 One question that many students ask is why is the growth or decay factor in an exponential equation equal to 1 plus the rate of growth of decay? So you have an equation this form where a is the initial amount and y is the final amount and x is the unit of time measured in years but any unit of time that suits your problem is just fine and then B is the growth or decay factor the growth or decay factor now if we say that the rate of growth let's call that P why is it that B is equal to 1 plus P where we will interpret P as positive if it represents growth and negative if it represents decay let's look at two examples one of growth and one of decay the foundation of your house is about 1200 termites the termites grow at a rate of about 2.4% per day what will the population of termites be in five days this is a y is equal to a b x now in this case we know a to b we know a to be 1200 and we know x to be five in five days and B well B will be its growth so it'll be 1 plus 0.024 we expressed this percent in decimal form so that this is just y is equal to 1200 times 1.024 raised to the fifth power and whatever number that is that will be the population of the of termites after five days let's consider one of a decay you buy a new computer for $2100 that's the initial amount $2100 the computer decreases it decreases by 45% annually that's the rate of decrease it wrote the rate of decay what will be its value in three years well this will be one minus because it is decreasing its decay minus 0.45 we're measuring time in years now raised to the third power in other words y will be equal to 2100 times 0.55 raised to the third power whatever number that is that will be the value of your computer after three years but what about the general case if you have an equation an exponential equation of this form why is it that the growth factor is equal to 1 plus p where p is the rate of growth or the rate of decay well one way to look at it is that after one unit of time say after one one year after one unit of time after one unit of time y will be equal to a times b to the first power that's just a times b that is just a times b and another way of looking at this is after one unit of time the final population will be a will be a plus plus whatever you had at the beginning times the growth or decay factor which we are calling p we factor out the a we see that this is just one plus b and now combine looking at these two parts of this equation we can see why since a times b equals a times one plus b that b is indeed one plus b thank you