arithmetic and geometric means

Question 1
A group of students were proceeding on an education excursion. One of the students told:
The arithmetic mean of 2 numbers is 32.5 and their geometric mean is 30. One of the numbers will be
a) 20 b) 42 c) 12 d) none of these.
Answer : a) 20
Solution :
Let the numbers be x and y
Arithmetic mean of the two numbers 32.5
Then, (x+y) / 2 = 32.5
So, x + y = 65 or y = 65 - x ...(1)
Geometric mean of x and y = √ xy = 30
Hence xy = (30)² = 900
Substituting the value of y from equation 1 in the above equation we get,
x (65-x) = 900
or x² -65x = 900
x² – 65x -900 = 0
(x-45)(x-20) = 0
Hence x = 45 or 20
Or y = 20 or 45
Question 2
A group of teachers and students were proceeding on an education excursion. One of the students told:
The arithmetic mean of 2 numbers is 50 and their geometric mean is 40. LCM of the numbers will be.
a) 150 b) 120 c) 100 d) 80
Answer : d) 80
Solution :
Arithmetic mean of two numbers = 50
Let the numbers be x and y
Then (x + y) / 2 =50
(x + y) = 100 or y = 100 - x ...(1)
Geometric mean of x and y = √ xy = 40
Hence xy = (40)² = 1600
Substituting the value of y from equation 1, we get,
x (100-x) = 1600
or x² -100x = 1600
x² – 100x -1600 = 0
(x-80)(x-20) = 0
Hence x = 80 or 20
Y = 20 or 80
if x = 80, then y = 20 and vice versa. LCM of 80 and 20 is 80, which is our answer.
Hence answer d) 80
Question 3
Ramasamy, Kuppuswamy and Govindaswamy are close friends. They use to discuss politics, sports, cinema, drama, literature and maths from time to time. Once Kuppuswamy had a doubt he posed this problem to the other two: “The arithmetic mean of 2 numbers is 39 and their geometric mean is 15. What will be the difference between the two numbers.
a) 72 b) 48 c) 56 d) 23
Answer : a) 72
Solution :
Arithmetic mean of two numbers = 39
Let the numbers be x and y
Then (x +y)/2 =39
(x + y) = 78 or y = 78 - x ...(1)
Geometric mean of x and y = √ xy = 15
Hence xy = (15)² = 225
Substituting the value of y from equation 1, we get
x (78-x) = 225
or x² -78x = 225
x² – 78x - 225 = 0
(x-75)(x-3) = 0
Hence x = 75 or 3
y = 3 or 75
One of the numbers will be 75 and the other will be 3. Therefore, difference = 75 - 3 = 72
Hence answer is choice a) 72.