If there are $74$ heads and $196$ legs, how many horses and humans are there? [closed]

I was going through some problems then I arrived at this question which I couldn't solve. Does anyone know the answer to this question?

One day, a person went to a horse racing area. Instead of counting the number of humans and horses, he counted $74$ heads and $196$ legs. How many humans and horses were there?

Solution 1:

A hypercentaur is a creature with two heads and six legs; an anticentaur is a creature with no head and two negative legs.

Since $74$ heads make for $37$ hypercentaurs, with $74\cdot 6/2=222$ legs, you have $(222-196)/2=13$ anticentaurs.

Since a hypercentaur is the same as a human on a horse, and an anticentaur is a human deprived of a horse, we have counted $37-13=24$ horses and $37+13=50$ humans.

Solution 2:

Just in case you don't like algebra and don't think in terms of centaurs, here's yet another approach (very close to the centaur version but omitting the mythology). Suppose for a moment that all 74 heads belong to humans. Then there would be $2\times 74=148$ legs. That's 48 legs short of the specified number 196. To get that many extra legs, we have to replace some of the humans with horses. Every time we replace a human with a horse, we gain two legs, so we should do $\frac{48}2=24$ such replacements. We started with 74 humans, and we need to replace 24 of them with horses, so at the end we have 24 horses and $74-24=50$ humans.

Solution 3:

Hint:

Let there be $x$ humans and $y$ horses.

Each human and horse have one head. Each human has two legs and each horse has four legs.

Then, $$x+y=74$$ $$2x+4y=196$$

which can be solved.