Solving the 100 Animals Riddle: A Comprehensive Guide to Understanding and Applying Algebraic Techniques

The classic 100 animals problem is a fascinating riddle that combines the fun of puzzles with the application of algebraic techniques. This article explores the step-by-step solution to this riddle, outlining the process and providing a detailed explanation of the algebraic equations used. By the end of this guide, you will not only learn how to solve this riddle but also enhance your algebra skills.

Understanding the Problem

A farmer buys 100 animals for a total of 1000. The prices are 50 for a cow, 10 for a sheep, and 0.5 for a rabbit. The question is, how many cows, sheep, and rabbits did the farmer buy? Let’s break down the problem and use algebra to find the solution.

Setting Up the Equations

To solve this problem, we need to set up a system of equations based on the information provided. Let’s define the variables as follows:

c number of cows s number of sheep r number of rabbits

Using the given information, we can derive the following two equations:

The total number of animals: c s r 100 The total cost of the animals: 50c 10s 0.5r 1000

To make the second equation easier to work with, let’s multiply it by 2 to eliminate the decimal:

100c 20s r 2000

Solving the System of Equations

Now we have the following system of equations:

c s r 100 100c 20s r 2000

Our next step is to eliminate r by subtracting the first equation from the second equation:

100c 20s r - (c s r) 2000 - 100

This simplifies to:

99c 19s 1900

Expressing s in Terms of c

Now we need to solve for s in terms of c:

19s 1900 - 99c

s (1900 - 99c) / 19

Expressing r in Terms of c

Next, we substitute s back into the first equation to solve for r:

c (1900 - 99c) / 19 r 100

Multiplying everything by 19 to eliminate the fraction:

19c 1900 - 99c 19r 1900

This simplifies to:

-80c 19r 0

19r 80c

r 80c / 19

Both s and r must be non-negative integers, so c must be an integer such that both (1900 - 99c) / 19 and 80c / 19 are integers.

Finding the Correct Value for c

To find the correct integer value for c, we can test several values:

If c 0:

s 1900 / 19 100 r 0

If c 1:

s (1900 - 99) / 19 1801 / 19 (not an integer)

We continue this process until we find the correct combination:

If c 19:

s (1900 - 99 * 19) / 19 (1900 - 1881) / 19 19 / 19 1 r (80 * 19) / 19 19 * 80 / 19 80

This gives us:

Cows: 19 Sheep: 1 Rabbits: 80

Verification

Let’s verify the solution:

Total animals: 19 1 80 100 Total cost: 50 * 19 10 * 1 0.5 * 80 950 10 40 1000

The solution is correct, and the farmer bought 19 cows, 1 sheep, and 80 rabbits.

{% post_image '100_animals_' %}Visual representation of the solution to the riddle

This solution not only raises the challenge level but also highlights the importance of algebraic techniques in solving complex problems. It’s a perfect exercise for students and teachers to reinforce their understanding of algebraic equations and problem-solving skills.

Keywords: 100 animals problem, algebraic equations, riddle solution