Correct Option (B)
Let the seven cash prizes be represented as an arithmetic progression. The problem states that each prize is Rs 20 less than its preceding prize. To simplify the calculation for the least value, let 'a' denote the value of the smallest (seventh) prize.
Based on this, the values of the seven prizes, in ascending order, are:
- Seventh prize: a
- Sixth prize: a + 20
- Fifth prize: a + 40
- Fourth prize: a + 60
- Third prize: a + 80
- Second prize: a + 100
- First prize: a + 120
The total sum of these seven prizes is given as Rs 700.
Sum = a + (a + 20) + (a + 40) + (a + 60) + (a + 80) + (a + 100) + (a + 120) = 700
Combining the 'a' terms and the constant terms:
7a + (20 + 40 + 60 + 80 + 100 + 120) = 700
7a + 420 = 700
Subtracting 420 from both sides of the equation:
7a = 700 - 420
7a = 280
Dividing by 7:
a = 280 / 7
a = 40
Therefore, the least value of the prize is Rs 40.
Incorrect Options:
The general formula for the sum of the seven prizes, where 'a' is the least prize, is 7a + 420. This sum must equal Rs 700.
- If the least value were Rs 30 (Option A): The sum would be 7(30) + 420 = 210 + 420 = 630. This does not equal Rs 700.
- If the least value were Rs 60 (Option C): The sum would be 7(60) + 420 = 420 + 420 = 840. This does not equal Rs 700.
- If the least value were Rs 80 (Option D): The sum would be 7(80) + 420 = 560 + 420 = 980. This does not equal Rs 700.
These calculations demonstrate that only a least prize value of Rs 40 yields a total sum of Rs 700, consistent with the problem statement.