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The A.M. of two positive numbers is 40 and their G. M. is 24. The numbers are
- a)(72, 8)
- b)(70, 10)
- c)(60, 20)
- d)none of these
Correct answer is option 'A'. Can you explain this answer?
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The A.M. of two positive numbers is 40 and their G. M. is 24. The numb...
Solution:
Given, A.M. of two positive numbers = 40
G.M. of two positive numbers = 24
Let the two positive numbers be a and b
We know that,
A.M. of two positive numbers = (a + b)/2 = 40
=> a + b = 80
G.M. of two positive numbers = √(ab) = 24
=> ab = 576
Now, we need to find the two positive numbers a and b.
Method 1: By substitution
We can substitute the value of b from the first equation in the second equation and solve for a.
=> a(80 - a) = 576
=> 80a - a^2 = 576
=> a^2 - 80a + 576 = 0
=> (a - 72)(a - 8) = 0
=> a = 72 or a = 8
If a = 72, then b = 80 - a = 8
If a = 8, then b = 80 - a = 72
Therefore, the two positive numbers are 72 and 8.
Method 2: By using the formula for A.M. and G.M.
We know that,
A.M. of two positive numbers = (a + b)/2
G.M. of two positive numbers = √(ab)
By substituting the given values, we get
(a + b)/2 = 40
=> a + b = 80
√(ab) = 24
=> ab = 576
Now, we can use the formula (a - b)^2 = (a + b)^2 - 4ab
=> (a - b)^2 = 80^2 - 4(576)
=> (a - b)^2 = 4096
=> a - b = ±64
We know that a + b = 80
Adding the above two equations, we get
2a = 144 or 2b = 16
=> a = 72 or b = 8
If a = 72, then b = 8
If b = 8, then a = 72
Therefore, the two positive numbers are 72 and 8.
Hence, the correct answer is option A) (72, 8).
Given, A.M. of two positive numbers = 40
G.M. of two positive numbers = 24
Let the two positive numbers be a and b
We know that,
A.M. of two positive numbers = (a + b)/2 = 40
=> a + b = 80
G.M. of two positive numbers = √(ab) = 24
=> ab = 576
Now, we need to find the two positive numbers a and b.
Method 1: By substitution
We can substitute the value of b from the first equation in the second equation and solve for a.
=> a(80 - a) = 576
=> 80a - a^2 = 576
=> a^2 - 80a + 576 = 0
=> (a - 72)(a - 8) = 0
=> a = 72 or a = 8
If a = 72, then b = 80 - a = 8
If a = 8, then b = 80 - a = 72
Therefore, the two positive numbers are 72 and 8.
Method 2: By using the formula for A.M. and G.M.
We know that,
A.M. of two positive numbers = (a + b)/2
G.M. of two positive numbers = √(ab)
By substituting the given values, we get
(a + b)/2 = 40
=> a + b = 80
√(ab) = 24
=> ab = 576
Now, we can use the formula (a - b)^2 = (a + b)^2 - 4ab
=> (a - b)^2 = 80^2 - 4(576)
=> (a - b)^2 = 4096
=> a - b = ±64
We know that a + b = 80
Adding the above two equations, we get
2a = 144 or 2b = 16
=> a = 72 or b = 8
If a = 72, then b = 8
If b = 8, then a = 72
Therefore, the two positive numbers are 72 and 8.
Hence, the correct answer is option A) (72, 8).
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