Log out Go to App
Go to App

Learning Materials

Features

Discover

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) \(-4\)

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

If \([1 /(\mathrm{b}-\mathrm{c})],[1 /(2 \mathrm{~b}-\mathrm{x})]\) and \([1 /(\mathrm{b}-\mathrm{a})]\) are in A. P., then \(\mathrm{a}-(\mathrm{x} / 2), \mathrm{b}-(\mathrm{x} / 2), \mathrm{c}-(\mathrm{x} / 2)\) are in (A) A. P. (B) G. P. (C) H. P. (D) A. G P.

If \(\mathrm{S}_{1}, \mathrm{~S}_{2}, \mathrm{~S}_{3}, \ldots \mathrm{S}_{\mathrm{n}}\) are the sums of infinite G. P.s. whose first terms are \(1,2,3, \ldots, \mathrm{n}\) and whose common ratios are \((1 / 2)\), \((1 / 3),(1 / 4), \ldots[1 /(\mathrm{n}+1)]\) respectively, then \(^{\mathrm{n}} \sum_{\mathrm{i}=1} \mathrm{~S}_{\mathrm{i}}=\) (A) \([\\{n(n+3)\\} / 2]\) (B) \([\\{n(n+4)\\} / 2]\) (C) \([\\{n(n-3)\\} / 2]\) (D) \([\\{n(n+1)\\} / 2]\)

If the \(\mathrm{H}\). M. of a and \(\mathrm{c}\) is \(\mathrm{b}\), G. \(\mathrm{M}\). of \(\mathrm{b}\) and \(\mathrm{d}\) is \(\mathrm{c}\) and \(\mathrm{A} . \mathrm{M} .\) of \(c\) and e is \(\mathrm{d}\), then the G. M. of a and e is (A) \(b\) (B) \(c\) (C) \(\mathrm{d}\) (D) ae

The sum of the series \(a-(a+d)+(a+2 d)-(a+3 d)+\ldots u p\) to 50 terms is \((\mathrm{A})-50 \mathrm{~d}\) (B) \(25 \mathrm{~d}\) (C) \(a+50 d\) (D) \(-25 \mathrm{~d}\)

\(\tan ^{-1}(1 / 3)+\tan ^{-1}(1 / 7)+\tan ^{-1}(1 / 13)+\ldots+\tan ^{-1}[1 /(9703)]\) (A) \(\begin{array}{lll}\text { (B) }(\pi / 6) & \text { (C) }(\pi / 3) & \text { (D) } \tan ^{-1}(0.98)\end{array}\)
See all solutions

Recommended explanations on English Textbooks

Essay Plan

Read Explanation

Phonology

Read Explanation

Synthesis Essay

Read Explanation

Email

Read Explanation

Sociolinguistics

Read Explanation

Language and Social Groups

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free