To best compare these fractions, it is helpful to reduce them to their simplest form.  is already as reduced as it can be.

For , you can factor out a 2 from the numerator and denominator to get to . Note that you could again divide by 2 to get , but here you can also be creative: if you were comparing to  and  as two options, you should see that  is a hair less than , which would reduce directly to .  So you can eliminate  by that comparison.

7007 and 6006 are each divisible by 1001 (something that should be somewhat evident by the similar way the numbers look). If you then factor out the 1001 you have  and the 1001s cancel, leaving .  Since you cannot have two correct answers, then  and  are a "tie" and both are incorrect. And when you see that  is just a bit larger than , a number that would also equal , you can see that  is correct.

Your first order of business on this problem should be to plug in the value of the constant  to simplify your work. When you do so, you have:

 which simplifies to .

Now you can plug in the two values of  you're given. When  the potential energy is .  When  the potential energy is . So between 2 and 3 seconds, the spring gains  joules.

It is important to recognize that this is an exponential rate of increase. If the population doubles every 3 days, then using x as the number of people who would be infected today, you can trace the growth exponentially using:

Today: x people

3 days from now: 2x people

6 days from now: 4x people (2x will have doubled to 4x)

9 days from now: 8x people (4x will have doubled to 8x)

So the correct ratio is 1:8.

Since  will be the same for each rider, you can take this ratio by plugging in their zone numbers.  Tamara's rate would be defined by  or  and Kenneth's rate would be defined as  or just . The ratio of their rates, then, is 16:1.

Whenever you're provided an equation for how to calculate an exponential outcome from an exponential proportion, it's helpful to try to solve the problem algebraically, as often the results are not directly intuitive to try to solve just conceptually.  Here you're given the current population, , and the way that that population can be calculated based on the number of days, . You can then set the population equal to  and perform the algebra:

You can divide both sides by  to simplify:

And then take the square root of both sides:

Multiply both sides by 2 and you have the answer, 

To calculate the answer to this problem, recognize that you're told how often the number of infected people doubles, so your first goal will be to determine how many times the number has doubled from 500 to 64,000. You can start by calculating what number 500 would have to be multiplied by to equal the current 64000:

So 

You're dealing with an exponential growth proportion in which you know that the virus doubles every two days. So you can express 128 as , meaning that since there were 500 people infected, the number has doubled 7 times.  Since it takes 2 days for each time the number doubles, you can calculate that it has been 14 days since there were 500 people infected.

Here you are dealing with an exponential rate of decay/loss, as you can see from the fact that the variable  is an exponent. What does that mean? That the loss of 27%  (in an exponential loss formula, the portion within parentheses is 1 - %lost; this one "keeps" 73% and subtracts the other 27%) per month compounds each month. Exponential increases and decreases mean that the increase or decrease is taken as a percent of the new value each time: if the forest started with 100 trees it would have lost 27 in the first year (27% of the original 100), and then the next year the 27% decrease would be taken from the new total of 73. Since the "new total" shrinks from year to year, the total number of trees lost will shrink, too, as it is a function of the number of trees to start that year. So here the answer is "the first year" as that is when the starting total will be greatest.

If the number of users has tripled each day, that reflects an exponential increase. If there were 1 user on day one, then on day two it would be 3 users, then on day three there would be 3x3=9 users, then on day four there would be 9x3=27 and so on. You can account for this by saying that the number of users will be equal to , where  is the original number of users and  is the number of days since the first day. Since you're looking for the number of users 7 days after the first day, you would calculate that as . Since , you'd have  users on day one and  users one week later, for a ratio of .

(And note: this ratio will work on any 7-day range of dates, not just from the first day to the 8th day. Whatever your "original" value is will triple each day for 7 days, so  times original is what the one-week-later number will be.)

Here, we'll want to keep in mind that the base we apply our percent change to changes each month. So, we'll want to take this step by step, and recognize that if the manufacturer sells 20% of its inventory each month, the remaining inventory is 80% of where it started that month. Thus, we can multiply each starting value by 80% as follows

Start of the year: 10,000

End of January: 8,000 (10,000*.8)

End of February: 6,400 (8,000*.8)

End of March: 5,120 (6,400*.8)

End of April: 4,096 (5,120*.8)

So, at the end of April, the manufacturer has 4,096 units of inventory. We could also attack this question by translating 80% into a fraction and cancelling factors once we set up the following equation

to once again arrive at 4096.

In this example, we'll want to keep in mind that the base we apply our percent change to will be different each month. So, one method of approach would be to take things step by step and apply the calculation of 125% of each months starting volume (a 25% increase) as follows

Beginning March: 64 ounces

End March: 80 ounces (64*1.25)

End April: 100 ounces (80*1.25)

End May: 125 ounces (100*1.25)

So, at the end of May, the tank will contain 125 ounces of the plant. 

We could also apply the fractional change to the original amount, using the following expression for the 125%, or change for each of the three months

Notice, with this setup, our 4*4*4 cancels with the 64 in the numerator, leaving us with 5*5*5, or again, 125 ounces.