The Number Of

[SheltieCute]

"More traffic lights would decrease (the number of) traffic accidents."
"More traffic lights would reduce (the number of) traffic accidents."
"Less traffic regulations would increase (the number of) traffic accidents."
"Higher tuition would decrease (the number of) students applying to university."
"He is going around trying to increase (the number of) business contacts."
"Careful proofreading would decrease (the number of) grammatical errors in published articles."
"Careful proofreading would reduce (the number of) grammatical errors in published articles."
"The new program would decrease (the number of) student loan payments."
"The new program would reduce (the number of) student loan payments."

When do I need "the number of" when using "increase", "decrease", and "reduce"?

[Raymott]

"More traffic lights would decrease (the number of) traffic accidents."
"More traffic lights would reduce (the number of) traffic accidents."
"Less traffic regulations would increase (the number of) traffic accidents."
"Higher tuition would decrease (the number of) students applying to university."
"He is going around trying to increase (the number of) business contacts."
"Careful proofreading would decrease (the number of) grammatical errors in published articles."
"Careful proofreading would reduce (the number of) grammatical errors in published articles."
"The new program would decrease (the number of) student loan payments."
"The new program would reduce (the number of) student loan payments."

When do I need "the number of" when using "increase", "decrease", and "reduce"?

I don't think there's an easy answer to that. It's mostly a matter of style.
The only examples from above where I'd say "the number of" was essential, are the last two sentences. Without it, the sentence would be ambiguous.
"The new program would reduce student loan payments." This could mean that the new program reduces the amount of any one student loan payment, without having any effect on the number of loan payments. Only by adding "the number of" can you make the meaning clear.