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Chapter 8: Problem 677

If A is the A. M. between a and b, then \([(A-2 b) /(A-a)]+[(A-2 a) /(A-b)]=\) (A) \(-8\) (B) 2 (C) 4 (D) \(-4\)

Short Answer

Expert verified
The simplified expression is \(-4\), so the correct option is (D) \(-4\).

Step by step solution

01

Define A.M of a and b

The arithmetic mean (A) of two numbers a and b is given by the formula: \(A = \frac{a + b}{2}\) We will use this formula to help simplify the given expression.
02

Write the given expression with defined A.M

We are given the expression: \(\frac{(A - 2b)}{(A - a)} + \frac{(A - 2a)}{(A - b)}\) Substitute the defined A.M (A) from Step 1: \(\frac{(\frac{a+b}{2} - 2b)}{(\frac{a+b}{2} - a)} + \frac{(\frac{a+b}{2} - 2a)}{(\frac{a+b}{2} - b)}\)
03

Simplify the expression

Now, we will simplify the expression by combining the terms in the numerators and denominators. \(\frac{(a - 3b)}{(b - a)} + \frac{(b - 3a)}{(a - b)}\)
04

Make the denominators same for both fractions

In order to add or subtract fractions, we need to have the same denominator. We can do this by multiplying the first fraction by \(-1\), to make the denominators the same. \(-1 \times \frac{(a - 3b)}{(b - a)} = \frac{(3b - a)}{(a - b)}\) Now we rewrite the expression with the same denominators for both fractions: \(\frac{(3b - a)}{(a - b)} + \frac{(b - 3a)}{(a - b)}\)
05

Add the fractions and simplify

Now that the denominators are the same, we can add the fractions: \(\frac{(3b - a) + ( b - 3a)}{(a - b)}\) Combine the terms in the numerator: \(\frac{(4b - 4a)}{(a - b)}\) Factor out common factor "4" from the numerator: \(4\cdot\frac{(b - a)}{(a - b)}\)
06

Simplify the expression further

Observe that we can rewrite the denominator as \(-(b - a)\). Then we can simplify the expression: \(4\cdot\frac{(b - a)}{-(b - a)}\) Now, cancel out the common terms in the numerator and denominator: \(4\cdot\frac{1}{-1}\) Finally, simplify: \(-4\) Hence, the correct option is: (D) \(-4\)

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Most popular questions from this chapter

If the harmonic mean and geometric mean of two numbers a and \(b\) are 4 and \(3 \sqrt{2}\) respectively then the interval \([a, b]=\) (A) \([3,6]\) (B) \([2,7]\) (C) \([4,5]\) (D) \([1,8]\)

If for the triangle whose perimeter is \(37 \mathrm{cms}\) and length of sides are in G. P. also the length of the smallest side is \(9 \mathrm{cms}\) then length of remaining two sides are and (A) 12,16 (B) 14,14 (C) 10,18 (D) 15,13

If \((1 / a),(1 / H),(1 / b)\) are in A. P. then \([(H+a) /(H-a)]\) \(+[(\mathrm{H}+\mathrm{b}) /(\mathrm{H}-\mathrm{b})]=\) (A) 2 (B) 4 (C) 0 (D) 1

If \([(b+c-a) / a],[(c+a-b) / b],[(a+b-c) / c]\) are in A. \(P\) then \(\mathrm{a}, \mathrm{b}, \mathrm{c}\) are in (A) G P. (B) A. P. (C) H. P. (D) A. G P.

If \(\mathrm{a}_{1}, \mathrm{a}_{2}, \ldots \mathrm{a}_{10}\) be in A. P., \(\left(1 / \mathrm{h}_{1}\right),\left(1 / \mathrm{h}_{2}\right) \ldots\left[1 / \mathrm{h}_{10}\right)\) be in A. \(P\) and \(a_{1}=h_{1}=2, a_{10}=h_{10}=3\) then \(a_{4} h_{7}=\) (A) \((1 / 6)\) (B) 6 (C) 3 (D) 2
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