Solving the Stump Pond Puzzle: A Guide to Fractional Length Problems

Have you ever come across a problem like the one in the pond where a stick is partially submerged in water and on land, and you need to find the total length of the stick? This article will walk you through different methods to solve this fascinating problem and provide a step-by-step solution. Whether you're a student, teacher, or simply curious about mathematics, you'll find this guide both informative and practical.

The Problem Statement

A stick is planted in a pond. One-third of the stick is under the ground, one-third is under the water, and the piece that protrudes from the water is 9 centimeters long. How long is the whole stick?

Method 1: Basic Fractional Analysis

First, let's break down the problem using basic fraction analysis.

We know that one-third of the stick is under the ground and one-third is under the water. That makes a total of 2/3 of the stick underwater and on land. So the remaining one-sixth of the stick (1 - 2/3 1/6) protrudes from the water and is 9 cm long.

To find the total length, we can set up the equation:

1/6 of the length 9 cm

Thus, the total length of the stick is:

6 × 9 cm 54 cm

Hence, the stick is 54 cm long.

Method 2: Algebraic Approach

Let's solve this using algebra. Denote the total length of the stick as x cm.

Step 1:

Identify the fractions of the stick:

1/2 of the stick is stuck under the ground

1/3 of the stick is under the water

1/6 of the stick protrudes from the water (since 1 - 1/2 - 1/3 1/6)

And we know that 1/6 of the stick 9 cm.

Step 2:

Set up the equation to find the total length:

1/6x 9

Multiply both sides by 6:

x 54 cm

Therefore, the stick is 54 cm long.

Alternative Solution Using Common Denominators

Another approach involves converting the fractions to a common denominator and then solving the problem.

Step 1:

Convert 1/2 and 1/3 to a common denominator:

1/2 × 1/3 6

7 units 7 × 6 42

Step 2:

Alternatively, if the remaining length is 8 feet and it represents 1/6 of the total length of the stick, then:

x 8 × 6 48 ft

Hence, the stick is 48 feet long.

Additional Insights

This problem involving fractional lengths is a classic example of how simple arithmetic and algebra can be used to solve real-life problems. The key takeaway is understanding the relationship between the fractions and the remaining length.

By breaking down the problem into smaller, manageable parts, you can easily find the solution. Remember, the important part is not just getting the right answer but understanding the process.

To deepen your understanding, try applying these techniques to similar problems. Practice is key to mastering mathematical concepts.