Understanding the Horses and Men Problem

This problem asks us to find the number of horses given information about horses, men, and the total number of legs walking on the ground. We are told that the number of horses and men is equal. We are also given how the men are divided into two groups: those riding horses and those walking and leading their horses. Finally, we have the total count of legs touching the ground.

Setting up the Variables

Let's use a variable to represent the unknown quantity. Since the number of horses and the number of men are equal, we can use one variable for both.

• Let $x$ be the number of horses.
• Then, the number of men is also $x$.

Analysing the Groups of Men and Horses

The problem states that half of the owners (men) are on their horses' backs, and the remaining half are walking and leading their horses.

• Number of men on horses = $\frac{x}{2}$
• Number of men walking and leading horses = $\frac{x}{2}$

All horses are on the ground, whether they are being ridden or led. So, all $x$ horses contribute to the legs on the ground.

Calculating Legs Walking on the Ground

Now let's count the legs from each group that are walking on the ground:

• Legs from men walking: There are $\frac{x}{2}$ men walking, and each man has 2 legs. Total legs from walking men = $\frac{x}{2} \times 2 = x$.
• Legs from horses being led: There are $\frac{x}{2}$ horses being led by the walking men. Each horse has 4 legs. Total legs from horses being led = $\frac{x}{2} \times 4 = 2x$.
• Legs from horses being ridden: There are $\frac{x}{2}$ horses being ridden. Each horse has 4 legs, and they are also on the ground. Total legs from horses being ridden = $\frac{x}{2} \times 4 = 2x$.

The total number of legs walking on the ground is the sum of the legs from all these sources.

Total legs on the ground = (Legs from walking men) + (Legs from horses being led) + (Legs from horses being ridden)

Total legs on the ground = $x + 2x + 2x = 5x$.

Solving the Equation

We are given that the total number of legs walking on the ground is 70. So, we can set up an equation:

\begin{equation*} 5x = 70 \end{equation*}

To find the value of $x$, we need to divide both sides of the equation by 5:

\begin{equation*} x = \frac{70}{5} \end{equation*}

\begin{equation*} x = 14 \end{equation*}

The value of $x$ is 14. Since $x$ represents the number of horses, there are 14 horses.

Step-by-Step Solution for the Horses and Men Problem

1. Identify the unknowns: Let the number of horses and men be $x$.
2. Divide the men into groups: $\frac{x}{2}$ riding, $\frac{x}{2}$ walking.
3. Count legs from walking men: $\frac{x}{2} \times 2 = x$ legs.
4. Count legs from horses being led: $\frac{x}{2} \times 4 = 2x$ legs.
5. Count legs from horses being ridden: $\frac{x}{2} \times 4 = 2x$ legs. (All horses contribute legs to the ground).
6. Sum the legs on the ground: Total legs = $x + 2x + 2x = 5x$.
7. Set up the equation: Given total legs = 70, so $5x = 70$.
8. Solve for x: $x = \frac{70}{5} = 14$.
9. State the answer: The number of horses is $x$, which is 14.

Breakdown of Legs on the Ground

Given $5x = 70$, we find $x = 14$. Therefore, there are 14 horses.

Revision Table: Key Concepts

Additional Information: Solving Word Problems

Word problems like the horses and men legs problem require careful reading and translating the given information into mathematical terms. Here are some tips for tackling such problems:

• Read Carefully: Understand what information is given and what you need to find.
• Assign Variables: Use variables for the unknown quantities.
• Break Down Information: Separate the problem into smaller parts (like different groups of people or animals).
• Formulate Equations: Write mathematical equations that represent the relationships described in the problem.
• Solve Equations: Use algebraic techniques to find the value(s) of the variable(s).
• Check Your Answer: Make sure your solution makes sense in the context of the original problem.

In this specific problem, understanding which legs are "walking on the ground" is key. It includes the legs of the walking men and the legs of *all* the horses, as all horses are on the ground.