You misunderstand 分 here.

分 does not mean "division" here. Here 分 means the "points" (test score), in an exam or a contest where the perfect score is ten.

分 can also means "perdec" (*per ten*, a tenth), as a synonym of 成.

> 三分  three points (out of ten), three perdec (3/10)
>
> 五分  five points (out of ten), five perdec (5/10)
>
> 十分 ①[original meaning] ten points (out of ten), perfect score, full marks ②[extended meaning] "perfectly", very, extremely
>
> 幾分  how many points (out of ten), how many perdec

Assuming that there is an exam called "love", how many points can I get from the exam?

>我愛你有幾分？
>
> How many points (out of ten) do I love you?
>
> How many perdec do I love you?
>
> = How much do I love you?


Extra examples:

> 三分天注定，七分靠打拼。 [famous lyrics]
>
> Three points depend on fate, while seven points depend on hardworking.
>
> 3/10 depends on fate, 7/10 depends on hardworking.
>
> [Chinese version of "nature versus nurture"]

> 逢人只說**三分**話，未可全拋一片心。[ancient proverb]
>
> Every time you meet a person, you should only tell them **three perdec (3/10)** of your mind. You should not "throw the entire heart to them" (tell them all your mind).

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**Disputes**

As 分 is also a traditional Chinese unit of length, which modernly equals 1/3 centimetres, some people believe that it is better to parse as the depth of love measured in this length unit, instead of points or perdecs.

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**Some knowledge about the Chinese culture of proportions**

Western culture prefers to use "percent" (*per hundred*, a hundredth), while Chinese culture traditionally uses "perdec" (*per ten*, a tenth), known as 分, 成, 折.

Examples:
> 三分  = three perdec (three tenths, 3/10)
>
> 七分 = seven perdec (seven tenths, 7/10)
>
> 八成  = eight perdec (eight tenths, 8/10)
>
> 九成 = nine perdec (nine tenths, 9/10)

折 is only used for discounts.

> 九折 = nine perdec remaining (i.e. 10% discount)
>
> 八折 = eight perdec remaining (i.e. 20% discount)
>
> 八五折 (8.5折) = 8.5 perdec remaining (i.e. 15% discount)