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The harmonic mean h of two numbers is four and their arithmetic mean a and the geometric mean g satisfy the equation to2 A + G square is equal to 27 then the numbers are?
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The harmonic mean h of two numbers is four and their arithmetic mean a...
Understanding the Problem
To find the two numbers given the harmonic mean (h), arithmetic mean (a), and geometric mean (g), we have the following information:
- Harmonic mean (h) = 4
- The equation relating a and g: 2a + g^2 = 27
Formulas for Means
- Harmonic Mean (h) of two numbers x and y is given by:
h = 2xy / (x + y)
- Arithmetic Mean (a) is:
a = (x + y) / 2
- Geometric Mean (g) is:
g = √(xy)
Using the Harmonic Mean
From the harmonic mean, we can express the relationship:
2xy / (x + y) = 4
This leads to:
xy = 2(x + y)
Expressing Relationships
Let:
- x + y = S (sum)
- xy = P (product)
Thus, we rewrite the harmonic mean relationship as:
P = 2S
Now substituting in the equation for arithmetic mean:
a = S / 2.
Using the Given Equation
Substituting into the equation 2a + g^2 = 27:
2(S / 2) + (√P)² = 27
This simplifies to:
S + P = 27.
Solving the Equations
We now have two equations:
1. P = 2S
2. S + P = 27
Substituting P from the first equation into the second:
S + 2S = 27
3S = 27
S = 9, thus P = 2 * 9 = 18.
Finding the Numbers
Now we have:
- S = 9
- P = 18
The two numbers are the roots of the equation:
t² - St + P = 0
t² - 9t + 18 = 0.
Solving this quadratic gives:
t = (9 ± √(9² - 4 * 18)) / 2
t = (9 ± √(81 - 72)) / 2
t = (9 ± 3) / 2.
Thus, the numbers are:
- t = 6
- t = 3.
Final Result
The two numbers are 6 and 3.
To find the two numbers given the harmonic mean (h), arithmetic mean (a), and geometric mean (g), we have the following information:
- Harmonic mean (h) = 4
- The equation relating a and g: 2a + g^2 = 27
Formulas for Means
- Harmonic Mean (h) of two numbers x and y is given by:
h = 2xy / (x + y)
- Arithmetic Mean (a) is:
a = (x + y) / 2
- Geometric Mean (g) is:
g = √(xy)
Using the Harmonic Mean
From the harmonic mean, we can express the relationship:
2xy / (x + y) = 4
This leads to:
xy = 2(x + y)
Expressing Relationships
Let:
- x + y = S (sum)
- xy = P (product)
Thus, we rewrite the harmonic mean relationship as:
P = 2S
Now substituting in the equation for arithmetic mean:
a = S / 2.
Using the Given Equation
Substituting into the equation 2a + g^2 = 27:
2(S / 2) + (√P)² = 27
This simplifies to:
S + P = 27.
Solving the Equations
We now have two equations:
1. P = 2S
2. S + P = 27
Substituting P from the first equation into the second:
S + 2S = 27
3S = 27
S = 9, thus P = 2 * 9 = 18.
Finding the Numbers
Now we have:
- S = 9
- P = 18
The two numbers are the roots of the equation:
t² - St + P = 0
t² - 9t + 18 = 0.
Solving this quadratic gives:
t = (9 ± √(9² - 4 * 18)) / 2
t = (9 ± √(81 - 72)) / 2
t = (9 ± 3) / 2.
Thus, the numbers are:
- t = 6
- t = 3.
Final Result
The two numbers are 6 and 3.
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Question Description
The harmonic mean h of two numbers is four and their arithmetic mean a and the geometric mean g satisfy the equation to2 A + G square is equal to 27 then the numbers are? for CA Foundation 2025 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 four and their arithmetic mean a and the geometric mean g satisfy the equation to2 A + G square is equal to 27 then the numbers are? covers all topics & solutions for CA Foundation 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The harmonic mean h of two numbers is four and their arithmetic mean a and the geometric mean g satisfy the equation to2 A + G square is equal to 27 then the numbers are?.
The harmonic mean h of two numbers is four and their arithmetic mean a and the geometric mean g satisfy the equation to2 A + G square is equal to 27 then the numbers are? for CA Foundation 2025 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 four and their arithmetic mean a and the geometric mean g satisfy the equation to2 A + G square is equal to 27 then the numbers are? covers all topics & solutions for CA Foundation 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The harmonic mean h of two numbers is four and their arithmetic mean a and the geometric mean g satisfy the equation to2 A + G square is equal to 27 then the numbers are?.
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