In a note collection, there is one new Rs 2000 note for every three old Rs 500 notes. If 10 more new Rs 2000 notes are added to the collection, the ratio of new Rs 2000 notes to old Rs 500 notes becomes 1 : 2. Based on this information, what is the total number of notes in the collection after the addition?

Introduction / Context:
This arithmetic reasoning question involves ratios and a simple linear relation between numbers of currency notes of two denominations. Initially the collection has a fixed ratio of new Rs 2000 notes to old Rs 500 notes. After adding some new notes, the ratio changes. Using this information, we determine the actual counts of each type of note and then calculate the total number of notes in the collection.

Given Data / Assumptions:

• Initially, the ratio of new Rs 2000 notes to old Rs 500 notes is 1 : 3.
• Then 10 more Rs 2000 notes are added to the collection.
• After this addition, the new ratio of Rs 2000 notes to Rs 500 notes becomes 1 : 2.
• Old Rs 500 notes are not added or removed in this operation.
• We must find the total count of notes (both denominations) after adding the 10 new Rs 2000 notes.

Concept / Approach:
Let the initial number of new Rs 2000 notes be n. Then, by the given ratio 1 : 3, the initial number of old Rs 500 notes is 3n. After adding 10 new notes, the new count of Rs 2000 notes is n + 10, while old Rs 500 notes remain 3n. The new ratio condition states (n + 10) : (3n) = 1 : 2. Solving this proportion gives the value of n. Using n, we can compute both types of notes and hence the new total.

Step-by-Step Solution:
Step 1: Let the initial number of Rs 2000 notes be n, so initial Rs 500 notes = 3n. Step 2: After adding 10 new Rs 2000 notes, the new count of Rs 2000 notes = n + 10. The old Rs 500 notes remain 3n. Step 3: The new ratio is (n + 10) : 3n = 1 : 2. Step 4: Write the proportion as (n + 10) / (3n) = 1 / 2. Cross-multiplying gives 2(n + 10) = 3n. Step 5: Simplify: 2n + 20 = 3n, so n = 20. Step 6: Initial numbers: Rs 2000 notes = 20, Rs 500 notes = 3 * 20 = 60. Step 7: After adding 10 new Rs 2000 notes, total Rs 2000 notes = 20 + 10 = 30, Rs 500 notes remain 60. Step 8: Total number of notes in the collection after addition = 30 + 60 = 90.

Verification / Alternative check:
Check the final ratio: new Rs 2000 notes : old Rs 500 notes = 30 : 60 = 1 : 2, which matches the condition. Also verify the initial ratio: 20 : 60 = 1 : 3, which matches the original statement. Both ratio checks confirm that the counts are consistent and that the total 90 is correct.

Why Other Options Are Wrong:
Option A 60 would imply far fewer notes and cannot satisfy both ratio conditions with the addition of exactly 10 notes. Option B 30 and option C 70 also fail to match the new ratio 1 : 2 after adding 10 new Rs 2000 notes and the original ratio 1 : 3 simultaneously. Only 90 fits both stages of the problem.

Common Pitfalls:
A common mistake is to treat ratios as fixed differences instead of proportional relations, which leads to linear addition or subtraction errors. Another error is to forget that only the Rs 2000 notes increase while Rs 500 notes remain constant. Keeping the before and after stages clearly separated and writing equations from the ratio definitions prevents confusion.

Final Answer:
The total number of notes in the collection after adding the 10 new Rs 2000 notes is 90.