In physics, you're not allowed to ignore the units; they come along for the ride on every sub-step of every calculation. From a mathematics perspective, consider the units to be variables, so instead of 5 meters + 10 seconds, you have 5x + 10y. Unless you arbitrarily assign x = y = 1, there is no way you're getting 15 out of this; at the end of the day, you still have 5x + 10y, and physics is uninterested in complex numbers. At the end of a computation, you need one number, and "5x + 10y" is two numbers. And here's the rub in physics: you're not allowed to assign values to those variables. They are basic, irreducable units; you can't say "meters are 1".

On the other hand, you're allowed to multiply units like nobody's business. "Furlongs per fortnight" is a silly phrase, but you can use it and everyone will (after some conversion) understand exactly what you're saying:

$$13440\frac{furlongs}{fortnight} \times \frac{1 mile}{8 furlongs} = 1680\frac{miles}{fortnight}$$

$$1680 \frac{miles}{fortnight} \times \frac{1 fortnight}{336 hours} = 5 \frac{miles}{hour}$$

(Cue a horde of angry physicists descending upon me for my use of Imperial units.)

On every step of these conversions, the units never went away; they stayed with the numbers. When you're computing something in physics, the numbers you play with are real things; they represent a quantity of something, and that something doesn't go away just because you think it's inconvenient. So if an asteroid goes 10 meters in 5 seconds, it looks like this:

$$10 meters \times \frac{1}{5 seconds} = 2 \frac{meters}{second}$$

You can multiply and divide quantities however you like, and you'll end up with some funky units, but your final number will be valid — though they may be difficult to relate to everything else. (For instance, the viscosity of a fluid is measured in (kilograms per (meter * second)), which isn't particularly intuitive but is useful in the particular places that viscosity is used.)

In physics, you're not allowed to ignore the units; they come along for the ride on every sub-step of every calculation. From a mathematics perspective, consider the units to be variables, so instead of 5 meters + 10 seconds, you have 5x + 10y. Unless you arbitrarily assign x = y = 1, there is no way you're getting 15 out of this; at the end of the day, you still have 5x + 10y, and physics is uninterested in complex numbers. At the end of a computation, you need one number, and "5x + 10y" is two numbers. And here's the rub in physics: you're not allowed to assign values to those variables. They are basic, irreducable units; you can't say "meters are 1".

On the other hand, you're allowed to multiply units like nobody's business. "Furlongs per fortnight" is a silly phrase, but you can use it and everyone will (after some conversion) understand exactly what you're saying:

$$13440\frac{\text{furlongs}}{\text{fortnight}} \times \frac{1 \; \text{mile}}{8 \; \text{furlongs}} = 1680\frac{\text{miles}}{\text{fortnight}}$$

$$1680 \frac{\text{miles}}{\text{fortnight}} \times \frac{1 \; \text{fortnight}}{336 \; \text{hours}} = 5 \frac{\text{miles}}{\text{hour}}$$

(Cue a horde of angry physicists descending upon me for my use of Imperial units.)

On every step of these conversions, the units never went away; they stayed with the numbers. When you're computing something in physics, the numbers you play with are real things; they represent a quantity of something, and that something doesn't go away just because you think it's inconvenient. So if an asteroid goes 10 meters in 5 seconds, it looks like this:

$$10 \; \text{meters} \times \frac{1}{5 \; \text{seconds}} = 2 \frac{\text{meters}}{\text{second}}$$

You can multiply and divide quantities however you like, and you'll end up with some funky units, but your final number will be valid — though they may be difficult to relate to everything else. (For instance, the viscosity of a fluid is measured in (kilograms per (meter * second)), which isn't particularly intuitive but is useful in the particular places that viscosity is used.)