Final answer:
Paul, Brendan, and Charlie's combined work rate for chopping a cord of firewood is 1/4 of a cord per hour. The reciprocal of their combined work rate is 4 hours, which is how long it will take them to chop one cord of firewood together.
Step-by-step explanation:
To determine how long it will take Paul, Brendan, and Charlie to chop one cord of firewood when working together, we need to calculate their combined work rate. To do this, we find how much of the work each person completes in one hour and add these fractions together.
Paul chops 1/10 of the cord in an hour since it takes him 10 hours to chop one cord. Similarly, Brendan chops 1/12 of the cord in an hour, and Charlie chops 1/15 of the cord in an hour. Therefore, their combined hourly work rate is (1/10) + (1/12) + (1/15).
To add the fractions, we need a common denominator, which is 60 in this case. This makes their combined work rate (6/60) + (5/60) + (4/60) = (15/60), which simplifies to 1/4. This means together they chop 1/4 of a cord in one hour.
To find out how long it will take them to chop one full cord, we take the reciprocal of their combined work rate: 1 / (1/4) = 4 hours.
Therefore, it will take 4 hours for the three boys to chop up one cord of firewood when working together.