Step 1: IMNCI uses three respiratory-rate thresholds because normal breathing rate falls with age. Identify which age band a 6-month-old belongs to.
Step 2: The bands are $<2$ months $\rightarrow 60/min$, $2$ to $12$ months $\rightarrow 50/min$, and $12$ months to $5$ years $\rightarrow 40/min$.
Step 3: Six months sits squarely in the 2-to-12-month group, so the fast-breathing cut-off is 50 breaths per minute. The lower value 40 applies only after the first birthday, and 60 applies only in young infants under 2 months.
\[\boxed{50 \text{ breaths/min}}\]