What is the value of the resistor

A loop circuit has a resistance of R1 and a current of 1.8 A. The current is reduced to 1.6 A when an additional 4.2 ohms resistor is added in series with R1. What is the value of R1? Assume the inter-
nal resistance of the source of emf is zero.

You have two unknowns, emf and R1. So you'll need two equations, which are given via two different scenarios. The first scenario says:

V = (1.8 A)R1

The second says:

V = (1.6 A)(R1 + 4.2)

Plug equation 1 into equation 1:

1.8R1 = 1.6R1 + 6.72
0.2R1 = 6.72
R1 = 33.6 Ω