Solving the Average Weight Problem

This question asks us to find the weight of P, given information about the average weight of a group including P and the average weight of the rest of the group.

Understanding the Problem Details

We are given two key pieces of information:

• The average weight of P and his three friends is 55 kg. This involves a total of 4 people (P + 3 friends).
• P's weight is 4 kg more than the average weight of just his three friends.

Setting Up Equations for Weight Calculation

Let's use variables to represent the unknown weights:

• Let \(W_P\) be the weight of P (in kg).
• Let \(W_1, W_2, W_3\) be the weights of the three friends (in kg).

From the first piece of information, the total weight of the four people is the average weight multiplied by the number of people:

\(\frac{W_P + W_1 + W_2 + W_3}{4} = 55\)

Multiplying both sides by 4, we get the total weight:

\(W_P + W_1 + W_2 + W_3 = 55 \times 4 = 220\) kg. (Equation 1)

From the second piece of information, P's weight is related to the average weight of the three friends:

Average weight of three friends \( = \frac{W_1 + W_2 + W_3}{3}\)

P's weight is 4 kg more than this average:

\(W_P = \frac{W_1 + W_2 + W_3}{3} + 4\) (Equation 2)

Solving for P's Weight

Let \(S_3\) represent the sum of the weights of the three friends: \(S_3 = W_1 + W_2 + W_3\).

Now we can rewrite our equations using \(S_3\):

• From Equation 1: \(W_P + S_3 = 220\)
• From Equation 2: \(W_P = \frac{S_3}{3} + 4\)

We can rearrange the second equation to express \(S_3\) in terms of \(W_P\):

Multiply by 3: \(3W_P = S_3 + 12\)

Subtract 12 from both sides: \(S_3 = 3W_P - 12\) (Equation 3)

Now, substitute the expression for \(S_3\) from Equation 3 into the first equation (\(W_P + S_3 = 220\)):

\(W_P + (3W_P - 12) = 220\)

Combine the terms with \(W_P\):

\(4W_P - 12 = 220\)

Add 12 to both sides:

\(4W_P = 220 + 12\)

\(4W_P = 232\)

Divide by 4 to find \(W_P\):

\(W_P = \frac{232}{4}\)

\(W_P = 58\)

So, P's weight is 58 kg.

Verification of P's Weight Calculation

Let's check if our answer fits the original conditions.

If \(W_P = 58\) kg, then from \(W_P + S_3 = 220\), we get \(58 + S_3 = 220\). So, \(S_3 = 220 - 58 = 162\) kg.

The sum of the weights of the three friends is 162 kg.

The average weight of the three friends is \(\frac{S_3}{3} = \frac{162}{3} = 54\) kg.

According to the problem, P's weight should be 4 kg more than this average: \(54 + 4 = 58\) kg.

Our calculated weight for P is 58 kg, which matches this condition. The average weight of P and his three friends is \(\frac{W_P + S_3}{4} = \frac{58 + 162}{4} = \frac{220}{4} = 55\) kg, which also matches the condition.

Therefore, P's weight is indeed 58 kg.

Revision Table: Key Concepts for Average Problems

Additional Information: Solving Average Weight Problems

Average weight problems often involve setting up relationships between the total weight of different groups of people or objects. Here are some tips:

• Always define your variables clearly, representing the quantities you need to find or use.
• Translate the given information into mathematical equations. Pay close attention to which group the average or sum refers (e.g., everyone vs. a subgroup).
• If you have multiple unknowns, you will likely need a system of equations. Use substitution or elimination methods to solve them.
• Remember that the sum of quantities is always the average multiplied by the count of quantities. This is a fundamental relationship in average problems.
• Always verify your final answer by plugging it back into the original problem statement to ensure all conditions are met.
• These types of word problems are common in quantitative aptitude tests and require careful reading and step-by-step problem-solving.