Question

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`?

Answer 1

A very nice question! Lots of maths involved, but easy to do with the use of a clever technique, Thanks Tony B

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

Explanation:

`-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`