the present age of the daughter is 20 years.
