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How do you compare 1/23 + 1/24 + 1/25 + 1/26 + 1/27 to 1/5 without long calculation?

1/5 = 5/25

1/23 + 1/24 > 2/25

1/26 + 1/27 < 2/25

???

you dont need to do complex calculation. Just compare

A = 1/23-1/25 and B = 1/25-1/27 A = 2/(23*25) and B = 2/(25*27)

clearly A > B

So ((1/23 + 1/24) - 2/25 ) is more than (2/25 - (1/26 + 1/27))

First two terms are increasing the sum from 1/5 and last two terms are decreasing it. But Contribution of first two terms is more than that of last two terms

We can say that 1/23 + 1/24 + 1/25 + 1/26 + 1/27 > 1/5

Last edited by durgesh79 on 08 Jul 2008, 23:54, edited 2 times in total.

How do you compare 1/23 + 1/24 + 1/25 + 1/26 + 1/27 to 1/5 without long calculation?

[...]

clearly 47/552 > 53/702 and also 47/552 > 2/25

so "1/23 + 1/24 + 1/25 + 1/26 + 1/27" > 1/5

I don't agree with your conclusion, sorry .

Your answer is right, but the "proof" here isn't one :

From 47/552 > 53/702 and 47/552 > 2/25, how do you know that 53/702 > 2/25? Because you're assuming that, and it is : - not prooved - and more importantly: not true

Last edited by Oski on 09 Jul 2008, 01:49, edited 2 times in total.

How do you compare 1/23 + 1/24 + 1/25 + 1/26 + 1/27 to 1/5 without long calculation?

[...]

clearly 47/552 > 53/702 and also 47/552 > 2/25

so "1/23 + 1/24 + 1/25 + 1/26 + 1/27" > 1/5

I don't agree with your conclusion, sorry .

Your answer is right, but the "proof" here isn't one :

From 47/552 > 53/702 and 47/552 > 2/25, how do you know that 53/702 > 2/25? Because you're assuming that, and it is : - not prooved - and more importantly: not true

Did I assume 53/702 > 2/25 ? Can you show me? hmm.......... Of course I assumed 47/552 > 2/25.
_________________

When we are comparing the sum against the middle fraction * the number of fractions, the sum will always be greater than the middle fraction * the number of fractions, provided the numerators are the same and the denominators are consecutive.

The reason behind this is if you start out with the middle, here \(\frac{1}{25}\) as the "average" we need to figure out which way the two fractions on either side tip the scales.