Understanding the Problem: Weight of Iron Balls

The question asks for the total weight of ten iron balls. We are told that each ball has the same weight. To find the total weight of ten balls, we first need to determine the weight of a single iron ball. We are given two statements and need to decide if either or both statements are sufficient to find the weight of one ball, and consequently, the total weight of ten iron balls.

Analyzing Statement 1: One-Fourth Weight

Statement 1 says: "One-fourth of each ball’s weight is 5 kg."

Let's denote the weight of one iron ball as \(w\) kg.

According to Statement 1, one-fourth of the weight of a ball is 5 kg. We can write this as an equation:

\(\frac{1}{4} \times w = 5\)

To find the weight \(w\), we can multiply both sides of the equation by 4:

\(w = 5 \times 4\)

\(w = 20\)

So, the weight of one iron ball is 20 kg.

Now that we know the weight of one ball, we can easily find the weight of ten iron balls:

Total weight of ten balls = \(10 \times w = 10 \times 20 = 200\) kg.

Since Statement 1 alone allows us to find the weight of one iron ball and thus the total weight of ten iron balls, Statement 1 is sufficient to answer the question.

Analyzing Statement 2: Comparing Weights of Balls

Statement 2 says: "The total weight of three iron balls is 20 kg more than the total weight of two iron balls."

Let's again denote the weight of one iron ball as \(w\) kg.

The total weight of three iron balls is \(3w\).

The total weight of two iron balls is \(2w\).

According to Statement 2, the weight of three balls is 20 kg more than the weight of two balls. We can write this as an equation:

\(3w = 2w + 20\)

To find the weight \(w\), we can subtract \(2w\) from both sides of the equation:

\(3w - 2w = 20\)

\(w = 20\)

So, the weight of one iron ball is 20 kg.

Again, now that we know the weight of one ball, we can find the weight of ten iron balls:

Total weight of ten balls = \(10 \times w = 10 \times 20 = 200\) kg.

Since Statement 2 alone allows us to find the weight of one iron ball and thus the total weight of ten iron balls, Statement 2 is sufficient to answer the question.

Conclusion on Sufficiency for Iron Ball Weight

We found that Statement 1 alone is sufficient to determine the weight of a single iron ball and the total weight of ten iron balls. We also found that Statement 2 alone is sufficient to determine the weight of a single iron ball and the total weight of ten iron balls.

Therefore, both Statement 1 alone and Statement 2 alone are sufficient to answer the question about the weight of ten iron balls.

Revision Table: Data Sufficiency Concepts

Data sufficiency questions test your ability to determine if the given statements provide enough information to answer a question, without necessarily solving the problem completely. Here are the common outcomes:

• Statement 1 Alone is Sufficient: Statement 1 provides enough information, but Statement 2 does not.
• Statement 2 Alone is Sufficient: Statement 2 provides enough information, but Statement 1 does not.
• Both Statements Together are Sufficient, but Neither Alone is Sufficient: Combining information from both statements answers the question, but neither statement by itself does.
• Each Statement Alone is Sufficient: Statement 1 alone provides enough information, AND Statement 2 alone provides enough information. This is the case in our problem about the iron ball weight.
• Neither Statement is Sufficient: Even combining both statements does not provide enough information to answer the question.

Additional Information: Solving Weight Problems

Problems involving weight and equal items often boil down to finding the weight of a single item. Once the weight of one item is known, the total weight of any number of identical items can be calculated by simple multiplication.

In this problem, both statements effectively provide a linear equation with one variable (the weight of a single ball). Solving this equation gives the weight of one ball, which is the key piece of information needed to answer the question.

Statement 1 uses a fraction of the weight: \(\frac{1}{4}w = 5\). This is a direct way to find \(w\).

Statement 2 uses a difference in total weights: \(3w = 2w + 20\). This equation simplifies to \(3w - 2w = 20\), which also directly gives \(w\). The difference in total weight between three balls and two balls is simply the weight of one ball: \(3w - 2w = w\). So, Statement 2 essentially says the weight of one ball is 20 kg.

Both statements independently lead to the conclusion that the weight of one iron ball is 20 kg, making them each sufficient to find the total weight of ten iron balls.