Question

A sum of Rs. 700 is used to give 7 cash prizes to students for academic performance. If each prize is Rs. 20 less than its preceding prize, the value of the first (highest) prize is:

• A. Rs. 140
• B. Rs. 160
• C. Rs. 100
• D. Rs. 180

Hint:

An alternative quick method:
The average of the 7 prizes is \(\frac{700}{7} = 100\).
In any odd-numbered AP, the average is exactly equal to the middle (4th) term.
So, the 4th prize is \(\text{Rs. } 100\).
Since terms decrease by 20, the 1st prize is:
\[ \text{1st prize} = \text{4th prize} + 3 \times 20 = 100 + 60 = 160 \]
This bypasses the algebraic formulas entirely.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
This question is from the topic of Arithmetic Progression (AP).
We are given the total sum of 7 cash prizes, and the rule that each consecutive prize decreases by a fixed amount (\(\text{Rs. } 20\)).
We need to find the value of the first (highest) prize.

Step 2: Key Formula or Approach:
Let the first prize be \(a\).
Since each prize is \(\text{Rs. } 20\) less than the preceding one, the common difference is:
\[ d = -20 \]
The number of terms (prizes) is:
\[ n = 7 \]
The sum of the AP is given by the formula:
\[ S_n = \frac{n}{2} [2a + (n-1)d] \]
Here, \(S_7 = 700\).

Step 3: Detailed Explanation:
Substitute the values into the AP sum formula:
\[ 700 = \frac{7}{2} [2a + (7-1)(-20)] \]
Divide both sides by 7:
\[ 100 = \frac{1}{2} [2a + 6(-20)] \]
Multiply both sides by 2:
\[ 200 = 2a - 120 \]
Isolate \(2a\):
\[ 2a = 200 + 120 \]
\[ 2a = 320 \]
Divide by 2:
\[ a = \frac{320}{2} = 160 \]
Thus, the value of the first (highest) prize is \(\text{Rs. } 160\).

Step 4: Final Answer:
The value of the first (highest) prize is \(\text{Rs. } 160\).