Question

In a locality, the houses are numbered in the following way: The house-numbers on one side of a road are consecutive odd integers starting from 301, while the house-numbers on the other side of the road are consecutive even numbers starting from 302. The total number of houses is the same on both sides of the road. If the difference of the sum of the house-numbers between the two sides of the road is 27, then the number of houses on each side of the road is:

• A. 27
• B. 52
• C. 54
• D. 26

Hint:

For problems involving sums of consecutive numbers, use the formula for the sum of an arithmetic progression: \[ S_n = \frac{n}{2} \left[ 2a + (n-1)d \right]. \]

Answer 1

The Correct Option is A

Step 1: Problem Setup. The house numbers on one side of the road are consecutive odd integers starting from 301. The house numbers on the other side are consecutive even integers starting from 302. Let the number of houses on each side of the road be \( n \). Step 2: Sum of odd-numbered houses. The sum of odd-numbered houses is: \[ 301 + 303 + 305 + \ldots + \text{(n terms)}. \] Using the formula for the sum of an arithmetic series, we have: \[ \text{Sum of odd-numbered houses} = \frac{n}{2} [2 \times 301 + (n - 1) \times 2]. \] Simplifying: \[ \text{Sum of odd-numbered houses} = \frac{n}{2} [602 + 2n - 2] = n[n + 300]. \] Step 3: Sum of even-numbered houses. The sum of even-numbered houses is: \[ 302 + 304 + 306 + \ldots + \text{(n terms)}. \] Using the same formula for the sum of an arithmetic series, we have: \[ \text{Sum of even-numbered houses} = \frac{n}{2} [2 \times 302 + (n - 1) \times 2]. \] Simplifying: \[ \text{Sum of even-numbered houses} = \frac{n}{2} [604 + 2n - 2] = n[n + 301]. \] Step 4: Difference of sums. According to the question, the difference of the sums of house numbers between the two sides of the road is 27. Thus: \[ n[n + 301] - n[n + 300] = 27. \] Simplifying: \[ n^2 + 301n - (n^2 + 300n) = 27, \] \[ 301n - 300n = 27, \] \[ n = 27. \] Step 5: Conclusion. The number of houses on each side of the road is \( \text{27} \).