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 arithmetic progressions:
1. Use the formula for the sum of an arithmetic series: \(S = n \times \frac{\text{First term} + \text{Last term}}{2}\).
2. Set up equations based on the problem's conditions to solve for unknowns.
3. Double-check calculations for consistency.

Answer 1

The Correct Option is A

Step 1: Representation of house numbers. On one side of the road, house numbers are consecutive odd integers starting from \(301\). For \(n\) houses, the numbers are: \[ 301, 303, 305, \ldots, (301 + 2(n-1)). \] The sum of these house numbers, \(S_{\text{odd}}\), is given by the formula for the sum of an arithmetic progression: \[ S_{\text{odd}} = n \times \frac{\text{First term} + \text{Last term}}{2}. \] The last term is: \[ 301 + 2(n-1) = 301 + 2n - 2 = 299 + 2n. \] \[ S_{\text{odd}} = n \times \frac{301 + (299 + 2n)}{2} = n \times \frac{600 + 2n}{2} = n \times (300 + n). \] On the other side of the road, house numbers are consecutive even integers starting from \(302\). For \(n\) houses, the numbers are: \[ 302, 304, 306, \ldots, (302 + 2(n-1)). \] The sum of these house numbers, \(S_{\text{even}}\), is: \[ S_{\text{even}} = n \times \frac{302 + (302 + 2(n-1))}{2}. \] \[ S_{\text{even}} = n \times \frac{302 + (300 + 2n)}{2} = n \times \frac{602 + 2n}{2} = n \times (301 + n). \] Step 2: Difference between the sums. The difference between the sums of the two sides of the road is given as \(27\): \[ S_{\text{even}} - S_{\text{odd}} = 27. \] Substitute the expressions for \(S_{\text{even}}\) and \(S_{\text{odd}}\): \[ n \times (301 + n) - n \times (300 + n) = 27. \] Simplify: \[ n(301 + n - 300 - n) = 27. \] \[ n \times 1 = 27. \] \[ n = 27. \]