Measure 99 Kg Sugar: Weighing Problem

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Let's dive into a classic weighing problem! We have a sack of sugar weighing 120 kg, a balance scale, and four weights: 5 kg, 9 kg, 12 kg, and 19 kg. The challenge is to figure out the fewest number of weighings needed to measure exactly 99 kg of sugar.

Understanding the Problem

Before we start trying different combinations, let's break down what we're trying to achieve. Our goal is to isolate 99 kg of sugar. We can use the weights to help us measure this amount accurately using the balance scale. Think of it like a puzzle where each weighing is a step closer to the solution. We want to find the most efficient path, using the minimum number of steps or weighings.

Key elements:

  • Starting point: 120 kg sack of sugar.
  • Goal: Accurately measure 99 kg of sugar.
  • Tools: Balance scale and weights of 5 kg, 9 kg, 12 kg, and 19 kg.
  • Objective: Minimize the number of weighings.

Exploring Possible Solutions

Now, let's explore some possible solutions. Remember, with a balance scale, we can place weights on either side to help us determine the desired amount. We need to find a way to use the given weights to isolate exactly 99kg. We need to be strategic about how we use them.

One Weighing?

Is it possible to get 99kg in a single weighing? This would mean we could somehow combine the given weights to directly measure 99kg. Let's see:

5 + 9 + 12 + 19 = 45. No way.

Two Weighings?

Let's see if we can do it in two weighings.

First Weighing: Let's try to obtain an intermediate weight to help in the second weighing. For instance, can we get close to a useful value?

Second Weighing: Then, use the intermediate weight to get 99kg.

Three Weighings?

If two weighings don't work, we may need three weighings. In this scenario, we are essentially creating intermediate weights that will eventually lead us to the target amount.

The Optimal Solution

Alright, guys, after thinking through the different possibilities, here's the most efficient solution. The key is to realize we can work with the difference between the total weight (120 kg) and the target weight (99 kg), which is 21 kg.

One Weighing:

  1. First Weighing: Place the 5 kg, 9 kg and 12kg weights on one side of the balance scale, and pour sugar from the sack onto the other side until the scale balances. This gives you 5 + 9 + 12 = 26 kg of sugar on the scale. Subtract the 5kg: 26 - 5 = 21kg. Now take the initial sack of sugar, and subtract 21kg: 120 - 21 = 99kg.

So, only one weighing is needed.

Why This Solution Works

This solution cleverly uses the weights to directly measure the difference between the initial amount and the target amount. By accurately measuring out 21 kg, we are left with exactly 99 kg. This approach avoids multiple intermediate steps, making it the most efficient solution.

Conclusion

Therefore, the minimum number of weighings required to obtain exactly 99 kg of sugar is just one. Option (b) is the correct answer. Remember, the key to solving these types of problems is to think strategically about how to use the given tools to achieve the desired outcome with the fewest steps possible.

Let's delve a bit deeper into the fascinating world of balance scale weighing problems! These problems, often encountered in mathematics and puzzles, involve determining an unknown weight or measuring a specific quantity using a balance scale and a set of known weights. The challenge lies in finding the most efficient method, requiring the fewest weighings, to achieve the desired result.

Core Concepts

Before tackling these problems, it's essential to grasp the fundamental concepts:

  • Balance Scale: A balance scale compares the weights of two objects placed on either side of a fulcrum. When balanced, the weights on both sides are equal.
  • Weights: Known quantities used to measure or determine unknown weights. They can be added or subtracted from either side of the balance scale.
  • Weighing: A single act of using the balance scale to compare weights. The goal is to minimize the number of weighings.
  • Objective: Typically, the objective is to determine an unknown weight or measure a specific quantity using the fewest possible weighings.

Problem-Solving Strategies

Here's a breakdown of effective strategies to solve balance scale weighing problems:

  1. Understand the Problem: Carefully read and understand the problem statement. Identify the knowns (weights, initial quantity) and the unknowns (target weight, unknown object).
  2. Consider the Balance Scale Principle: Remember that the balance scale compares weights. You can add or subtract weights from either side to find the difference or equivalence.
  3. Look for Differences: Sometimes, focusing on the difference between the initial quantity and the target quantity can simplify the problem. As seen in the initial problem, weighing the difference (21kg) directly led to the solution.
  4. Strategic Use of Weights: Think carefully about how to combine the given weights. Can you create intermediate weights that will help you in subsequent weighings?
  5. Binary Thinking (for some problems): In certain problems, especially those involving identifying a single heavier or lighter item among a set of items, binary thinking can be useful. This involves dividing the items into groups and comparing them to narrow down the possibilities.
  6. Systematic Approach: Start by exploring simple solutions (one weighing, two weighings) before moving to more complex scenarios. A systematic approach will help you avoid missing the optimal solution.
  7. Optimization: Once you find a solution, ask yourself if it's the most efficient. Can you achieve the same result with fewer weighings?

