The ratio of two numbers is 3 to 2 and the difference of their squares is 20. Find the numbers.
two numbers: x and y
x/y = 3/2 or y = 2x/3
x^2 - y^2 = 20
x^2 - 4x^2/9 = 20
9x^2 - 4x^2 = 180
5x^2 = 180
x^2 = 36
x = ±6
if x = 6, then y = 4
if x = -6 then y = -4
To find the numbers, we can set up a system of equations based on the given information.
Let's assume the two numbers are represented by x and y.
According to the information given, the ratio of the two numbers is 3:2. Therefore, we can write the equation as:
x/y = 3/2 ----(Equation 1)
Additionally, we are given that the difference of their squares is 20. So we can write the equation as:
x^2 - y^2 = 20 ----(Equation 2)
We now have a system of two equations. To solve for x and y, we can use the method of substitution.
First, we can use Equation 1 to express x in terms of y:
x = (3/2)y
Substituting this expression for x into Equation 2, we get:
((3/2)y)^2 - y^2 = 20
Simplifying this equation, we have:
(9/4)y^2 - y^2 = 20
(9y^2/4) - y^2 = 20
(9y^2 - 4y^2)/4 = 20
5y^2/4 = 20
5y^2 = 4 * 20
5y^2 = 80
Dividing both sides by 5, we have:
y^2 = 16
Taking the square root of both sides, we get:
y = ±4
Now, we substitute the found value of y back into Equation 1 to solve for x.
Using y = 4: x = (3/2)*4 = 6
Using y = -4: x = (3/2)*(-4) = -6
Therefore, the two numbers are 6 and 4, or -6 and -4.