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:

When working with sequences and series, especially in real-world contexts like numbering systems, setting up the series formula correctly is crucial for solving the problem efficiently.

Answer 1

The Correct Option is A

Let \( n \) be the number of houses on each side of the road. The house numbers on the odd side start from 301 and end at \( 301 + 2(n-1) \), and on the even side start from 302 and end at \( 302 + 2(n-1) \). Sum of odd-numbered houses: \[ S_{\text{odd}} = \frac{n}{2} \left[2 \times 301 + (n-1) \times 2\right] = n \left[301 + (n-1)\right] \] Sum of even-numbered houses: \[ S_{\text{even}} = \frac{n}{2} \left[2 \times 302 + (n-1) \times 2\right] = n \left[302 + (n-1)\right] \] The difference between the sums of the house numbers is given to be 27: \[ S_{\text{even}} - S_{\text{odd}} = n \left[(302 + (n-1)) - (301 + (n-1))\right] = n \times 1 = n = 27 \] Thus, the number of houses on each side of the road is \( \boxed{27} \).