Types of Weighing Problems

Balance scale weighing problems come in various forms. Here are a few common types:

  • Measuring a Specific Quantity: These problems involve measuring a specific amount of a substance (like sugar, as in the initial problem) from a larger quantity.
  • Identifying a Counterfeit Coin: These classic problems involve identifying a single counterfeit coin (which may be heavier or lighter) among a set of genuine coins.
  • Determining an Unknown Weight: These problems involve finding the weight of an unknown object using a set of known weights.
  • Division Problems: These may involve dividing a quantity into specified ratios using the balance scale.

Example: Identifying a Counterfeit Coin

Let's illustrate these strategies with an example:

Problem: You have 12 coins, and one of them is counterfeit. The counterfeit coin is either heavier or lighter than the genuine coins. Using a balance scale, what is the minimum number of weighings needed to identify the counterfeit coin and determine whether it is heavier or lighter?

Solution:

  1. First Weighing: Divide the coins into three groups of four coins each (Group A, Group B, Group C). Place Group A on one side of the balance scale and Group B on the other side.

    • Scenario 1: The scale balances. This means the counterfeit coin is in Group C. We've eliminated 8 coins with one weighing!
    • Scenario 2: The scale tips. This means the counterfeit coin is in either Group A or Group B, and we know whether it's heavier or lighter (depending on which side went down).

  2. Second Weighing (if Scenario 1 occurred): Take three coins from Group C and weigh them against three known genuine coins.

    • If the scale balances: The remaining coin in Group C is the counterfeit.
    • If the scale tips: You've identified the counterfeit coin and whether it's heavier or lighter.

  3. Third Weighing (if Scenario 1 occurred and the previous weighing didn't balance): Weigh the suspected counterfeit coin against a genuine coin to confirm whether it is heavier or lighter.

  4. Second Weighing (if Scenario 2 occurred): Take the group of coins that contains the counterfeit. Weigh 3 of the coins against each other. You can now determine whether the remaining coin is heavier or lighter.

In this case, the minimum number of weighings needed is three. The key is to divide the coins into groups and use each weighing to eliminate possibilities.

Tips for Success

  • Practice: The more weighing problems you solve, the better you'll become at recognizing patterns and applying the right strategies.
  • Draw Diagrams: Visualizing the problem with diagrams can often help you understand the relationships between the weights and the balance scale.
  • Don't Give Up: Weighing problems can be challenging, but with persistence and a systematic approach, you can find the solution!

Balance scale weighing problems aren't just abstract mathematical exercises. They are excellent tools for honing crucial problem-solving skills that are valuable in many aspects of life.

Analytical Thinking

These problems force you to think analytically. You must break down the problem into smaller, manageable steps, identify relevant information, and develop a logical approach to find the solution. Analytical thinking is essential in fields like science, engineering, and data analysis.

Logical Reasoning

Balance scale problems require you to use logical reasoning to deduce information from each weighing. You must carefully consider the possible outcomes and draw conclusions based on the evidence. Logical reasoning is critical in law, philosophy, and any field that requires critical thinking.

Strategic Thinking

Finding the most efficient solution to a weighing problem demands strategic thinking. You must plan your moves carefully, anticipate the consequences of each action, and optimize your approach to achieve the desired outcome. Strategic thinking is crucial in business, leadership, and game theory.

Creative Problem Solving

Sometimes, the most effective solution to a weighing problem requires creative thinking. You may need to think outside the box, explore unconventional approaches, and develop innovative strategies to overcome the challenge. Creative problem-solving is highly valued in art, design, and entrepreneurship.

Quantitative Skills

Weighing problems often involve quantitative reasoning, such as calculating weights, comparing quantities, and determining ratios. Strengthening these skills is beneficial in finance, economics, and any field that involves numerical data.

Real-World Applications

While balance scale problems may seem abstract, the problem-solving skills they cultivate have numerous real-world applications:

  • Diagnosis: Doctors use analytical and logical reasoning to diagnose illnesses by evaluating symptoms and test results.
  • Troubleshooting: Engineers use strategic thinking to troubleshoot complex systems and identify the root cause of problems.
  • Decision Making: Business leaders use creative problem-solving to make strategic decisions in uncertain environments.
  • Negotiation: Negotiators use logical reasoning and strategic thinking to reach mutually beneficial agreements.
  • Research: Scientists use analytical and quantitative skills to conduct research and draw conclusions from data.

By practicing balance scale weighing problems, you not only sharpen your mathematical abilities but also cultivate valuable problem-solving skills that will serve you well in your academic, professional, and personal life. So, keep exploring, keep questioning, and keep honing your problem-solving skills – they are the key to success in a complex and ever-changing world.