Step 1 :First, we need to calculate the ratio of pizza places per person for both the north and south side of town. The ratio is calculated by dividing the number of pizza places by the total population of the area.
Step 2 :For the north side of town, the total population is \(225\) people/block * \(15\) blocks = \(3375\) people. The ratio of pizza places per person is then \(36\) pizza places / \(3375\) people = \(0.010666666666666666\).
Step 3 :For the south side of town, the total population is \(275\) people/block * \(22\) blocks = \(6050\) people. The ratio of pizza places per person is then \(37\) pizza places / \(6050\) people = \(0.006115702479338843\).
Step 4 :We then compare these ratios to the desired ratio of less than \(0.01\). If the ratio for an area is less than \(0.01\), then the Pretty Pizza Company should consider opening a new pizza place in that area.
Step 5 :Final Answer: The Pretty Pizza Company should open a new pizza place on the \(\boxed{\text{South side of town}}\).