# The arithmetic mean of two numbers is 10 and their geometric mean is 8. What are the two numbers ?

A

15, 5

B

12, 8

C

16, 4

D

18, 2

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## To solve the problem, we need to find two numbers based on their arithmetic mean and geometric mean.1. Understanding the Means: - The arithmetic mean (AM) of two numbers A and B is given by the formula: AM=A+B2 - According to the problem, the arithmetic mean is 10: A+B2=102. Setting Up the Equation for AM: - Multiply both sides of the equation by 2 to eliminate the fraction: A+B=20(Equation 1)3. Geometric Mean (GM): - The geometric mean of two numbers A and B is given by: GM=√A⋅B - According to the problem, the geometric mean is 8: √A⋅B=84. Setting Up the Equation for GM: - Square both sides to eliminate the square root: A⋅B=64(Equation 2)5. Substituting Equation 1 into Equation 2: - From Equation 1, we can express B in terms of A: B=20−A - Substitute B in Equation 2: A⋅(20−A)=646. Expanding the Equation: - Distributing A: 20A−A2=64 - Rearranging gives us a standard quadratic equation: A2−20A+64=07. Solving the Quadratic Equation: - We can use the quadratic formula A=−b±√b2−4ac2a: - Here, a=1, b=−20, and c=64. - Calculate the discriminant: b2−4ac=(−20)2−4⋅1⋅64=400−256=144 - Now, applying the quadratic formula: A=20±√1442=20±122 - This gives us two potential solutions: A=322=16andA=82=48. Finding Corresponding Values of B: - If A=16: B=20−16=4 - If A=4: B=20−4=169. Conclusion: - The two numbers are 16 and 4.Final Answer:The two numbers are 16 and 4.

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Updated on:8/8/2024

### Knowledge Check

• Question 1 - Select One

## The arithmetic mean between two numbers is A and the geometric mean is G. Then these numbers are:

AA2G2±A
B12(A2G2+A)
CA±A2G2
D2(A±A2G2)
• Question 2 - Select One

## If the arithmetic mean of two numbers is 7 and the geometric mean of the same two numbers is 2√10. Then find the numbers x and y respectively, such that x>y.

A4, 10
B2, 5
C5, 2
D10, 4
• Question 3 - Select One

## If the arithmetic mean of two numbers is 5 and geometric mean is 4 then the numbers are

A4,6
B4,7
C3,8
D2,8

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