Step 1: Match the age band to its threshold. For 2 to 12 months the alarming respiratory rate is 50 or more per minute; at 3 months a count of 52 crosses this line, confirming fast breathing.
Step 2: Check for severity markers. There is no lower chest wall indrawing and no general danger sign, so the case sits in the mildest pneumonia category.
Step 3: Fast breathing alone, with a clear chest wall and no danger signs, is labelled pneumonia, treated with home oral antibiotics.
\[\boxed{\text{Pneumonia}}\]