# How many men and how many horses are in the stable?

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In a stable there are men and horses. In all, there are 22 heads and 72 feet. How many men and how many horses are in the stable?

8 men and 14 horses. But I guess the way to answer it is probably more important than the answer itself.

Suppose we use “x” as the number of men and “y” as the number of horses. If there are 22 heads and knowing that men and horses only have one head each, we can say that

x + y = 22.

We shall call this Equation 1.

Now, knowing that there are 72 feet, that men have 2 feet and horses have 4, we can write that down as

(number of men × number of feet per man) + (number of horses × number of feet per horse = 72; or simply

2x + 4y = 72.

This Equation 2.

The two equations are what we call a system of linear equations. Since we have two equations and two unknowns (i.e. men and horses), we can solve for the exact value of the unknowns using algebra. Here, we’ll use the method of substitution.

Taking first Eq. 1 and subtracting y from both sides of the equation, we get another equation that only has x on the left hand side shown below. We’ll call this Equation 3 and use it later.

x + y – y = 22 – y

x = 22 – y. (Equation 3)

Now, we can substitute the expression in the right hand side of Eq. 3 (which is just another version of Eq. 1) as ‘x’ in Eq. 2,

2x + 4y = 72 (Equation 2)

2*(22-y) + 4y = 72 (substituting Eq. 3 to Eq. 2)

44 – 2y + 4y = 72.

Simplifying and solving for y, we get

-2y + 4y = 72 – 44

2y = 28

y = 14.

Thus, y, or the number of horses, is 24. If we plug this value into either equation, we should get the value of x to be 8.

x = 22 – y (Eq. 3)

x = 22 – 14 (using y=14)

x = 8.