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If the arithmetic and geometric means of...

If the arithmetic and geometric means of two numbers are 10, 8 respectively, then one number exceeds the other number by

A

8

B

10

C

12

D

16

Text Solution

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The correct Answer is:
To solve the problem, we need to find the two numbers \( A \) and \( B \) given that their arithmetic mean is 10 and their geometric mean is 8. We will then determine how much one number exceeds the other. ### Step-by-Step Solution: 1. **Use the formula for Arithmetic Mean (AM)**: \[ \text{AM} = \frac{A + B}{2} \] Given that the arithmetic mean is 10, we can set up the equation: \[ \frac{A + B}{2} = 10 \] Multiplying both sides by 2 gives: \[ A + B = 20 \quad \text{(Equation 1)} \] 2. **Use the formula for Geometric Mean (GM)**: \[ \text{GM} = \sqrt{AB} \] Given that the geometric mean is 8, we can set up the equation: \[ \sqrt{AB} = 8 \] Squaring both sides gives: \[ AB = 64 \quad \text{(Equation 2)} \] 3. **Substitute Equation 1 into Equation 2**: From Equation 1, we can express \( B \) in terms of \( A \): \[ B = 20 - A \] Now substitute \( B \) into Equation 2: \[ A(20 - A) = 64 \] Expanding this gives: \[ 20A - A^2 = 64 \] Rearranging the equation results in: \[ A^2 - 20A + 64 = 0 \] 4. **Solve the quadratic equation**: We can use the quadratic formula \( A = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \): Here, \( a = 1, b = -20, c = 64 \). \[ A = \frac{20 \pm \sqrt{(-20)^2 - 4 \cdot 1 \cdot 64}}{2 \cdot 1} \] \[ A = \frac{20 \pm \sqrt{400 - 256}}{2} \] \[ A = \frac{20 \pm \sqrt{144}}{2} \] \[ A = \frac{20 \pm 12}{2} \] This gives us two possible values for \( A \): \[ A = \frac{32}{2} = 16 \quad \text{or} \quad A = \frac{8}{2} = 4 \] 5. **Find the corresponding values of \( B \)**: If \( A = 16 \): \[ B = 20 - 16 = 4 \] If \( A = 4 \): \[ B = 20 - 4 = 16 \] 6. **Determine how much one number exceeds the other**: The difference between the two numbers is: \[ |A - B| = |16 - 4| = 12 \] ### Final Answer: One number exceeds the other by **12**.

To solve the problem, we need to find the two numbers \( A \) and \( B \) given that their arithmetic mean is 10 and their geometric mean is 8. We will then determine how much one number exceeds the other. ### Step-by-Step Solution: 1. **Use the formula for Arithmetic Mean (AM)**: \[ \text{AM} = \frac{A + B}{2} \] ...
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