By increasing the price of an entry ticket to a fair in the ratio 17 : 19, the number of visitors decreases in the ratio 11 : 10. In what ratio has the total collection changed (old collection to new collection)?

Introduction / Context:
This question combines the ideas of ratio, price change and demand change to find the effect on total revenue or collection. Ticket price and visitor count are both changed in given ratios, and we must determine how the total collection changes. Such problems are important for understanding the combined effect of price and quantity changes in simple economic scenarios.

Given Data / Assumptions:

• Price of entry ticket increases in the ratio 17 : 19.
• Number of visitors decreases in the ratio 11 : 10.
• Total collection is defined as ticket price multiplied by number of visitors.
• We are to find the ratio of old collection to new collection.

Concept / Approach:
Let the original price be P and the original number of visitors be N. The new price then becomes (19/17) * P, and the new number of visitors becomes (10/11) * N. Total collection originally is P * N. New collection is the product of new price and new visitors. The ratio of new collection to old collection is obtained by multiplying the price ratio and visitor ratio. Then we interpret this ratio and compare to the options which describe whether the collection has increased or decreased and by what ratio.

Step-by-Step Solution:
1) Let original ticket price be P and original number of visitors be N. 2) New price = (19/17) * P because price increases in the ratio 17 : 19. 3) New visitors = (10/11) * N because visitors decrease in the ratio 11 : 10. 4) Original collection = P * N. 5) New collection = (19/17 * P) * (10/11 * N) = (19 * 10) / (17 * 11) * P * N. 6) Simplify the factor: (19 * 10) / (17 * 11) = 190 / 187. 7) So new collection : old collection = 190 : 187. 8) This means new collection is slightly larger than old collection, since 190 > 187. 9) Therefore old collection : new collection = 187 : 190.
Verification / Alternative check:
We can assume convenient numbers to verify. Let original price P = Rs 17 and original visitors N = 11, so original collection = 17 * 11 = 187. New price becomes Rs 19 and new visitors become 10, so new collection = 19 * 10 = 190. The ratio of old to new collection is 187 : 190, consistent with our calculation. This confirms that the total collection has increased and not decreased.

Why Other Options Are Wrong:
Option A states that the collection decreased in the ratio 190 : 187, which is incorrect both in direction (it is actually an increase) and ratio order. Options C and D use ratios 170 : 209 or 209 : 170, which do not match the factor 190 : 187 obtained from multiplying the price and visitor ratios. Option E assumes no change in collection, which is also wrong because the product factor is not equal to 1. Only option B correctly describes the change as an increase with old collection : new collection = 187 : 190.

Common Pitfalls:
Learners often invert ratios or confuse which ratio is old : new versus new : old. Another common mistake is to subtract the ratios instead of multiplying them, or to treat ratios as simple percentage changes without performing proper multiplication. Always express both price and quantity changes as multiplicative factors, multiply them to get the revenue factor, and then convert that into a ratio in the correct order required by the question.

Final Answer:
The total collection has increased in the ratio 187 : 190 (old collection : new collection).