Which of the following is the value of 1/(2^(1/2) + 1)

Question

Which of the following is the value of  \frac{1}{\sqrt{2} + 1} + \frac{1}{\sqrt{3} + \sqrt{2} } + \frac{1}{2 + \sqrt{3} } ?

A.  \frac{1}{3 + 2\sqrt{2} + 2\sqrt{3} }

B.  \frac{3}{3 + 2\sqrt{2} + 2\sqrt{3} }

C. 1/3

D. 3/7

E. 1

chetan2u · Accepted Answer

The square root in the denominator should get you thinking on how to remove them.

\frac{1}{\sqrt{2} + 1} + \frac{1}{\sqrt{3} + \sqrt{2} } + \frac{1}{2 + \sqrt{3} }

\frac{1*(\sqrt{2} - 1)}{(\sqrt{2} +1)(\sqrt{2} - 1)} + \frac{1*(\sqrt{3} - \sqrt{2} )}{(\sqrt{3} + \sqrt{2}) (\sqrt{3} - \sqrt{2} ) } + \frac{1 (2 - \sqrt{3}) }{(2 + \sqrt{3}) (2 - \sqrt{3}) }

\frac{(\sqrt{2} - 1)}{2-1} + \frac{(\sqrt{3} - \sqrt{2} )}{3-2 } + \frac{(2 - \sqrt{3}) }{4-3 }

{(\sqrt{2} - 1)}+ {(\sqrt{3} - \sqrt{2} )}+ {(2 - \sqrt{3}) }

2-1

1

E

You could also guess
 \frac{1}{\sqrt{2} + 1} + \frac{1}{\sqrt{3} + \sqrt{2} } + \frac{1}{2 + \sqrt{3} }

\frac{1}{1.4 + 1} + \frac{1}{1.7 + 1.4} + \frac{1}{2+1.7}

\frac{1}{2.4 } + \frac{1}{3.1} + \frac{1}{3.7} 
The first fraction is more than 1/3 just less than 1/2, second term is around 1/3. Just these two combined should give you answer greater than 2/3.
Only E is left.

JeffTargetTestPrep · Answer

We can make the denominators “disappear” if we remove the radicals from the denominators.

How can we remove those radicals without changing the values of the fractions? By multiplying each fraction by a corresponding fraction that is equal to one:

(1/ ) × (√2–1)/(√2–1) = (√2–1)/(2–1) = √2 – 1

(1/ ) × (√3–√2)/(√3–√2) = (√3–√2)/(3–2) = √3 – √2

(1/ ) × (2–√3)/(2–√3) = (2–√3)/(4–3) = 2 – √3

So, we have:

√2 – 1 + √3 – √2 + 2 – √3 = 1

Answer: E

MonishBhawale · Answer

componendo dividendo does the trick here

Rohanx9 · Answer

The quickest way IMO is to approximate, especially if you think multiple manual calculations of rationalising denominators could lead to errors. The only background know is the square root of 2 and square root of 3 are approx. 1.4 and 1.7.

First term =  \frac{1}{(1.4+1)} = \frac{1}{2.4} = \frac{1}{(5/2) }= \frac{2}{5}

Second term =  \frac{1}{(1.7+1.4)} = \frac{1}{3.1} = \frac{1}{3}

Third term =  \frac{1}{(2+1.7)} = \frac{1}{3.7} = \frac{1}{4}

2/5+1/3+1/4 =59/60 =0.983 pretty close to 1. 
There aren't any options that are remotely close to 1 so E must be the answer.

Aboyhasnoname · Answer

I think the most efficient approach would be to infer....

You see... 1/2 is 0.5

Now there are three terms here... 1/ 1.xx +1 .. around 1/2.xx ..Next is 1/ 1.xx +1.yy ..Same around 1/2.xx ....last term 1/2+ 1.xxx ...around 1/3.xxx 
So we can get an estimate that the Answer should be very close to 1......

A.  \frac{1}{3 + 2\sqrt{2} + 2\sqrt{3} }

This term is very small..Eliminate

B.
 \frac{3}{3 + 2\sqrt{2} + 2\sqrt{3} } 
This will be also very small as 3/7.xxx ..So very small

C. 1/3
Clearly smaller than 1

D. 
3/7 also

E. 
1.

This would be my answer...

Bunuel

Which of the following is the value of 1/(2^(1/2) + 1)

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GMAT Problem Solving (PS) Questions

Which of the following is the value of 1/(2^(1/2) + 1)

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