In this tutorial, we will cover the topic of percents in all its glory.

Comprehension of this math concept is critical for two reasons: First, it is one member of the part-to-whole relationship trifecta (decimals and fractions are the other two) and, as such, shows up in a wide variety of GMAT problems. Second, our primary goal is to be able to do all but the most difficult percent problems in sixty to ninety seconds, thus saving extra time to apply to more difficult problems.

First, we discuss the basics and next chapter we will discuss efficient ways to calculate and estimate percents and convert between percents, fractions and decimals. Once we have laid this foundation, we’ll cover percent change, successive percents, mixtures, unspecified values, and other advanced percent topics.

“Percent” literally translates as "out of one hundred" or "per 100" and is one way of expressing a part-to-whole relationship. "Part-to-whole" is just the fancy math expression for how many parts or portions you have out of a whole thing. For example, two pieces of pizza out of a whole eight-slice pie would be 2/8, which also equals 25/100, or 25%.

Using a percentage allows us to express this part-to-whole relationship as a whole number instead of as a fraction or decimal; for example “45% of the population” means we are talking about 45 out of every 100 people. In decimal form, this number would be 0.45 and in fraction form it would be 45/100. All three forms tell us the same piece of information.

For easier percent calculations, we can set up simple equations or proportions. The Manhattan GMAT Fractions, Decimals, & Percents preparation guide details the Percent Table technique (chapter 3); here we will explore a second technique, word translation.

Let’s solve the problem “What is 20% of 60?” Every word or symbol in this sentence needs to be translated into math. Try to write the equation yourself before continuing below. (Note that next chapter we’ll take a look at faster ways to solve this problem, but it’s important to learn the fundamentals so that you are less likely to make a mistake when you use “Fast Math” methods.)

“What” always represents a variable; let’s use x. The verb “to be” (is, was, are) always represents an equals sign. For any percentage, we write a fraction with the given number over 100. “Of” always represents multiplication. Now, we can write: x = 20/100 * 60 and we do the math to solve for x. (The answer is 12.)

Try another one: 8 is what percent of 40?

The translated equation is: 8 = x/100 * 40. Therefore x = 20. Note that the answer is just 20, and not 20%. That is because “percent” is already accounted for in the original question. The only unknown is “what” – which represents just the number in the numerator of our fraction.

Let’s try a slightly more complicated problem. x% of y is 5 and y% of 180 is 27. What is x?

Now we have multiple variables for our percentages. Can we still use our word translation method here? Sure!

We have: x/100 * y = 5 and y/100 * 180 = 27

Let’s solve the second equation first, since it has only one variable. We get y = 15 (remember – not 15%!) so we plug y = 15 into the first equation and we get x = 33 and 1/3 (again, with no percentage sign).

Practice both the above method and the Percent Table method from the preparation guide to see which one is best for you.

Generally speaking, those who are prone to making mistakes on percent problems or those who aren’t “math people” may prefer the more methodical Percent Table technique. Those who are proficient with basic algebra may prefer the Word Translation technique.

For those of you who want to increase speed so you can get through percent problems more quickly, go to the next chapter for our Fast Math lesson on percent calculations / estimations. In many instances, the Fast Math technique will allow you to bypass the Word Translation or Percent Table methods altogether!

Next, we move on to 'fast math for percents' and converting among percents, fractions and decimals.

Whenever we’re talking about a given number or amount, we can say that it equals 100%. On the GMAT, this realization is important because it can help us to calculate or estimate an answer very quickly.

Given 100% of a number, it is very easy to calculate 50%, 10%, 5% and 1% of that number. These four building blocks can then be used to calculate or estimate any whole-number percentage in a very short time. We’ll learn this method by example.

Let’s start with the number 90. First, we create a quick chart as follows:

90 = 100% 9 = 10%0.9 = 1%

45 = 50%4.5 = 5%

Note that to calculate 10% we just move the decimal one place to the left (9) and for 1% we move the decimal two places to the left (0.9). For 50%, we halve the number (45) and for 5% we move the decimal for the 50% figure one place to the left (4.5).

Now, we can calculate any whole percentage between 1 and 100; all we have to do is add or subtract the necessary amounts to get to the percentage that we want.

For example:

6% of 90 = 5% + 1%, or 4.5 + 0.9 = 5.418% of 90 = 10% + 10% - 1% - 1%, or 9 + 9 – 0.9 – 0.9 = 16.264% of 90 = 50% + 10% + 5% - 1%, or 45 + 9 + 4.5 – 0.9 = 57.6

And if we only have to estimate, rather than calculate exact numbers, it’s even easier – we can estimate that last one to be 45 + 9 + 5 – 1 = 58.

We can cut our time even further by calculating only the building blocks that we need. For example, if we’re asked for 18%, we’d only calculate 10% and 1%; we can ignore 50% and 5%. With practice, you’ll be able to do the entire calculation in ten to fifteen seconds and, if you work at it until it gets to be second nature, you might find you can approach five seconds!

