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If a and b are positive and
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Updated on: 25 Jul 2021, 21:41
Updated on: 25 Jul 2021, 21:41
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If a and b are positive and \(\frac{ab}{x}=\sqrt{a}\) then \(\frac{x}{b^2}=\)
A. \(\sqrt{a}\)
B. \(\frac{\sqrt{a}}{b}\)
C. \(\frac{a}{b^2}\)
D. \(\sqrt{ab}\)
E. \(\sqrt{\frac{a}{b}}\)
Kudos for the right answer and explanation
A. \(\sqrt{a}\)
B. \(\frac{\sqrt{a}}{b}\)
C. \(\frac{a}{b^2}\)
D. \(\sqrt{ab}\)
E. \(\sqrt{\frac{a}{b}}\)
Kudos for the right answer and explanation
Retired Moderator
Joined: 10 Apr 2015
Posts: 6218
Given Kudos: 136
Re: If a and b are positive and
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27 Jan 2020, 16:21
27 Jan 2020, 16:21
1
Carcass wrote:
If a and b are positive and \(\frac{ab}{x}=\sqrt{a}\) then \(\frac{x}{b^2}=\)
A. \(\sqrt{a}\)
B. \(\frac{\sqrt{a}}{b}\)
C. \(\frac{a}{b^2}\)
D. \(\sqrt{ab}\)
E. \(\sqrt{\frac{a}{b}}\)
Kudos for the right answer and explanation
A. \(\sqrt{a}\)
B. \(\frac{\sqrt{a}}{b}\)
C. \(\frac{a}{b^2}\)
D. \(\sqrt{ab}\)
E. \(\sqrt{\frac{a}{b}}\)
Kudos for the right answer and explanation
Rather than attempt a lot of algebraic manipulations, we can simply find values of a, b and x that satisfy the given equation and then test the answer choices...
If \(\frac{ab}{x}=\sqrt{a}\), then it COULD be the case that a = 4, b = 3 and x = 6, since these values satisfy the given equation.
The question asked us to find the value of \(\frac{x}{b^2}\)
So, plug a = 4, b = 3 and x = 6 into the expression to get: \(\frac{x}{b^2}=\frac{6}{3^2}=\frac{6}{9}=\frac{2}{3}\)
So, when a = 4, b = 3 and x = 6, \(\frac{x}{b^2}=\frac{2}{3}\)
This means the correct answer will be the one that evaluates to equal 2/3 when we plug in a = 4, b = 3 and x = 6....
A. \(\sqrt{4}\) = 2. NO GOOD. We want the expression to evaluate to be 2/3
B. \(\frac{\sqrt{4}}{3}\) = 2/3. Perfect!
C. \(\frac{4}{3^2}\) = 4/9. NO GOOD. We want the expression to evaluate to be 2/3
D. \(\sqrt{(4)(3)}\) = √12. NO GOOD. We want the expression to evaluate to be 2/3
E. \(\sqrt{\frac{4}{3}}\) = 2/√3. NO GOOD. We want the expression to evaluate to be 2/3
Answer: B
Cheers,
Brent
Retired Moderator
Joined: 10 Apr 2015
Posts: 6218
Given Kudos: 136
Re: If a and b are positive and
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27 Jan 2020, 16:36
27 Jan 2020, 16:36
1
Carcass wrote:
If a and b are positive and \(\frac{ab}{x}=\sqrt{a}\) then \(\frac{x}{b^2}=\)
A. \(\sqrt{a}\)
B. \(\frac{\sqrt{a}}{b}\)
C. \(\frac{a}{b^2}\)
D. \(\sqrt{ab}\)
E. \(\sqrt{\frac{a}{b}}\)
Kudos for the right answer and explanation
A. \(\sqrt{a}\)
B. \(\frac{\sqrt{a}}{b}\)
C. \(\frac{a}{b^2}\)
D. \(\sqrt{ab}\)
E. \(\sqrt{\frac{a}{b}}\)
Kudos for the right answer and explanation
Alternate solution....
Given: \(\frac{ab}{x}=\sqrt{a}\)
Square both sides to get: \(\frac{a^2b^2}{x^2}=a\)
Divide both sides by \(a^2\) to get: \(\frac{b^2}{x^2}=\frac{a}{a^2}\)
Simplified to get: \(\frac{b^2}{x^2}=\frac{1}{a}\)
Multiply both sides by \(x\) to get: \(\frac{b^2}{x}=\frac{x}{a}\)
Flip both fractions to get: \(\frac{x}{b^2}=\frac{a}{x}\)
NOTE: Since none of the answer choices include the variable x, we still need to replace the x (on the right side of the equation) with an equivalent expression that equals x
So at this point we're going to solve the original equation for x.
Take: \(\frac{ab}{x}=\sqrt{a}\)
Multiply both sides of the equation by x to get: \(ab=x\sqrt{a}\)
Divide both sides of the equation by \(\sqrt{a}\) to get: \(\frac{ab}{\sqrt{a}}=x\)
Simplify the left side of the equation to get: \(b\sqrt{a}=x\)
Great! We've now found an expression that is equivalent to x
So, take: \(\frac{x}{b^2}=\frac{a}{x}\)
And replace x with \(b\sqrt{a}\) to get: \(\frac{x}{b^2}=\frac{a}{b\sqrt{a}}\)
Simplify to get: \(\frac{x}{b^2}=\frac{\sqrt{a}}{b}\)
Answer: B
