The voltage across resistors in a series circuit is proportional to their resistances. This rule works because the current in each resistor of a series circuit is the same. Sometimes solder splashes or contamination on the surface of a circuit board affords an alternate path for current to flow.
Generally a troubleshooter does not pull out a calculator every time they make a measurement and calculate the current in each resistor using ohms law. They are usually capable checking that the voltage drop across resistors in series is proportional to the resistors indicated value.
Intuitive thinking is hard to teach but one aspect can be described. Assume you have a 6K ohm resistor in series with a 1.5k ohm resistor. You can visualize this circuit as being equivalent to 1.5k resistor in series with four 1.5k resistors. Obviously 20% of the voltage will be dropped across 1.5k resistor.
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