Question

Rajiv is a student at a business school. After every test, he calculates his cumulative average score. QT and OB were his last two tests. Scoring 83 in QT increased his average by 2. Scoring 75 in OB further increased his average by 1. If his next test is Reasoning and he scores 51 in it, his new average will be

• A. 63
• B. 62
• C. 61
• D. 60

Hint:

Let his average and number of tests before QT be A and n, then write one equation for how 83 changes the average and another for how 75 changes it again.

Answer 1

The Correct Option is A

Step 1: Recall the surplus rule for averages.
When a new score is added to a group and the average changes, the new score always equals the new average plus (the number of old items) times (the rise in average). This is because the extra marks the new score contributes above the new average must exactly balance the boost every old item's average also received. Using this rule avoids expanding totals by hand.

Step 2: Apply the rule to QT.
Let A be the average and n the number of tests before QT. The average rises by 2, so:
\[ 83 = (A + 2) + n(22) \implies A + 2n = 81 \quad \text{...(ii)} \]
This is the same relationship as adding directly, just reached through the surplus idea instead of expanding brackets.

Step 3: Apply the rule to OB.
Before OB, there are $(n+1)$ tests at average $(A+2)$. The average rises by 1 more, so:
\[ 75 = (A + 3) + (n+1)(11) \implies 75 = A + n + 4 \implies A + n = 71 \quad \text{...(ii)} \]

Step 4: Solve for n and A.
Subtracting (ii) from (ii): $n = 81 - 71 = 10$, so $A = 71 - 10 = 61$.

Step 5: Apply the surplus rule once more for the Reasoning test.
Before Reasoning, there are $n + 2 = 12$ tests at average $A + 3 = 64$. Let the new average be $x$ after adding the score 51. By the same rule:
\[ 51 = x + 12(x - 64) \] \[ 51 = 13x - 768 \] \[ 13x = 819 \implies x = 63 \]

Final Answer:
Rajiv's average after the Reasoning test is 63.\[ \boxed{63} \]