The length of the wire affects the amount of juice it can carry. Same as a hose with water, longer it is the more resistance and the less water flows through it. End result with water is you get a dribble out the end of the hose & you're disappointed if you make the hose too long and too small. End result with wiring is you overheat the wire and start a fire.
So, you got a long run there. You need to up the wire size over the 'regular' wire ratings - they asume less than 75 foot lengths and you are double that. Just simple how it is, how you have to do it.
Yup, that's expensive, but if you don't the wire overheats and melts and burns, and that is _more_ expensive.
John T has it pretty good, you are getting by with those instructions, don't try to cut corners even more!
Sounds like you are gonna be enjoying this livestock project with the kids, and livestock need water. Mixing livestock, water, electricity, and kids - be careful you get the grounding right. Spending a bit extra on the right wire should be worth it. :)
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