How much heat transfer occurs from a system, if its internal energy decreased by 150 j while it was doing 30.0 j of work?

In this case we apply the concepts in thermodynamics. When it comes to intensive properties like internal energy (U), heat transfer (Q) and work (W), there are already derived relationships for this which were based on differential equations. The equation is

ΔU = Q + W

But we should remember to incorporate their signs because these parameters are dependent on these. According to Van Ness, Smith and Abbott, the sign convention for decreasing is negative. For work, the sign is positive if it is work ON the system, but negative if it is work done BY the system. So, in this case,

ΔU = - 150 J

W = - 30 J

Then, we can determine Q.

-150 J = Q - 30 J

Q = - 120 J

This means that 120 J of heat was released by the system to the surroundings.