Question

A sum of Rs. 280 is to be used to award four prizes. If each prize after the first prize is Rs. 20 less than its preceding prize, then the value of the fourth prize is

• A. \(20\)
• B. \(40\)
• C. \(60\)
• D. \(80\)
• E. \(10\)

Hint:

In decreasing A.P., always take common difference as negative and write all terms clearly.

Answer 1

The Correct Option is B

Concept: The prizes form an Arithmetic Progression (A.P.) because each prize is a fixed amount less than the previous one. If the first term is \(a\) and common difference is \(d\), then the terms are: \[ a,\; a+d,\; a+2d,\; a+3d \]

Step 1: Identify the common difference. Since each prize is Rs. 20 less than the previous: \[ d = -20 \]

Step 2: Write all four prizes explicitly. \[ \text{1st prize} = a \] \[ \text{2nd prize} = a - 20 \] \[ \text{3rd prize} = a - 40 \] \[ \text{4th prize} = a - 60 \]

Step 3: Use the total sum condition. \[ a + (a-20) + (a-40) + (a-60) = 280 \] Now combine like terms step-by-step: \[ = a + a + a + a - 20 - 40 - 60 \] \[ = 4a - (20+40+60) \] \[ = 4a - 120 \] Thus: \[ 4a - 120 = 280 \]

Step 4: Solve for \(a\). \[ 4a = 280 + 120 \] \[ 4a = 400 \] \[ a = 100 \]

Step 5: Find the fourth prize. \[ \text{Fourth prize} = a - 60 = 100 - 60 = 40 \]