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: Name the unknowns.
Let A be Rajiv's average before the test QT, and let n be the number of tests he had already taken before QT. So his total marks before QT were \(nA\).

Step 2: Use the QT information.
After QT, he has taken \(n+1\) tests and scored 83 in it, and his average has become \(A+2\). Total marks after QT equal total marks before plus 83, and also equal the new average times the new count:
\[ nA + 83 = (n+1)(A+2) \]
Expand the right side: \((n+1)(A+2) = nA + 2n + A + 2\). Cancel \(nA\) from both sides:
\[ 83 = 2n + A + 2 \implies A + 2n = 81 \quad \text{...(ii)} \]

Step 3: Use the OB information.
After OB, he has taken \(n+2\) tests, scored 75 in it, and his average has risen by 1 more, to \(A+3\). So:
\[ (n+1)(A+2) + 75 = (n+2)(A+3) \]
Expand both sides: the left is \(nA + 2n + A + 2 + 75\), the right is \(nA + 3n + 2A + 6\). Cancel \(nA\) and simplify:
\[ 2n + A + 77 = 3n + 2A + 6 \implies n + A = 71 \quad \text{...(ii)} \]

Step 4: Solve equations (ii) and (ii) together.
Subtract (ii) from (ii): \((A+2n) - (A+n) = 81 - 71\), which gives \(n = 10\). Then from (ii), \(A = 71 - 10 = 61\).

Step 5: Track the totals to reach the final average.
Before QT: 10 tests, total \(= 10 \times 61 = 610\). After QT: 11 tests, total \(= 610 + 83 = 693\), average \(= 693/11 = 63\) (matches \(A+2=63\), a good check). After OB: 12 tests, total \(= 693 + 75 = 768\), average \(= 768/12 = 64\) (matches \(A+3=64\), another good check). After Reasoning: 13 tests, total \(= 768 + 51 = 819\), average \(= 819/13 = 63\).

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