How did the teacher know that Bobby had cheated?

Mrs. Jones gave this homework assignment to her students:

Find the pattern behind these wrong-looking equations: $$56+43+2+16=5+85+6+1+99$$ $$27+16+117=6+76+117$$ And then come up with two wrong-looking equations of your own that satisfy the same pattern.

Bobby was stumped, but he remembered that Mrs. Jones had been teaching this course almost the exact same way for 30 years, and that she taught his dad too. He went home, dug through his dad's old papers and copied his dad's (correct) solution to a very similar-looking assignment as his own:

$$7+8+107=102+92+7+16$$ $$35+109+53=5+44+10+53$$

When Mrs. Jones first checked Bobby's answers, she just thought they were wrong. But after thinking about what error he could have made, she realized what he had done.

What was the pattern behind the equations, and what did Mrs. Jones realize about Bobby's answers?

• Bobby should get a 100 A+ and a pat on the back for being able to find a math assignment 20-30 years later! :)
– JLee
Commented Aug 7, 2022 at 21:59
• @JLee rot13(Jub fnvq vg jnf n zngu pynff?) Commented Aug 8, 2022 at 16:49
• @JLee Bobby should get therapy because his dad is obviously a hoarder (although possibly an amazingly organized one) and growing up with that has got to be rough. Commented Aug 8, 2022 at 21:11

The numbers are

atomic numbers:

so the examples become

BA TC HE S = B AT C H ES

CO S TS = C OS TS

including some elements that went by generic names back then because no one had synthesized them yet,

became

N O UNS = NO U N S

BR UNE I = B RU NE I

• But Ts was on his dad’s paper… Commented Aug 7, 2022 at 22:11
• @msh210 Whoops, good catch. I just tweaked the wording a bit to hopefully address that problem without invalidating this (intended) answer. Commented Aug 7, 2022 at 22:26
• Hang on ... what "answer" is this puzzle proposing that Bobby gave? Commented Aug 8, 2022 at 11:35
• If the answer was "they spell out the same names in Atomic letters" ... then the answer is correct. Commented Aug 8, 2022 at 11:36
• @Brondahl The question was "come up with two wrong-looking equations of your own". If Bobby copied his dad's solution, it wouldn't be valid anymore.
– jpa
Commented Aug 8, 2022 at 14:11