The product of two numbers is 108, and the quotient is 12. What are the two numbers.
a=first number
b=second number
a*b=108 Divide with b
a=108/b
a/b=12
(108/b)/b=12
108/b^2=12
108=12*b^2 Divide with 12
9=b^2
b=sqroot(9)
b=3
a=108/b
a=108/3
a=36
a=36 b=3
a*b=36*3=108
a/b=36/3=12
You also can write:
b=sqroot(9)
b= -3
a=108/b
a=108/ -3
a= -36
a= -36 b= -3
a*b=(-36)*(-3)=108
a/b= -36/ -3=12
sqroot(9)= 3 OR -3
To find the two numbers, we can set up a system of equations based on the given information and solve it simultaneously.
Let's assume the two numbers are x and y. According to the problem, the product of the two numbers is 108:
x * y = 108 ---- Equation 1
The quotient is given as 12, which means one number divided by the other gives 12:
x / y = 12 ---- Equation 2
To solve this system of equations, we can use the method of substitution. Rearrange Equation 2 to solve for x:
x = 12y
Now, substitute this expression for x in Equation 1:
(12y) * y = 108
Simplify the equation:
12y^2 = 108
Divide both sides of the equation by 12:
y^2 = 9
Taking the square root of both sides:
y = ±√9
y = ±3
Now that we have the value of y, we can substitute it back into Equation 2 to find the corresponding value of x.
For y = 3:
x = 12 * 3 = 36
For y = -3:
x = 12 * -3 = -36
Therefore, the two numbers are 36 and -36 or 3 and -3, depending on whether we consider the positive or negative value of y.