Question

Without using a calculator, order the following expressions from least to greatest.

11/3, square root 20, square root 17

Answer 1

`\displaystyle \large{(11)/(3) < √(17) < √(20)}`

Explanation:

( 1 ) 11/3

`\displaystyle \large{(11)/(3)}` can be converted to mixed fractions to
`\displaystyle \large{3(2)/(3)}`

Therefore, 11/3 is at least around 3, almost 4. It could be more than 3.5 by approximation but not exactly 4.

( 2 ) √20

`\displaystyle \large{√(20)=√(5\cdot 4)}\\\displaystyle \large{√(20)=√(5\cdot 2\cdot 2)}\\\displaystyle \large{√(20)=2√(5)}`

We know that
`\displaystyle \large{√(4)=2}` so
`\displaystyle \large{√(5) > √(4)}` which means approximately,
`\displaystyle \large{√(20)}` can be either at least 4.3 up to 4.4

( 3 ) √17

We know that
`\displaystyle \large{√(16) = 4}` so
`\displaystyle \large{√(17)}` can be at least 4.1 up to 4.2, but it’s obvious that the square root of 17 is less than square root of 20 since
`\displaystyle \large{√(a) > √(b)}` for a > b

Hence, from least to greatest:

`\displaystyle \large{(11)/(3) < √(17) < √(20)}`