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The arithmetic mean of two numbers is smaller by 24 than the larger of the two numbers and the GM of the same numbers exceeds by 12 the smaller of the numbers. Find the numbers.
- a)6 and 54
- b)8 and 56
- c)12 and 60
- d)7 and 55
Correct answer is option 'A'. Can you explain this answer?
Most Upvoted Answer
The arithmetic mean of two numbers is smaller by 24 than the larger of...
If a and b are two numbers, then their Arithmetic mean is given by (a + b)/2 while their geometric mean is given by (ab)0.5.
Using the options to meet the conditions we can see that for the numbers in the first option (6 and 54) the AM being 30, is 24 less than the larger number while the GM being 18, is 12 more than the smaller number.
Option (a) is correct.
Using the options to meet the conditions we can see that for the numbers in the first option (6 and 54) the AM being 30, is 24 less than the larger number while the GM being 18, is 12 more than the smaller number.
Option (a) is correct.
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The arithmetic mean of two numbers is smaller by 24 than the larger of...
To solve this problem, let's assume the two numbers as x and y.
Given:
1) The arithmetic mean of the two numbers is smaller by 24 than the larger number.
This can be expressed as:
(x + y) / 2 = y - 24
2) The geometric mean of the two numbers exceeds by 12 the smaller number.
This can be expressed as:
√(xy) = x + 12
We need to solve these two equations to find the values of x and y.
Solving the first equation:
(x + y) / 2 = y - 24
x + y = 2y - 48
x = y - 48
Substituting this value of x in the second equation:
√(xy) = x + 12
√(y(y - 48)) = y - 48 + 12
√(y^2 - 48y) = y - 36
Squaring both sides of the equation:
y^2 - 48y = (y - 36)^2
y^2 - 48y = y^2 - 72y + 1296
-48y = -72y + 1296
24y = 1296
y = 1296 / 24
y = 54
Substituting this value of y back into the first equation:
x = y - 48
x = 54 - 48
x = 6
Therefore, the two numbers are 6 and 54, which matches with option A.
Given:
1) The arithmetic mean of the two numbers is smaller by 24 than the larger number.
This can be expressed as:
(x + y) / 2 = y - 24
2) The geometric mean of the two numbers exceeds by 12 the smaller number.
This can be expressed as:
√(xy) = x + 12
We need to solve these two equations to find the values of x and y.
Solving the first equation:
(x + y) / 2 = y - 24
x + y = 2y - 48
x = y - 48
Substituting this value of x in the second equation:
√(xy) = x + 12
√(y(y - 48)) = y - 48 + 12
√(y^2 - 48y) = y - 36
Squaring both sides of the equation:
y^2 - 48y = (y - 36)^2
y^2 - 48y = y^2 - 72y + 1296
-48y = -72y + 1296
24y = 1296
y = 1296 / 24
y = 54
Substituting this value of y back into the first equation:
x = y - 48
x = 54 - 48
x = 6
Therefore, the two numbers are 6 and 54, which matches with option A.
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The arithmetic mean of two numbers is smaller by 24 than the larger of the two numbers and the GM of the same numbers exceeds by 12 the smaller of the numbers. Find the numbers.a)6 and 54b)8 and 56c)12 and 60d)7 and 55Correct answer is option 'A'. 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 The arithmetic mean of two numbers is smaller by 24 than the larger of the two numbers and the GM of the same numbers exceeds by 12 the smaller of the numbers. Find the numbers.a)6 and 54b)8 and 56c)12 and 60d)7 and 55Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for CAT 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The arithmetic mean of two numbers is smaller by 24 than the larger of the two numbers and the GM of the same numbers exceeds by 12 the smaller of the numbers. Find the numbers.a)6 and 54b)8 and 56c)12 and 60d)7 and 55Correct answer is option 'A'. Can you explain this answer?.
The arithmetic mean of two numbers is smaller by 24 than the larger of the two numbers and the GM of the same numbers exceeds by 12 the smaller of the numbers. Find the numbers.a)6 and 54b)8 and 56c)12 and 60d)7 and 55Correct answer is option 'A'. 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 The arithmetic mean of two numbers is smaller by 24 than the larger of the two numbers and the GM of the same numbers exceeds by 12 the smaller of the numbers. Find the numbers.a)6 and 54b)8 and 56c)12 and 60d)7 and 55Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for CAT 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The arithmetic mean of two numbers is smaller by 24 than the larger of the two numbers and the GM of the same numbers exceeds by 12 the smaller of the numbers. Find the numbers.a)6 and 54b)8 and 56c)12 and 60d)7 and 55Correct answer is option 'A'. Can you explain this answer?.
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