There are three laws that apply when looking at DC circuits:
- Kirchoffs First and Second (see wikipedia)
- Ohms Law
The statement above that "The 7.63mA is being used by the LED" is not right. Don't even think it.
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The voltage round the circuit must sum to zero. That means 6.12 into the LED and resistor are balanced by the voltages across the resistor and LED.
The LED is a non linear device - it does not conduct until the voltage across it rises to about 1.7V. Thereafter the voltage across it stays nearly constant. So 6.12 = voltage across LED + voltage across resistor. Ohms Law gives the values for the resistor. -
The current round the circuit is constant. The same current flows through the battery, the resistor and the LED. The current flowing into any point of your circuit is the same as that flowing out of it. E.g. the current into the LED is the same as the current out of it.
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The other things to remember are that components are approximate. Meters are approximate. Also, especially when looking at a voltage divider, the resistance of your multimeter might be significant. If you had a voltage divider of 2 10MOhm resistors, and your multimeter had a resistance of 10 MOhm (which one of mine has), you change the circuit from 10MOhm + 10 MOhm to 10MOhm + 5 MOhm when you put the meter on. By using Kirchoffs Laws and Ohms law you could work out the effect of that.
Luckily, for logic circuits, you can often approximate them as DC circuits in one state or another.