# List the following numbers in increasing order: sqrt13-sqrt5; 4sqrt38-((sqrt3)^-1); sqrt2-61/152; 3π-sqrt5/32; π

embizze | High School Teacher | (Level 1) Educator Emeritus

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List the following numbers in increasing order: `sqrt(13)-sqrt(5)` ,`4sqrt(38)-(sqrt(3))^(-1)` ,`sqrt(2)-61/152` ,`3pi-(sqrt(5))/(32)` ,`pi` .

We can find the decimal approximations with a calculator:

`sqrt(13)-sqrt(5)~~1.369483298`

`4sqrt(38)-1/sqrt(3)~~24.08030574`

`sqrt(2)-61/152~~1.012897773`

`3pi-(sqrt(5))/32~~9.354900836`

`pi~~3.141592654`

Thus in increasing order we have:

`sqrt(2)-61/152,sqrt(13)-sqrt(5),pi,3pi-(sqrt(5))/32,4sqrt(38)-1/sqrt(3)`

If this was a number sense type question, you might reason as follows:

`sqrt(2)-61/152` will be near 1 since `sqrt(2)~~1.4` and `61/152~~60/150=.4`

`sqrt(13)-sqrt(5)` will be near 1.5 as `3<sqrt(13)<4` and `2<sqrt(5)<3`

`pi~~3.14` is a well known approximation.

`3pi-(sqrt(5))/32` will be near 9 as `3pi~~9.42` and the fraction is small.

`4sqrt(38)-1/sqrt(3)~~24` since `6<sqrt(38)<7` so `24<4sqrt(38)<28` and the fraction being subtracted is close to 1/2.