   SEARCH HOME Math Central Quandaries & Queries  Question from kashish, a student: why we cannot add or subtract "speed" / directly in any given distance , speed , time questions? THERE IS NO SUCH RULE. The rule that one should master is that

ONLY LIKE QUANTITIES MAY BE ADDED TOGETHER.

This rule is often stated negatively as, "you can't add apples and oranges."

EXAMPLE WHERE YOU CAN ADD SPEEDS TOGETHER:
The speed of a boat traveling down a river is added to the speed of the river when determining how fast the boat is moving; for example, if the boat moves at 10 miles per hour in still water, and the river is moving at 2 miles per hour, then the boat will move at 10 + 2 = 12 mph downstream, and at 10 - 2 = 8 mph upstream.

EXAMPLE WHERE ADDING TOGETHER SPEEDS MAKES NO SENSE:
If you peddle a bicycle up a hill at 8 mph and back down at 24 mph, then to find the average speed you must NOT add 8 to 24 -- it's not the speeds that are being combined. To make the problem precise, let's say the distance travelled up is 1 mile, in which case the time elapsed is 1/8 of an hour. The distance down would also be 1 mile, but the time elapsed would be 1/24 of an hour. The distances are combined to make 1+1 = 2 miles travelled in all; the times are also combined to make 1/8 + 1/24 = 3/24 + 1/24 = 4/24 of an hour (or, if you prefer, 7.5 minutes plus 2.5 minutes makes 10 minutes). Thus the average speed for the round trip would be

(total distance)/(total time elapsed) = 2/(1/6) = 12 miles per hour.

Chris     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.