How do you order the following from least to greatest without a calculator #-sqrt10, -10/3, -3, -2.95, -3 1/4, -3.5 times 10^0#?

#-sqrt10, -10/3, -3, -2.95, -3 1/4, -3.5 xx 10^0#?

The boring way of doing this would be to just use a calculator to work out each one. There is a lot of basic maths in this question, and it is possible without a calculator.

To start, notice that:

• they are all negative values! The smallest is therefore the one furthest to the left on the number line. The one that looks the biggest is actually the smallest.
• They are in different forms - make them the same if possible
• They all lie close to #3#

#-sqrt10, -10/3, -3, -2.95, -3 1/4, -3.5 xx 10^0#
#-sqrt10, -3 1/3, -3, -2.95, -3 1/4, -3 1/2 xx cancel(10^0)^1#

They are now easy to arrange, except for #-sqrt10#
(The bigger the denominator, the smaller the fraction, and the numerators are all 1)

The biggest number is clearly #-2.95", followed by "-3#

#-3 1/2, -3 1/3, -3 1/4, -3, -2.95#

Change the others into improper fractions.

#-7/2, -10/3, -13/4, -3/1, -2.95#

Instead of working with #-sqrt10#, if we square it we will have just 10!

Let's square them all, then we will be able to see where 10 fits in.

#49/4, 100/9, 169/16, 9, (2.95)^2#

(This is now in the reverse order, because they are all positive)

We can see that the first 3 numbers are all bigger than 10, so 10 lies before 9.

The final order is

#-3 1/2, -3 1/3, -3 1/4, -sqrt10, -3, -2.95#

But they must be in their original form.

#-3.5 xx 10^0, -10/3, -3 1/4, -sqrt10 -2.95#