Page:Calculus Made Easy.pdf/73

 What do we mean by rate? In both these cases we are making a mental comparison of something that is happening, and the length of time that it takes to happen. If the motor-car flies past us going $$10$$ yards per second, a simple bit of mental arithmetic will show us that this is equivalent–while it lasts–to a rate of $$600$$ yards per minute, or over $$20$$ miles per hour.

Now in what sense is it true that a speed of $$10$$ yards per second is the same as $$600$$ yards per minute? Ten yards is not the same as $$600$$ yards, nor is one second the same thing as one minute. What we mean by saying that the rate is the same, is this: that the proportion borne between distance passed over and time taken to pass over it, is the same in both cases.

Take another example. A man may have only a few pounds in his possession, and yet be able to spend money at the rate of millions a year–provided he goes on spending money at that rate for a few minutes only. Suppose you hand a shilling over the counter to pay for some goods; and suppose the operation lasts exactly one second. Then, during that brief operation, you are parting with your money at the rate of $$1$$ shilling per second, which is the same rate as £$$3$$ per minute, or £$$180$$ per hour, or £$$4320$$ per day, or £$$1,576,800$$ per year! If you have £$$10$$ in your pocket, you can go on spending money at the rate of a million a year for just $$5\tfrac{1}{4}$$ minutes.

It is said that Sandy had not been in London