“I'll run it by April and see what she thinks.”
You're trying to think of a name for a band that you're starting with a group of musicians. You and some of the other band members come up with a name that you all like, but one of the members isn't there. You want to know what she thinks of the name, so you say this.
I'll run it by April and see what she thinks.
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To "run an idea by someone" means to tell them about the idea and find out what they think of it. Another example, which can be used in a work setting:
Make sure to run it by James before you send it out.
In the example above, the speaker uses "it" to talk about the name that the band members chose. "It" is the correct pronoun to use if they're just talking about the name. However, there are some situations where "this" could also be used. For example, if the name was written down on a piece of paper that the speaker was holding, he could point to it and say:
I'll run this by April and see what she thinks.
The word "see" can be used to mean "find out" or "learn" in situations like this.
This phrase can be used in casual or business settings:
Show this to the agency and see what they think.
I wrote a story, and I wanted to see what you thought of it.