Finding the Heaviest Object Among Five

This question asks us to compare the weights of five objects, P, Q, R, S, and T, based on given relationships and determine which object is the heaviest.

Analyzing the Given Weight Relationships

We are provided with the following information about the weights of the five objects:

• Q is twice as heavy as P.
• S is twice as heavy as Q.
• R is half as heavy as T.
• T is equally as heavy as Q.

Establishing Mathematical Relationships

Let's represent the weight of each object by its letter (P, Q, R, S, T). We can translate the given statements into mathematical equations or comparisons:

• Q is twice as heavy as P: \(Q = 2P\)
• S is twice as heavy as Q: \(S = 2Q\)
• R is half as heavy as T: \(R = \frac{1}{2}T\)\( or \)\(T = 2R\)
• T is equally as heavy as Q: \(T = Q\)

Comparing the Weights

We need to compare the weights of all five objects. A good way to do this is to express the weight of each object in terms of a single object's weight. From the relationships, we see that T is equal to Q (\(T = Q\)). Let's use Q as the reference point.

• From \(Q = 2P\)\(, we get \)\(P = \frac{1}{2}Q\).
• We already know \(Q = Q\).
• From \(S = 2Q\), S is twice the weight of Q.
• From \(T = Q\), T has the same weight as Q.
• From \(R = \frac{1}{2}T\)\( and \)\(T = Q\)\(, we substitute T with Q: \)\(R = \frac{1}{2}Q\).

So, comparing the weights relative to Q:

• Weight of P = \(\frac{1}{2}Q\)
• Weight of Q = \(1Q\)
• Weight of R = \(\frac{1}{2}Q\)
• Weight of S = \(2Q\)
• Weight of T = \(1Q\)

Let's summarize this in a table for clarity:

Now we can clearly see the weights relative to Q. To find the heaviest object, we look for the object with the largest positive coefficient when expressed in terms of Q (assuming weights are positive).

• Coefficient for P is \(\frac{1}{2}\).
• Coefficient for Q is \(1\).
• Coefficient for R is \(\frac{1}{2}\).
• Coefficient for S is \(2\).
• Coefficient for T is \(1\).

Comparing the coefficients (\(\frac{1}{2}, 1, \frac{1}{2}, 2, 1\)\(), the largest value is \)\(2\). This corresponds to the object S.

Conclusion: Identifying the Heaviest Object

Based on our analysis of the weight relationships, the object S has a weight of \(2Q\)\(, while the other objects weigh \)\(Q\)\(, \)\(\frac{1}{2}Q\)\(, or \)\(Q\). Therefore, S is the heaviest among the five objects P, Q, R, S, and T.

Revision Table: Comparing Object Weights

From this table, it is evident that S has the largest weight coefficient (2) when compared to Q.

Additional Information: Solving Comparison Problems

Comparison problems involving weights, heights, or other quantities often require careful reading and systematic organization of the given information. Here are some tips for solving such problems:

• Identify the items being compared: In this case, they are the five objects P, Q, R, S, and T.
• List all given relationships: Write down each piece of information provided about how the items compare.
• Translate relationships into a consistent form: Use mathematical equations or inequalities. For example, "A is twice as heavy as B" can be written as \(A = 2B\)\( or \)\(A > B\)\(. "C is half as heavy as D" is \)\(C = \frac{1}{2}D\)\( or \)\(C < D\)\(. "E is equally as heavy as F" is \)\(E = F\).
• Find a common reference point: If possible, express all quantities in terms of one item, as we did with Q in this problem. This makes direct comparison easier.
• Use substitution: Substitute known equalities into other relationships to simplify them and find connections between items that weren't directly compared initially.
• Compare the values: Once all quantities are expressed in a comparable form (like relative to Q), identify the minimum, maximum, or required order based on the values.
• Check your work: Reread the original problem and verify that your final answer is consistent with all the given conditions.

These techniques can be applied to various types of comparison problems.