A sum of ₹ 700 is to be used to give seven cash prizes to students of a school for their overall academic performance. If each prize is ₹ 20 less than its preceding prize, find the value of each of the prizes.
The seven prizes are: ₹ 160, ₹ 140, ₹ 120, ₹ 100, ₹ 80, ₹ 60, and ₹ 40.
Explanation
This problem involves an Arithmetic Progression where the prizes decrease by a fixed amount. The total sum of 7 prizes equals ₹ 700, and the common difference is -20 (since each prize is ₹ 20 less than the preceding one). Using the sum formula for AP, we can find the first term and then calculate all seven prizes.
Solution Steps
Step 1: Let the first prize be ₹ a. Since each prize is ₹ 20 less than its preceding prize, this forms an AP with common difference d = -20 and number of terms n = 7.
Step 2: Using the sum formula: S = n/2 × [2a + (n-1)d]. Substituting the values: 700 = 7/2 × [2a + (7-1)(-20)]
Step 3: Simplifying: 700 = 7/2 × [2a - 120]. Multiplying both sides by 2: 1400 = 7 × (2a - 120)
Step 4: Solving: 1400 = 14a - 840, therefore 14a = 2240, giving a = ₹ 160.
Step 5: The seven prizes are: ₹ 160, ₹ 140, ₹ 120, ₹ 100, ₹ 80, ₹ 60, ₹ 40.
Verification: 160 + 140 + 120 + 100 + 80 + 60 + 40 = ₹ 700 ✓