Anton is four years older than Liza. The product of his age three years from now and Liza's age two years ago is 90 years.Find their present ages
A = L + 4
(A + 3) * (L - 2) = 90
Solve for A and L
Let's solve this step by step:
Step 1: Assign variables to the unknowns.
Let's assume Liza's current age as "x" years.
Anton's current age would be "x + 4" years because Anton is four years older than Liza.
Step 2: Express the given conditions algebraically.
According to the problem, the product of Anton's age three years from now and Liza's age two years ago is 90.
So, (Anton's age + 3) * (Liza's age - 2) = 90
Step 3: Substitute the variables.
Substituting the values in terms of "x", we get:
(x + 4 + 3) * (x - 2) = 90
Simplifying the equation:
(x + 7) * (x - 2) = 90
Step 4: Expand and simplify the equation.
Using the distributive property, we get:
x^2 + 7x - 2x - 14 = 90
Combining like terms, we have:
x^2 + 5x - 14 = 90
Step 5: Rearrange the equation.
Move 90 to the left side:
x^2 + 5x - 14 - 90 = 0
Simplifying further, we get:
x^2 + 5x - 104 = 0
Step 6: Solve the quadratic equation.
Factoring the quadratic equation or using the quadratic formula, we can find that x = 8 or x = -13.
Since age cannot be negative, we discard x = -13.
Therefore, Liza's current age is x = 8 years.
Step 7: Find Anton's current age.
Anton's current age is x + 4 = 8 + 4 = 12 years.
So, Liza is currently 8 years old, and Anton is currently 12 years old.
To find the present ages of Anton and Liza, let's assign variables to their ages.
Let's say Anton's age is 'A' and Liza's age is 'L'.
Given that "Anton is four years older than Liza," we can express this relationship as A = L + 4.
Now, let's break down the second piece of information: "The product of Anton's age three years from now and Liza's age two years ago is 90 years."
The age of Anton three years from now would be (A + 3), and Liza's age two years ago would be (L - 2). The product of these values is (A + 3) * (L - 2).
According to the problem, the product of (A + 3) * (L - 2) is 90, so we can write the equation as:
(A + 3) * (L - 2) = 90
Since we know A = L + 4, let's substitute that into the equation:
(L + 4 + 3) * (L - 2) = 90
Simplifying the equation:
(L + 7) * (L - 2) = 90
Expanding:
L^2 - 2L + 7L - 14 = 90
L^2 + 5L - 14 = 90
Rearranging the equation:
L^2 + 5L - 104 = 0
Now, we need to solve this quadratic equation. We can factor it or use the quadratic formula to find the values of L.
Factoring the equation may not be feasible as there are no factors of -104 that add up to 5. Therefore, let's use the quadratic formula:
L = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = 1, b = 5, and c = -104:
L = (-5 ± √(5^2 - 4 * 1 * -104)) / (2 * 1)
L = (-5 ± √(25 + 416)) / 2
L = (-5 ± √441) / 2
L = (-5 ± 21) / 2
We have two possible solutions for L:
L1 = (-5 + 21) / 2 = 16 / 2 = 8
L2 = (-5 - 21) / 2 = -26 / 2 = -13
Since the age cannot be negative, we discard -13 as a possible solution.
Therefore, Liza's age is 8 years.
Now, using the relationship A = L + 4:
A = 8 + 4 = 12
Anton's age is 12 years.
So, Anton is 12 years old and Liza is 8 years old.