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The harmonic mean H of two numbers is 4 and their arithmetic mean A and the geometric mean G satisfy the equation 2A G² = 27, then the numbers are 33. (a)(1, 3) (c) (6,3) (b)(9,5) (d) (12, 7)?
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The harmonic mean H of two numbers is 4 and their arithmetic mean A an...
Solution:
Harmonic Mean:
The harmonic mean H of two numbers a and b is given by:
H = 2ab/(a+b)
Let the two numbers be a and b.
Given, H = 4
Hence, 4 = 2ab/(a+b)
Multiplying both sides by (a+b)/2, we get:
2ab = 4(a+b)
ab = 2(a+b) ........(1)
Arithmetic Mean:
The arithmetic mean A of two numbers a and b is given by:
A = (a+b)/2
Hence, a+b = 2A
Substituting this in equation (1), we get:
ab = 2(2A)
ab = 4A ........(2)
Geometric Mean:
The geometric mean G of two numbers a and b is given by:
G = sqrt(ab)
Hence, ab = G²
Substituting this in equation (2), we get:
G² = 4A
2A G² = 8A
Given, 2A G² = 27
Hence, 8A = 27
A = 27/8
Substituting this value of A in equation (2), we get:
ab = 27/2
Using the formula for G, we get:
G = sqrt(ab) = sqrt(27/2)
G = 3sqrt(2)
Using the formula for H, we get:
H = 2ab/(a+b)
Substituting the values of ab, H, A and G, we get:
4 = 2(27/2)/(a+b)
a+b = 27/4
We have two equations:
a+b = 27/4
ab = 27/2
Solving these equations, we get:
a = 9, b = 5
Hence, the two numbers are 9 and 5.
Therefore, the correct answer is (b) (9,5).
Harmonic Mean:
The harmonic mean H of two numbers a and b is given by:
H = 2ab/(a+b)
Let the two numbers be a and b.
Given, H = 4
Hence, 4 = 2ab/(a+b)
Multiplying both sides by (a+b)/2, we get:
2ab = 4(a+b)
ab = 2(a+b) ........(1)
Arithmetic Mean:
The arithmetic mean A of two numbers a and b is given by:
A = (a+b)/2
Hence, a+b = 2A
Substituting this in equation (1), we get:
ab = 2(2A)
ab = 4A ........(2)
Geometric Mean:
The geometric mean G of two numbers a and b is given by:
G = sqrt(ab)
Hence, ab = G²
Substituting this in equation (2), we get:
G² = 4A
2A G² = 8A
Given, 2A G² = 27
Hence, 8A = 27
A = 27/8
Substituting this value of A in equation (2), we get:
ab = 27/2
Using the formula for G, we get:
G = sqrt(ab) = sqrt(27/2)
G = 3sqrt(2)
Using the formula for H, we get:
H = 2ab/(a+b)
Substituting the values of ab, H, A and G, we get:
4 = 2(27/2)/(a+b)
a+b = 27/4
We have two equations:
a+b = 27/4
ab = 27/2
Solving these equations, we get:
a = 9, b = 5
Hence, the two numbers are 9 and 5.
Therefore, the correct answer is (b) (9,5).
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The harmonic mean H of two numbers is 4 and their arithmetic mean A and the geometric mean G satisfy the equation 2A G² = 27, then the numbers are 33. (a)(1, 3) (c) (6,3) (b)(9,5) (d) (12, 7)?
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The harmonic mean H of two numbers is 4 and their arithmetic mean A and the geometric mean G satisfy the equation 2A G² = 27, then the numbers are 33. (a)(1, 3) (c) (6,3) (b)(9,5) (d) (12, 7)? for CA Foundation 2024 is part of CA Foundation preparation. The Question and answers have been prepared according to the CA Foundation exam syllabus. Information about The harmonic mean H of two numbers is 4 and their arithmetic mean A and the geometric mean G satisfy the equation 2A G² = 27, then the numbers are 33. (a)(1, 3) (c) (6,3) (b)(9,5) (d) (12, 7)? covers all topics & solutions for CA Foundation 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The harmonic mean H of two numbers is 4 and their arithmetic mean A and the geometric mean G satisfy the equation 2A G² = 27, then the numbers are 33. (a)(1, 3) (c) (6,3) (b)(9,5) (d) (12, 7)?.
The harmonic mean H of two numbers is 4 and their arithmetic mean A and the geometric mean G satisfy the equation 2A G² = 27, then the numbers are 33. (a)(1, 3) (c) (6,3) (b)(9,5) (d) (12, 7)? for CA Foundation 2024 is part of CA Foundation preparation. The Question and answers have been prepared according to the CA Foundation exam syllabus. Information about The harmonic mean H of two numbers is 4 and their arithmetic mean A and the geometric mean G satisfy the equation 2A G² = 27, then the numbers are 33. (a)(1, 3) (c) (6,3) (b)(9,5) (d) (12, 7)? covers all topics & solutions for CA Foundation 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The harmonic mean H of two numbers is 4 and their arithmetic mean A and the geometric mean G satisfy the equation 2A G² = 27, then the numbers are 33. (a)(1, 3) (c) (6,3) (b)(9,5) (d) (12, 7)?.
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