Let the initial number of cats be $5x$ and the initial number of dogs be $3x$, based on the ratio 5:3.

Since 48 new animals arrived, the total number of animals after they arrived is given by:

$5x + 3x + 48 = 800$

Simplifying, we have:

$8x + 48 = 800$

Subtract 48 from both sides:

$8x = 752$

Now, divide by 8:

$x = 94$

Now that we know $x = 94$, we can find the initial number of cats:

$5x = 5 \times 94 = 470$

Answer: There were 470 cats at the pet shelter at first.

Would you like more details, or have any other questions?

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Here are 5 related questions to explore further:

1. If the number of dogs also increased, what would be the new ratio of cats to dogs?
2. What would the initial count of animals be if the ratio were changed to 4:3 with the same conditions?
3. If 30 more dogs arrive after the 48 animals, how does that affect the ratio?
4. How would the calculation change if the ratio were 6:4?
5. What is the percentage of cats in the initial animal count?

Tip: Ratios can often simplify complex problems by turning them into proportional relationships.