# 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?

Aug 12, 2016

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

$- 3.5 \times {10}^{0} , - \frac{10}{3} , - 3 \frac{1}{4} , - \sqrt{10} - 2.95$

#### Explanation:

$- \sqrt{10} , - \frac{10}{3} , - 3 , - 2.95 , - 3 \frac{1}{4} , - 3.5 \times {10}^{0}$?

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 $3$

$- \sqrt{10} , - \frac{10}{3} , - 3 , - 2.95 , - 3 \frac{1}{4} , - 3.5 \times {10}^{0}$
$- \sqrt{10} , - 3 \frac{1}{3} , - 3 , - 2.95 , - 3 \frac{1}{4} , - 3 \frac{1}{2} \times {\cancel{{10}^{0}}}^{1}$

They are now easy to arrange, except for $- \sqrt{10}$
(The bigger the denominator, the smaller the fraction, and the numerators are all 1)

The biggest number is clearly $- 2.95 \text{, followed by } - 3$

$- 3 \frac{1}{2} , - 3 \frac{1}{3} , - 3 \frac{1}{4} , - 3 , - 2.95$

Change the others into improper fractions.

$- \frac{7}{2} , - \frac{10}{3} , - \frac{13}{4} , - \frac{3}{1} , - 2.95$

Instead of working with $- \sqrt{10}$, 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.

$\frac{49}{4} , \frac{100}{9} , \frac{169}{16} , 9 , {\left(2.95\right)}^{2}$

(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

$- 3 \frac{1}{2} , - 3 \frac{1}{3} , - 3 \frac{1}{4} , - \sqrt{10} , - 3 , - 2.95$

But they must be in their original form.

$- 3.5 \times {10}^{0} , - \frac{10}{3} , - 3 \frac{1}{4} , - \sqrt{10} - 2.95$