How much agarose powder would you need to use (include units! ) if you only needed 40 ml of agarose gel and the package recipe states ‘use 6 grams of agarose powder dissolved in 500 ml of water’?

The rule of three is a way to solve problems of proportionality between three known values and one unknown.  This is an operation of finding the fourth term of a proportion knowing the other three.

We can use the rule of three to solve this problem in this way,

500 mL →  6 grams

40 mL  →     x

This is the same as saying: if in 500 mL there should be 6 grams of agarose powder then in 40 mL how many grams should there be?

So,

→ x = 0.48 g = 480 mg

Then, you need to dissolved 480 mg of agarose powder in 40 mL of water to meet the recipe.