In some code, letters P, Q, R, S, T represent numbers 4, 5, 10, 12, 15. It is not known which letter represents which number. If Q – S = 2S and T = R + S + 3, then what is the value of P + R – T?

Correct Option (B)

The given letters P, Q, R, S, T represent the numbers 4, 5, 10, 12, 15. The specific mapping of letters to numbers is initially unknown.

From the first given equation, Q – S = 2S, we can simplify it to Q = 3S.

We must identify pairs of numbers (S, Q) from the set {4, 5, 10, 12, 15} that satisfy the condition Q = 3S:

• If S = 4, then Q = 3 × 4 = 12. Both 4 and 12 are present in the given set of numbers. This is a possible assignment.
• If S = 5, then Q = 3 × 5 = 15. Both 5 and 15 are present in the given set of numbers. This is also a possible assignment.
• No other number from the set, when assigned to S, yields a Q value that is also present in the set (e.g., if S=10, Q=30, which is not available).

Next, we utilize the second equation, T = R + S + 3, and evaluate the two viable scenarios for (S, Q):

Scenario 1: S = 4 and Q = 12

• The numbers remaining for P, R, and T are {5, 10, 15}.
• Substitute S=4 into the equation T = R + S + 3: T = R + 4 + 3 ⇒ T = R + 7.
• We need to find R and T from the remaining set {5, 10, 15} that satisfy T = R + 7.
• If R = 5, then T = 5 + 7 = 12. However, the number 12 has already been assigned to Q in this scenario. Therefore, this specific assignment for R and T is invalid.
• Other potential assignments for R from {5, 10, 15} do not yield a corresponding T value within the remaining set (e.g., if R=10, T=17; if R=15, T=22).
• Consequently, Scenario 1 is not a valid solution path.

Scenario 2: S = 5 and Q = 15

• The numbers remaining for P, R, and T are {4, 10, 12}.
• Substitute S=5 into the equation T = R + S + 3: T = R + 5 + 3 ⇒ T = R + 8.
• We need to find R and T from the remaining set {4, 10, 12} that satisfy T = R + 8.
• If R = 4, then T = 4 + 8 = 12. Both 4 and 12 are available in the remaining set {4, 10, 12}. This assignment is valid.
• Other potential assignments for R from {4, 10, 12} do not yield a corresponding T value within the remaining set (e.g., if R=10, T=18; if R=12, T=20).
• Thus, Scenario 2 is the correct solution path, establishing R=4 and T=12.

Based on Scenario 2, the confirmed assignments are:

• S = 5
• Q = 15
• R = 4
• T = 12

The only remaining letter is P, and the only remaining number is 10. Therefore, P = 10.

Finally, we calculate the value of the expression P + R – T:

P + R – T = 10 + 4 – 12 = 14 – 12 = 2.

Incorrect Options:

The problem provides sufficient information to determine a unique mapping of letters to numbers, leading to a specific value for P + R – T. The systematic evaluation of the given equations and number set yields a definite result of 2. Therefore, options 1 (1), 3 (3), and 4 (Cannot be determined due to insufficient data) are incorrect.