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 two same-length A.P.s differ termwise by a constant (here, every even number is exactly $1$ more than the preceding odd), the difference of their sums equals that constant times the number of terms: $1\times n=n$.

Answer 1

The Correct Option is A

Step 1: Write the two sequences. Odd side (start at 301, common difference $2$): \[ 301,\; 303,\; \ldots,\; 301+2(n-1)=2n+299. \] Even side (start at 302, common difference $2$): \[ 302,\; 304,\; \ldots,\; 302+2(n-1)=2n+300. \] Step 2: Compute the sums on both sides using A.P. sum. Sum of $n$ odd-side numbers: \[ S_{\text{odd}}=\frac{n}{2}\Big(301+(2n+299)\Big)=\frac{n}{2}(2n+600)=n(n+300). \] Sum of $n$ even-side numbers: \[ S_{\text{even}}=\frac{n}{2}\Big(302+(2n+300)\Big)=\frac{n}{2}(2n+602)=n(n+301). \] Step 3: Use the given difference. Difference between sums (even minus odd): \[ S_{\text{even}}-S_{\text{odd}}=n(n+301)-n(n+300)=n. \] Given this difference equals 27, \[ n=27 \;\Rightarrow\; \boxed{27}. \]