In one of the pans of the beam balance, masses of [tex]3 \, \text{kg}, 900 \, \text{g}, 90 \, \text{g},[/tex] and [tex]5 \, \text{g}[/tex] are placed. What amount of mass should be placed on the other side of the beam balance to make it balanced?

To solve this problem, we need to figure out the total mass on one side of the beam balance. We'll add up all the masses given in kilograms (kg) and then determine how much should be placed on the other side to balance it.

Step-by-step Solution:

1. Convert All Masses to Kilograms:
- We have several masses in grams that need to be converted to kilograms. Since there are 1000 grams in a kilogram, we can convert each mass to kilograms.
- [tex]$3 \, \text{kg}$[/tex] is already in kilograms.
- [tex]$900 \, \text{g} = \frac{900}{1000} \, \text{kg} = 0.9 \, \text{kg}$[/tex].
- [tex]$90 \, \text{g} = \frac{90}{1000} \, \text{kg} = 0.09 \, \text{kg}$[/tex].
- [tex]$5 \, \text{g} = \frac{5}{1000} \, \text{kg} = 0.005 \, \text{kg}$[/tex].

2. Calculate the Total Mass on One Side of the Beam Balance:
- To find the total mass, simply add up all the converted masses:
[tex]\[
3 \, \text{kg} + 0.9 \, \text{kg} + 0.09 \, \text{kg} + 0.005 \, \text{kg} = 3.995 \, \text{kg}
\][/tex]

3. Determine the Mass Needed on the Other Side:
- For the beam balance to be balanced, the mass on the other side should be equal to the total mass we just calculated.
- Therefore, the mass that should be placed on the other side is [tex]$3.995 \, \text{kg}$[/tex].

So, you need to place [tex]$3.995 \, \text{kg}$[/tex] on the other side of the beam balance to make it balanced.