If A.M. and G.M. of two positive numbers a and b are 10 and 8, respectively, find the numbers.
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Step-by-Step Solution
Step 1: Formulate equations from given information
The problem states that the arithmetic mean (A.M.) of two positive numbers a and b is 10, and their geometric mean (G.M.) is 8. We can write these as two equations. The formula for A.M. is (a+b)/2 and for G.M. is ab.
Step 2: Simplify the equations
From equation (1), we multiply both sides by 2 to get a+b=20. From equation (2), we square both sides to eliminate the square root, resulting in ab=64. These simplified equations will be easier to work with.
Step 3: Express one variable in terms of the other
From equation (3), we can express b in terms of a. This allows us to substitute b into the other equation, reducing the system to a single variable.
Step 4: Substitute and form a quadratic equation
Substitute the expression for b from equation (5) into equation (4). This will give us a quadratic equation in terms of a. Rearranging the terms, we get a2−20a+64=0.
Step 5: Solve the quadratic equation for 'a'
We can solve this quadratic equation by factoring. We need two numbers that multiply to 64 and add up to -20. These numbers are -4 and -16. So, the equation factors into (a−4)(a−16)=0, giving us two possible values for a: 4 or 16.
Step 6: Find the corresponding values for 'b'
Now, we use equation (5), b=20−a, to find the corresponding values for b. If a=4, then b=16. If a=16, then b=4. Both pairs satisfy the original conditions.