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
• E.

Hint:

For problems involving sums of consecutive numbers, use the arithmetic progression formula: \[ S_n = \frac{n}{2} \left[2a + (n-1)d\right], \] where \( a \) is the first term, \( n \) is the number of terms, and \( d \) is the common difference.

Answer 1

The Correct Option is A

Step 1: Problem setup.
House numbers on one side are consecutive odd integers starting from 301. On the other side, they are consecutive even integers starting from 302. Let \( n \) represent the number of houses on each side.
Step 2: Sum of odd-numbered houses.
Using the arithmetic progression formula, the sum is:
\[ n[n + 300]. \]
Step 3: Sum of even-numbered houses.
Similarly, the sum of even-numbered houses is:
\[ n[n + 301]. \]
Step 4: Calculating the difference in sums.
Given that the difference between the sums is 27:
\[ 301n - 300n = 27 \implies n = 27. \]
Step 5: Final conclusion.
The number of houses on each side of the road is \( 27 \).