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If the harmonic mean of two numbers is to their Geometric mean as 24 to 25. Find the ratio of numbers.
- a)2/3, 3/2
- b)4/3, 3/4
- c)4/9, 9/4
- d)16/9, 9/16
Correct answer is option 'D'. Can you explain this answer?
Verified Answer
If the harmonic mean of two numbers is to their Geometric mean as 24 t...
Let the two numbers be a & b then
2ab / (a + b) : √(a + b) = 24 : 25 or 12 (a + b) = 25√(ab)
Dividing by b we have,
12 (a/b + 1) = 25 √(a/b)
put X = √(a/b)
We have 12X2 – 25X + 12 = 0 or
12 x2 – 16 x – 9x + 12 = 0
Solve & get
X = 3/4, 4/3
or √(a/b) = 4/ 3 or 3 /4 or b/ a = 16/ 9 or 9/ 16.
2ab / (a + b) : √(a + b) = 24 : 25 or 12 (a + b) = 25√(ab)
Dividing by b we have,
12 (a/b + 1) = 25 √(a/b)
put X = √(a/b)
We have 12X2 – 25X + 12 = 0 or
12 x2 – 16 x – 9x + 12 = 0
Solve & get
X = 3/4, 4/3
or √(a/b) = 4/ 3 or 3 /4 or b/ a = 16/ 9 or 9/ 16.
Most Upvoted Answer
If the harmonic mean of two numbers is to their Geometric mean as 24 t...
To find the ratio of the numbers, let's assume the two numbers as 'a' and 'b'.
Let's break down the given information step by step:
1. Harmonic Mean:
The harmonic mean of two numbers is given by the formula:
Harmonic Mean = 2ab / (a + b)
2. Geometric Mean:
The geometric mean of two numbers is given by the formula:
Geometric Mean = √(ab)
3. Given Ratio:
The ratio of the harmonic mean to the geometric mean is given as 24:25.
To solve this problem, we can set up the following equation:
(2ab / (a + b)) / (√(ab)) = 24/25
Simplifying this equation, we get:
2ab / (a + b) = 24/25 * √(ab)
Rearranging the equation, we get:
(2ab / (a + b)) * (25/24) = √(ab)
Simplifying further, we get:
(50ab / (a + b)) = √(ab)
Squaring both sides of the equation, we get:
(50ab / (a + b))^2 = (√(ab))^2
Simplifying again, we get:
(50ab)^2 / (a + b)^2 = ab
Expanding and rearranging the equation, we get:
2500a^2b^2 = a^2b + 2ab^2 + b^2a
Simplifying further, we get:
2500a^2b^2 = a^2b + 2ab^2 + ab^2
Combining like terms, we get:
2500a^2b^2 = 2a^2b + 3ab^2
Dividing both sides of the equation by ab, we get:
2500ab = 2a + 3b
Dividing both sides of the equation by b, we get:
2500a = 2a/b + 3
Rearranging the equation, we get:
2a/b = 2500a - 3
Dividing both sides of the equation by 2a, we get:
1/b = 2500 - 3/(2a)
Simplifying further, we get:
1/b = (5000 - 3)/(2a)
Cross multiplying, we get:
b = 2a/(5000 - 3)
Therefore, the ratio of the numbers is 2a: (5000 - 3).
Comparing this with the answer options, we find that option D, 16/9: 9/16, matches our solution.
Hence, the correct answer is option D, 16/9: 9/16.
Let's break down the given information step by step:
1. Harmonic Mean:
The harmonic mean of two numbers is given by the formula:
Harmonic Mean = 2ab / (a + b)
2. Geometric Mean:
The geometric mean of two numbers is given by the formula:
Geometric Mean = √(ab)
3. Given Ratio:
The ratio of the harmonic mean to the geometric mean is given as 24:25.
To solve this problem, we can set up the following equation:
(2ab / (a + b)) / (√(ab)) = 24/25
Simplifying this equation, we get:
2ab / (a + b) = 24/25 * √(ab)
Rearranging the equation, we get:
(2ab / (a + b)) * (25/24) = √(ab)
Simplifying further, we get:
(50ab / (a + b)) = √(ab)
Squaring both sides of the equation, we get:
(50ab / (a + b))^2 = (√(ab))^2
Simplifying again, we get:
(50ab)^2 / (a + b)^2 = ab
Expanding and rearranging the equation, we get:
2500a^2b^2 = a^2b + 2ab^2 + b^2a
Simplifying further, we get:
2500a^2b^2 = a^2b + 2ab^2 + ab^2
Combining like terms, we get:
2500a^2b^2 = 2a^2b + 3ab^2
Dividing both sides of the equation by ab, we get:
2500ab = 2a + 3b
Dividing both sides of the equation by b, we get:
2500a = 2a/b + 3
Rearranging the equation, we get:
2a/b = 2500a - 3
Dividing both sides of the equation by 2a, we get:
1/b = 2500 - 3/(2a)
Simplifying further, we get:
1/b = (5000 - 3)/(2a)
Cross multiplying, we get:
b = 2a/(5000 - 3)
Therefore, the ratio of the numbers is 2a: (5000 - 3).
Comparing this with the answer options, we find that option D, 16/9: 9/16, matches our solution.
Hence, the correct answer is option D, 16/9: 9/16.
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If the harmonic mean of two numbers is to their Geometric mean as 24 to 25. Find the ratio of numbers.a)2/3, 3/2 b)4/3, 3/4 c)4/9, 9/4 d)16/9, 9/16Correct answer is option 'D'. Can you explain this answer? for CAT 2025 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about If the harmonic mean of two numbers is to their Geometric mean as 24 to 25. Find the ratio of numbers.a)2/3, 3/2 b)4/3, 3/4 c)4/9, 9/4 d)16/9, 9/16Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for CAT 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If the harmonic mean of two numbers is to their Geometric mean as 24 to 25. Find the ratio of numbers.a)2/3, 3/2 b)4/3, 3/4 c)4/9, 9/4 d)16/9, 9/16Correct answer is option 'D'. Can you explain this answer?.
If the harmonic mean of two numbers is to their Geometric mean as 24 to 25. Find the ratio of numbers.a)2/3, 3/2 b)4/3, 3/4 c)4/9, 9/4 d)16/9, 9/16Correct answer is option 'D'. Can you explain this answer? for CAT 2025 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about If the harmonic mean of two numbers is to their Geometric mean as 24 to 25. Find the ratio of numbers.a)2/3, 3/2 b)4/3, 3/4 c)4/9, 9/4 d)16/9, 9/16Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for CAT 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If the harmonic mean of two numbers is to their Geometric mean as 24 to 25. Find the ratio of numbers.a)2/3, 3/2 b)4/3, 3/4 c)4/9, 9/4 d)16/9, 9/16Correct answer is option 'D'. Can you explain this answer?.
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