House Numbering Setup

The problem describes house numbers on two sides of a road:

• Side 1: Consecutive odd integers starting from 301 (301, 303, 305, ...).
• Side 2: Consecutive even integers starting from 302 (302, 304, 306, ...).

Let '$n$' be the number of houses on each side. The total number of houses is the same on both sides.

Sum Calculation for Side 1 (Odd Numbers)

This is an arithmetic progression with:

• First term, $a_1 = 301$.
• Common difference, $d = 2$.
• Number of terms = $n$.

The sum of an arithmetic series is given by $S_n = \frac{n}{2}[2a_1 + (n-1)d]$.

Sum for Side 1 ($S_{odd}$): $ S_{odd} = \frac{n}{2}[2(301) + (n-1)2] $ $ S_{odd} = \frac{n}{2}[602 + 2n - 2] $ $ S_{odd} = \frac{n}{2}[600 + 2n] $ $ S_{odd} = n(300 + n) = 300n + n^2 $

Sum Calculation for Side 2 (Even Numbers)

This is an arithmetic progression with:

• First term, $b_1 = 302$.
• Common difference, $d = 2$.
• Number of terms = $n$.

Sum for Side 2 ($S_{even}$): $ S_{even} = \frac{n}{2}[2(302) + (n-1)2] $ $ S_{even} = \frac{n}{2}[604 + 2n - 2] $ $ S_{even} = \frac{n}{2}[602 + 2n] $ $ S_{even} = n(301 + n) = 301n + n^2 $

Difference Calculation

The difference between the sums is given as 27.

Difference $= S_{even} - S_{odd}$

$ (301n + n^2) - (300n + n^2) = 27 $ $ 301n + n^2 - 300n - n^2 = 27 $ $ n = 27 $

Conclusion

The number of houses on each side of the road is 27.