Measuring 1 Litre of Water Using 5 Litre and 3 Litre Cans: A Comprehensive Guide

Have you ever come across a situation where you need to measure an exact 1 litre of water, but the only vessels you have are a 5-litre and a 3-litre can? Don't worry; this article will walk you through the step-by-step solution to this classic water measurement problem. We will also discuss some mathematical principles involved and explore additional techniques that can be applied when faced with similar challenges.

Introduction to the Water Measurement Problem

The problem of measuring 1 litre of water using a 5-litre and a 3-litre can can be approached in several ways. The most common method involves a series of carefully planned pouring actions. Let's dive into the detailed steps:

Measuring 1 Litre Using 5 Litre and 3 Litre Cans

Fill the 3-litre can completely from the water source.

Pour the water from the 3-litre can into the 5-litre can.

Now, the 5-litre can contains 3 litres of water, and the 3-litre can is empty.

Fill the 3-litre can again completely.

Pour water from the 3-litre can into the 5-litre can until the 5-litre can is full.

Since the 5-litre can already has 3 litres, it can only take 2 more litres. Therefore, you will pour 2 litres from the 3-litre can into the 5-litre can.

After the previous step, the 3-litre can will have 1 litre of water left, as you started with 3 litres and poured 2 litres into the 5-litre can.

Now, the 3-litre can has exactly 1 litre of water.

Understanding the Mathematical Principles

These types of water measurement problems are based on mathematical operations. Let's represent the cans as variables:

x for the 5-litre can

y for the 3-litre can

Using this notation, we can represent the problem as follows:

x : 0

y : 0

Fill the 3-litre can completely: y : 3

Pour from 3-litre to 5-litre: x : 3

Fill the 3-litre can again: y : 3

Pour from 3-litre to 5-litre until 5-litre is full: x : 5, y : 1

The resulting value in the 3-litre can is 1 litre.

Expanding the Problem: Mathematical Constraints

Some mathematical constraints are involved in solving water measurement problems. For example, every achievable value must be a sum of multiples of 4 and 6, and their inverses. This is why the sum of 1 litre is possible with our 5-litre and 3-litre containers. However, if we had 4-litre and 6-litre containers, we would face a different set of constraints. Let's explore this further:

Measuring with 4-litre and 6-litre Containers

Label the operations for the 4-litre and 6-litre containers as follows:

x : 0

y : 0

Fill 4-litre can: x : 4

Fill 6-litre can: y : 6

Transfer from 4-litre to 6-litre: x : 0, y : 4

Transfer from 6-litre to 4-litre: y : 0, x : 4

Transfer from 4-litre to 6-litre until 6-litre is full: x : 2, y : 6

Transfer from 6-litre to 4-litre until 4-litre is full: x : 4, y : 4

These operations can be represented mathematically as:

x : 0y : 0

x : 4y : 0

x : 0y : 4

y : : 4

x : 2y : 6

y : 4x : 4

Alternative Solutions: Using a Balance Scale

A unique approach to solving these measurement problems is to use a balance scale. Here’s how it can be done:

Place the empty cans on the balance scale and add an equal amount of dirt to each side to balance them.

When pouring water from one can to another, monitor the balance; when the scale returns to balance, both cans will have the same amount of water.

Fill the 6-litre can, pour water into the 4-litre can until the balance returns to equilibrium.

Empty the 4-litre can into the 6-litre can, fill the 4-litre can, and pour the water into the 6-litre can until it's full.

Now you have exactly 1 litre of water in the 4-litre can.

Mathematically, the operations can be represented as:

x : 0y : 0

y : 6

x : x - 6y : x

x : 4y : 3

y : 6x : 1

Using this method, you can achieve the same result with water and a balance scale, making it a versatile approach.

Conclusion

Measuring 1 litre of water using 5-litre and 3-litre cans is a fascinating problem that involves both logic and mathematics. By understanding the mathematical constraints and exploring alternative methods like using a balance scale, you can master this challenge and similar ones.