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# Largest value

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Manager
Joined: 29 Aug 2007
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Largest value [#permalink]  19 Sep 2007, 18:38
I came across this question and thought of posting it for Fraction lovers! or not

Which of the following represents the greatest value

1. Sqrt ( 2)/ Sqrt (3) + Sqrt (3)/ Sqrt(4) + Sqrt( 4)/ Sqrt(5) + Sqrt (5)/ Sqrt(6)

2. 2/3 + 4/5 + 3/4 + 4/5 + 5/6

3. (2^2 / 3 ^2)+ (3 ^ 2 / 4 ^2) + (4^2 / 5^2) + (5^2/6^2)

4. 1- (1/3) + (4/5) - (3/4)

5. 1- (3/4) + (4/5) + (1/3)
Director
Joined: 08 Jun 2007
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Re: Largest value [#permalink]  19 Sep 2007, 18:48
beatgmat wrote:
I came across this question and thought of posting it for Fraction lovers! or not

Which of the following represents the greatest value

1. Sqrt ( 2)/ Sqrt (3) + Sqrt (3)/ Sqrt(4) + Sqrt( 4)/ Sqrt(5) + Sqrt (5)/ Sqrt(6)

2. 2/3 + 4/5 + 3/4 + 4/5 + 5/6

3. (2^2 / 3 ^2)+ (3 ^ 2 / 4 ^2) + (4^2 / 5^2) + (5^2/6^2)

4. 1- (1/3) + (4/5) - (3/4)

5. 1- (3/4) + (4/5) + (1/3)

Its A.
Observe that all fractions are less than 1.

I guess the best way to solve this is start comparing the first two and start eliminating the lower.

So between A and B , A is bigger as its the square root of fractions. Between A and C , C is small because sqaures of fractions will be smaller.
Now B is bigger than D and E. And B is smaller than A .
So A .
Manager
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4: 1-1/3+4/5-3/4 = 4/5-1/12 = 43/60
5: 1-3/4+4/5+1/3 = 83/60. Therefore, 5>4 hence 4 is eliminated.

Since all fractions are less than 1, the square of the fraction is less than the fraction itself. Therefore, 3 is less than 2 and hence 3 is eliminated.

2 includes the fractions 5/6 (approximately 0.8) and 4/5 = 0.8. Therefore, 2 is greater than 1.6 and hence, 2>5, since 83/60 approx. 1.4

Between 1 and 2, 1 is greater as the sq.root of a function that is less than 1 is greater than the function itself.

Example: sqrroot(4/9) / (4/9) = 9sqrroot(4) / 4sqroot(9) = sqroot(9) / sqroot(4) >1
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