A sum of Rs 1890 is to be distributed as 9 cash prizes to customers of a supermarket for their overall purchases. Each prize is Rs 30 less than the preceding one. What is the least value among these nine prizes?

Introduction / Context:
This question is based on arithmetic progression, where we have a fixed number of terms, a constant difference between consecutive terms and a given total sum. The prizes form a decreasing arithmetic sequence since each prize is Rs 30 less than the previous one. We are asked to find the smallest prize amount, which is the last term of this sequence when arranged from highest to lowest.

Given Data / Assumptions:

• Total prize money = Rs 1890.
• Number of prizes n = 9.
• Each prize is Rs 30 less than the preceding one, so the common difference in a decreasing sequence is −30.
• We can equivalently consider an increasing sequence with difference +30 from the least prize upwards.

Concept / Approach:
Let the least prize be L. Then the next prizes are L + 30, L + 60, and so on, up to L + 8 × 30. These 9 values form an arithmetic progression with first term L, common difference 30 and 9 terms. The sum of an arithmetic progression is given by S = n/2 × [2a + (n − 1)d], where a is the first term and d is the common difference. We plug in the known values and solve for L.

Step-by-Step Solution:
Write the nine prizes as: L, L + 30, L + 60, …, L + 8 × 30.Here, a = L, d = 30 and n = 9.Apply the sum formula: S = n/2 × [2a + (n − 1)d] = 9/2 × [2L + 8 × 30].We are given S = 1890, so 1890 = 9/2 × (2L + 240).Multiply both sides by 2 to clear the fraction: 3780 = 9(2L + 240).Divide both sides by 9: 420 = 2L + 240, so 2L = 180 and L = 90.

Verification / Alternative check:
If the least prize is Rs 90, the prizes are 90, 120, 150, 180, 210, 240, 270, 300 and 330.Sum these: pair them symmetrically (90 + 330) + (120 + 300) + (150 + 270) + (180 + 240) + 210 = 420 + 420 + 420 + 420 + 210 = 1890.This matches the given total, confirming that L = 90 is correct.

Why Other Options Are Wrong:
A least prize of 80 or 85 would produce a different total sum, not equal to 1890, when the 9-term arithmetic progression is summed.Similarly, 95 as the least prize would give an overall sum larger than 1890 due to every term being higher.

Common Pitfalls:
Some students mistakenly take the highest prize as the first term without converting carefully between increasing and decreasing sequences.Another pitfall is forgetting the factor n/2 in the sum formula and using S = n(2a + (n − 1)d), which gives an incorrect equation for L.

Final Answer:
The least of the nine prizes, and hence the smallest amount, is Rs 90.