Weight of P is twice that of Q. Weight of Q is half of R. Weight of R is 3 times of T. Weight of T is half of S. The weight of Q is greater than the weight of how many persons among P, R, S and T?
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Solving the Weight Comparison Problem (P, Q, R, S, T)
This problem involves understanding the relationships between the weights of five different persons: P, Q, R, S, and T. We are given a series of statements comparing their weights, and we need to determine how many persons among P, R, S, and T have a weight less than the weight of Q.
Understanding the Given Weight Relationships
Let's list the relationships provided in the question:
- Weight of P is twice that of Q: \(P = 2 \times Q\)
- Weight of Q is half of R: \(Q = \frac{1}{2} \times R\)
- Weight of R is 3 times of T: \(R = 3 \times T\)
- Weight of T is half of S: \(T = \frac{1}{2} \times S\)
Expressing All Weights in Terms of One Variable (Q)
To compare the weights easily, let's express the weight of each person (P, R, S, T) in terms of the weight of Q. We already have P in terms of Q and Q in terms of R. Let's use these to find R, T, and S in terms of Q.
- P in terms of Q: This is directly given: \(P = 2Q\).
- R in terms of Q: We know \(Q = \frac{1}{2}R\). To find R, we multiply both sides by 2: \(R = 2Q\).
- T in terms of Q: We know \(R = 3T\) and we just found \(R = 2Q\). So, \(2Q = 3T\). To find T, divide by 3: \(T = \frac{2Q}{3}\).
- S in terms of Q: We know \(T = \frac{1}{2}S\) and we just found \(T = \frac{2Q}{3}\). So, \(\frac{2Q}{3} = \frac{1}{2}S\). To find S, multiply by 2: \(S = 2 \times \frac{2Q}{3} = \frac{4Q}{3}\).
Listing the Weights in Terms of Q
Now we have all weights expressed relative to Q (assuming Q > 0, as weight must be positive):
- Weight of P: \(P = 2Q\)
- Weight of Q: \(Q = Q\)
- Weight of R: \(R = 2Q\)
- Weight of S: \(S = \frac{4}{3}Q\)
- Weight of T: \(T = \frac{2}{3}Q\)
| Person | Weight in terms of Q | Numerical Coefficient of Q |
|---|---|---|
| P | \(2Q\) | \(2\) |
| Q | \(Q\) | \(1\) |
| R | \(2Q\) | \(2\) |
| S | \(\frac{4}{3}Q\) | \(\approx 1.33\) |
| T | \(\frac{2}{3}Q\) | \(\approx 0.67\) |
Comparing the Weight of Q with P, R, S, and T
We need to find out how many persons among P, R, S, and T have a weight less than the weight of Q (\(Q\)). We compare the coefficients of Q for each person with the coefficient for Q (which is 1).
- Compare Q (\(1Q\)) with P (\(2Q\)): Since \(1 < 2\), \(Q < P\). Q's weight is NOT greater than P's.
- Compare Q (\(1Q\)) with R (\(2Q\)): Since \(1 < 2\), \(Q < R\). Q's weight is NOT greater than R's.
- Compare Q (\(1Q\)) with S (\(\frac{4}{3}Q\)): Since \(\frac{4}{3} = 1.33...\) and \(1 < 1.33...\), \(Q < S\). Q's weight is NOT greater than S's.
- Compare Q (\(1Q\)) with T (\(\frac{2}{3}Q\)): Since \(\frac{2}{3} = 0.66...\) and \(1 > 0.66...\), \(Q > T\). Q's weight IS greater than T's.
Conclusion on Weight Comparison
Based on the comparisons, the weight of Q is greater than the weight of T only among the persons P, R, S, and T.
Therefore, the weight of Q is greater than the weight of 1 person among P, R, S, and T.
Revision Table: Weight Comparisons
| Comparison | Result | Is Q's weight greater? |
|---|---|---|
| Q vs P (\(Q\) vs \(2Q\)) | \(Q < P\) | No |
| Q vs R (\(Q\) vs \(2Q\)) | \(Q < R\) | No |
| Q vs S (\(Q\) vs \(\frac{4}{3}Q\)) | \(Q < S\) | No |
| Q vs T (\(Q\) vs \(\frac{2}{3}Q\)) | \(Q > T\) | Yes |
Additional Information on Comparative Weight Problems
Problems like this test your ability to translate word problems into mathematical expressions and then use those expressions for comparison. Here are some key takeaways:
- Define Variables: Assign variables to the quantities you are comparing (like P, Q, R, S, T for weights).
- Write Equations: Convert the relationships given in the text into mathematical equations. Pay close attention to phrases like "twice that of", "half of", "3 times of", etc.
- Standardize Units/Basis: To compare different quantities, it's often easiest to express all of them in terms of a single base quantity, as we did by expressing all weights in terms of Q.
- Perform Comparisons Systematically: Once all quantities are on a common basis, compare them one by one as required by the question. In this case, we compared Q's weight to the weights of P, R, S, and T.
- Careful Counting: Ensure you count only the specific cases asked for in the question (persons whose weight is *less than* Q, among the specified group P, R, S, T).
These types of quantitative reasoning questions are common in various exams and help assess logical thinking and mathematical skills.
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