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Find the number whose arithmetic mean is 12.5 and geometric mean is 10.
- a)20 and 5
- b)10 and 5
- c)5 and 4
- d)None of these
Correct answer is option 'A'. Can you explain this answer?
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Most Upvoted Answer
Find the number whose arithmetic mean is 12.5 and geometric mean is 10...
Arithmetic Mean and Geometric Mean
Arithmetic Mean:
The arithmetic mean is the sum of a set of numbers divided by the number of items in the set. It is also known as the average of the set. Formula for arithmetic mean is:
Arithmetic Mean = (Sum of numbers) / (Number of numbers)
Geometric Mean:
The geometric mean is the nth root of the product of n numbers. Formula for geometric mean is:
Geometric Mean = (Product of numbers)^(1/n)
Solution:
Let the two numbers be a and b. Then,
Arithmetic Mean = (a + b)/2 = 12.5
Geometric Mean = √(a * b) = 10
Squaring both sides of the equation for geometric mean, we get:
(a * b) = 100
Substituting the value of (a + b) from the equation for arithmetic mean, we get:
(a + b)^2 = 4 * (a * b) + 4 * (12.5)^2
Simplifying this equation, we get:
a^2 + 2ab + b^2 = 4ab + 4 * 156.25
a^2 - 2ab + b^2 = 4 * 156.25
(a - b)^2 = 625
(a - b) = ± 25
Since a + b = 25, we can solve for a and b using simultaneous equations:
a + b = 25
a * b = 100
Solving for a and b, we get:
a = 20, b = 5 or a = 5, b = 20
Therefore, the numbers are 20 and 5, and their arithmetic mean is 12.5 and geometric mean is 10.
Hence, the correct answer is option A – 20 and 5.
Arithmetic Mean:
The arithmetic mean is the sum of a set of numbers divided by the number of items in the set. It is also known as the average of the set. Formula for arithmetic mean is:
Arithmetic Mean = (Sum of numbers) / (Number of numbers)
Geometric Mean:
The geometric mean is the nth root of the product of n numbers. Formula for geometric mean is:
Geometric Mean = (Product of numbers)^(1/n)
Solution:
Let the two numbers be a and b. Then,
Arithmetic Mean = (a + b)/2 = 12.5
Geometric Mean = √(a * b) = 10
Squaring both sides of the equation for geometric mean, we get:
(a * b) = 100
Substituting the value of (a + b) from the equation for arithmetic mean, we get:
(a + b)^2 = 4 * (a * b) + 4 * (12.5)^2
Simplifying this equation, we get:
a^2 + 2ab + b^2 = 4ab + 4 * 156.25
a^2 - 2ab + b^2 = 4 * 156.25
(a - b)^2 = 625
(a - b) = ± 25
Since a + b = 25, we can solve for a and b using simultaneous equations:
a + b = 25
a * b = 100
Solving for a and b, we get:
a = 20, b = 5 or a = 5, b = 20
Therefore, the numbers are 20 and 5, and their arithmetic mean is 12.5 and geometric mean is 10.
Hence, the correct answer is option A – 20 and 5.
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Community Answer
Find the number whose arithmetic mean is 12.5 and geometric mean is 10...
Option a is correct
a+b/2=12•5 then a+b=25
√ab =10 then ab=100
hence 20+5=25
20×5=100
a+b/2=12•5 then a+b=25
√ab =10 then ab=100
hence 20+5=25
20×5=100
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Find the number whose arithmetic mean is 12.5 and geometric mean is 10.a)20 and 5b)10 and 5c)5 and 4d)None of theseCorrect answer is option 'A'. Can you explain this answer?
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