What is the ratio of the smaller number to the larger number?
Algebra
Level
2
The arithmetic mean and geometric mean of two numbers are 25 and 20, respectively. What is the ratio of the smaller number to the larger number?
The answer is 0.25.
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1 solution
x = first number
y = second number
GM = √(xy)
20 = √(xy) [equation 1]
AM = (x+y)/2
25 = (x+y)/2
50 = x+y
x = 50 - y [equation 2]
Substitute equation 2 in equation 1.
20 = √(xy)
20 = √[(50-y)y]
20 = √(50y-y²)
square both sides
400 = 50y-y²
y² - 50y + 400 = 0
y = 40 and y = 10
if y = 10
x = 50 - y
x = 50 - 10 = 40
if y = 40
x = 50 - 10
x = 50 - 40 = 10
smaller number = 10
larger number = 40
ratio = 10/40 = 1/4 = 0.25