Question

The sum of four numbers A, B, C, and D is 10,600. Without A, the average of B, C, and D is 1,000. Without B, the average of A, C, and D is 3,220. Without C, the average of A, B, and D is 3,180. Find the value of D.

• A. 880
• B. 7600
• C. 1000
• D. 1120

Hint:

Questions involving averages of subsets are easiest to solve by converting every average into a sum.

Answer 1

The Correct Option is C

Step 1: Use the total.
We know $A+B+C+D=10600$. We will peel off one unknown at a time using the averages.
Step 2: Find A from the first average.
Without A, the average of $B,C,D$ is $1000$, so $B+C+D=3000$. Then \[ A=10600-3000=7600 \]
Step 3: Find C+D from the second average.
Without B, $A+C+D=3\times 3220=9660$. Subtracting $A=7600$ gives \[ C+D=2060 \]
Step 4: Find B+D from the third average.
Without C, $A+B+D=3\times 3180=9540$. Subtracting $A=7600$ gives \[ B+D=1940 \]
Step 5: Recover B.
From $B+C+D=3000$ and $C+D=2060$, we get $B=3000-2060=940$.
Step 6: Recover D.
Put $B=940$ into $B+D=1940$: \[ 940+D=1940 \implies D=1000 \]
\[ \boxed{1000} \]