If the ball begins at a height of 250 millimeters after the first bounce, we can apply the change for each bounce to arrive at the height after the 5th bounce by multiplying each height by  .

So, the heights after each bounce are as follows:

After the first bounce - 625 

After the second bounce - 250 (625 * )

After the third bounce - 100 (250 * )

After the fourth bounce - 40 (100 * )

After the fifth bounce - 16 (40 * )

Keep in mind, our starting value is already after the first bounce, so we're only applying our multiplier of  four times. The common mistake in this question is to assume that because we're looking for the "fifth bounce," we should apply the multiplier five times. Be sure to pay close attention to the wording and, if needed, take things step by step!

In any percent problem, it is crucial that you know exactly which value the percent is to be taken of. Here Portia is overcharged by 20%, meaning that she paid 20% more than the item should have cost. That means that the amount she paid, $480, equals 20% more than she should have paid:

, where  is the price she should have paid.

To solve for , divide both sides by :

This means that she should have paid $400 but instead paid $480, so she overpaid by $80.

Note that a common mistake on percent problems is taking the percent of the wrong value: here you might be tempted to simply take 20% of 480, for example. The SAT test makers know that, and will often set up problems so that you're given the result of the percent change (here the result is Portia paying $480) and need to work your way back to the start value as you did here. Always pause to ensure that you know which value (or variable) gets multiplied by the percent change.

This problem rewards students who do the work rather than just looking at a problem and going with what “looks right.” Whenever you see a problem that asks you to find the percent difference between two unknowns, you can probably pick your own numbers to make the problem easier since the problem is asking for a relationship rather than a value.

In this case, since you are dealing with percents, it is a good idea to start with the number 100. If you increase 100 by 10%, you will get

 after the first increase.

During the second week, you are told, the price of the stock increases by 10% of the new value. Be careful here to take the percent of the correct value, 110, rather than the original value. If you increase 110 by 10%, you get:

.

To find the percent increase from there, you find the difference and then divide by the original before multiplying by 100: 

Remember that when you are dealing with taking the percent of a number and then taking the percent of the result, that you must be careful to take the percent of the correct number. In this case, that means increase $60 by 50% and then taking 20% off that new number. What you cannot do is either take off 20% of 60 or just increase 60 by 30% rather than doing each step in order. Doing either will lead to a trap that the Testmaker has left for you.

The first step you’re given is to increase $60 by 50%. This is the same as taking 150% of $60, or as multiplying it by 1.5. If you do this you get;

From there, you now need to find what 20% off 90 is in order to get your solution. You can do this by first finding 10% of 90 by moving the decimal point one place to the left to get 9 and then doubling it to get 18.

Subtracting 18 from 90 gives that a customer would pay $72.

With percent change problems it is important to recognize that you take the percentage of the original value. The formula is , and then multiply by 100 to convert to a percent.

Recognizing that, the percent change here is . This is the type of calculation you should be able to do quickly in your head: to go from 80 to 100, a number must increase by one-quarter of itself, which is 25%.

This percents problem can be set up by noting the height of the tree () in two ways: 

 (the tree is 80% taller than Dan)

 (the tree is 20% taller than Alex)

Now you can set the equations equal, since each defines the value of :

With this, the denominators cancel (you can just multiply both sides of the equation by 5), leaving:

Divide both sides by  and you'll have your answer: 

, which simplifies to: .

A critical element of all percent problems is that you must pay particular attention to what the percent is being taken OF. Here, recognize that you're given two percents in terms of the beginning (start of 2006) value, but you're asked for the percent change for the year 2007, meaning that you're looking for the percent increase in terms OF the start of 2007 value.

If you say that the initial value at the beginning of 2006 was , then you know that at the end of 2006 / beginning of 2007 the value was , and at the end of 2007 it was . What you're being asked, then is the percent change from when it was at  to when it was . The difference is , so you can use the calculation . This means that you're looking at a fraction of , which translates to 12.5%.

Alternatively, you could pick numbers to make this easier. If you say that the land was worth $100 to start, grew to $160 by the end of the first year, and then was worth $180 at the end of the second year, then you know that it gained $20 from a base value of $160 over the course of the second year, again for a 12.5% increase.

You can set this problem up algebraically by recognizing what you're really solving for, which is essentially a ratio of doughnuts () to total (). You're given that the ratio of glazed () to doughnuts () is 70/100, and you're given that glazed () to total () is 21/100. Knowing that, you can set up: 

And since  has to be equal in each of those equations, you can set the equations equal: 

From there, your goal is to solve for , so you'll divide both sides by  to get: 

And then multiply both sides by  to cancel the fraction on the left and move all the numbers to the right. Now you have: 

. If  out of  is  out of , that means that the answer is .

With percent problems, the key is often to make sure that you take the percent of the correct value. In this case, the initial 25% off means that customers will pay 75% of the original price. Then for the second discount, keep in mind that the discount is taken off of the sale price, not of the original price. So that's 20% off of the 75% that they did pay, which can be made easier by looking at what the customer does pay: 80% of the 75% sale price. Using fractions, that means they pay:  of the original price, which nets to  of the original price, or 60%.