Question

Directions: A statement is followed by three conclusions. Choose the answer from the options below.
(AA) Using the given statement, only conclusion I can be derived.
(BB) Using the given statement, only conclusion II can be derived.
(CC) Using the given statement, only conclusion III can be derived.
(DD) Using the given statement, all conclusions can be derived.
(EE) Using the given statement, none of the three conclusions I, II and III can be derived.

A, B, C and D are whole numbers such that:
\(A + B + C = 118\)
\(B + C + D = 156\)
\(C + D + A = 166\)
\(D + A + B = 178\)
Conclusion I: A is the smallest number and A = 21.
Conclusion II: D is the largest number and D = 88.
Conclusion III: B is the largest number and B = 56.

• A. Using the given statement, only conclusion I can be derived.
• B. Using the given statement, only conclusion II can be derived.
• C. Using the given statement, only conclusion III can be derived.
• D. Using the given statement, all conclusions can be derived.

Hint:

Add all four given equations together first; that sum equals 3 times (A+B+C+D), which lets you find each variable one at a time.

Answer 1

The Correct Option is B

Step 1: Set up the four equations.
We are told:
\[ A + B + C = 118 \quad \text{(ii)} \]
\[ B + C + D = 156 \quad \text{(ii)} \]
\[ C + D + A = 166 \quad \text{(iii)} \]
\[ D + A + B = 178 \quad \text{(iv)} \]
Each equation leaves out exactly one of the four letters, which is the key to solving this quickly.

Step 2: Add all four equations.
Adding (ii), (ii), (iii) and (iv), each of \(A\), \(B\), \(C\) and \(D\) appears in exactly 3 of the 4 equations, so the left side becomes \(3(A+B+C+D)\).
The right side is \(118 + 156 + 166 + 178 = 618\).
So \(3(A+B+C+D) = 618\), which gives \(A + B + C + D = 206\). Call this equation (vv).

Step 3: Find each letter by subtracting one equation from (vv) at a time.
Equation (ii) leaves out \(D\), so subtracting (ii) from (vv) leaves just \(D\): \(D = 206 - 118 = 88\).
Equation (ii) leaves out \(A\), so subtracting (ii) from (vv) leaves just \(A\): \(A = 206 - 156 = 50\).
Equation (iii) leaves out \(B\), so subtracting (iii) from (vv) leaves just \(B\): \(B = 206 - 166 = 40\).
Equation (iv) leaves out \(C\), so subtracting (iv) from (vv) leaves just \(C\): \(C = 206 - 178 = 28\).

Step 4: Check the numbers add up correctly.
\(A + B + C + D = 50 + 40 + 28 + 88 = 206\), which matches (vv), so the values are consistent.
Checking equation (ii): \(A + B + C = 50 + 40 + 28 = 118\), correct.

Step 5: Test each conclusion against the values \(A=50, B=40, C=28, D=88\).
Conclusion I says A is smallest and A = 21. The actual smallest value is \(C = 28\), not \(A\), and \(A\) actually equals \(50\), not \(21\). So Conclusion I is false.
Conclusion II says D is largest and D = 88. Comparing all four values \(50, 40, 28, 88\), the biggest one is \(88\), belonging to \(D\). So Conclusion II is true.
Conclusion III says B is largest and B = 56. But \(B = 40\), not \(56\), and the largest value is \(D = 88\), not \(B\). So Conclusion III is false.

Step 6: Match with the answer options.
Only Conclusion II turned out to be true, and Conclusions I and III were both false, so the option claiming only conclusion II can be derived is correct. This rules out the only I, only III, and all conclusions options.

Final Answer:
Only Conclusion II can be derived: D is the largest number, and D = 88.
\[ \boxed{D = 88} \]