We can also use our charts for a reverse problem, like one of our problems from last week: 8 is what percent of 40? If 40 = 100%, then 4 = 10%. Therefore, 8 = 20%.

What if we’re not told the original number (the one that is equivalent to 100%) but instead asked to find it? For example, 8 is 16% of what number? Our Fast Math method is to find 1% and multiply by 100 to get 100%. The number divided by the percent, or 8/16, will always give us the 1% figure and then we just multiply by 100. 8/16 * 100 = 1/2 * 100 = 50.

Look for opportunities to practice this in everyday life – when you’re calculating that 18% tip, for example, or trying to figure out what percent of your salary to put into your 401(k). When you buy something, look at the amount listed for the sales tax and use that to estimate the percentage rate you were charged; then use your answer to estimate your total bill and see if you’re right.

Before you know it, you’ll be a percentage-calculating whiz who can ace the GMAT, run rings around your bosses, and impress your friends at parties.

One other type of calculation you must be adept at on the GMAT is converting between any two of the three part-to-whole relationships: fractions, decimals and percents.

The Manhattan GMAT FDP preparation guide includes a chart of the most common conversions (chapter 4); memorizing these will save you a lot of time and reduce potential opportunities for mistakes. For other conversions, use the following methods:

Percent to Decimal: move the decimal point two places to the left. For example, 52% = 0.52.

Percent to Fraction: place the percent number in the numerator and 100 in the denominator; simplify. For example, 52% = 52/100 = 13/25.

Decimal to Percent: move decimal point two places to the right. For example, 1.2 = 120%.

Decimal to Fraction: first convert from decimal to percent, then follow the directions to convert from percent to fraction. For example, 1.2 = 120% = 120/100 = 6/5.

Fraction to Decimal: divide the numerator by the denominator (this will probably involve long division). For example, 5/6 = 0.833 repeating.

Fraction to Percent: first convert fraction to decimal, then follow the directions to convert from decimal to percent. For example, 5/6 = 0.833 repeating = 83 and 1/3 %.

One thing to note to help you keep percent-to-decimal conversion in perspective. Zero is the decimal equivalent of 0% and 1 is the decimal equivalent of 100%, so any decimals between 0 and 1 represent the percentage numbers between 0% and 100%. Decimals greater than 1 represent percentages greater than 100%.

Join us in the next chapter as we delve into more complicated percent problems dealing with increases, decreases and successive changes. We have already examined the fundamentals of calculating percents. Next, we will discuss some of the more complicated percent concepts tested on the GMAT: percent increase, percent decrease and successive percentage change.

Percent increase or decrease is one way to represent a change in a given number. (Note that, here, we are talking about a single change. Multiple changes will be covered below in the successive percentage change section.)

Percent increase is the percentage that the original number increases and percent decrease is the percentage that the original number decreases.

We can use a very simple formula for either type of problem:

Increase (or Decrease) = Change / Original

Let’s try it out. You’ve had your eye on a $100 shirt at the store, but you think it’s too expensive. Finally, it goes on sale for $70. What is the percent decrease?

The “change” is always the difference between our starting and ending points. In this case, it’s 100 – 70 = 30. The “original” is our starting point; in this case, it’s 100. 30/100 = 0.3 = 30%. Don’t forget to convert your initial answer, which is in decimal form, to a percent by moving the decimal point two places to the right.

Always remember that your denominator is the original number or your starting point. The most common mistake made on this type of problem is using the smaller number for percent decrease or the larger number for percent increase.

This is actually exactly the opposite of what you want to do! Percent decrease means you’re going from a larger number to a smaller one, so the larger number is your starting point. And percent increase, of course, means you’re going from a smaller number to a larger one, so the smaller number is your starting point.

Always think of the denominator as your starting point number and you won’t get mixed up.

Let’s try another one. Elise makes $30 a week from her steady babysitting job. She earns a raise and now makes $35 a week. What is the percent increase?

Increase = 5/30 = 1/6 or .166 repeating ? .166 = 16.66% or 16 and 2/3 %. By the way, how did we know that 1/6 would equal .166 or 16.6 % Because we memorized it from the Common Equivalents chart in chapter 4 of my Manhattan GMAT Fractions, Decimals, & Percents Strategy Guide. If you don’t have this memorized, you’d have to do long division here, which is not something to waste time on during your actual GMAT exam.

What about when we have multiple changes happening in one problem? These are called successive percentage change problems and our process is almost exactly the same.

Try this:

Two years ago, the population of Sunnydale was an anemic 150 people. Last year, the population increased by 20% and this year the population is expected to increase another 10%. How many residents is Sunnydale expected to have at the end of this year?

The key to successive change problems is that you must do each step separately. You cannot just add the percentages together; if you do, you will always get it wrong! Let’s look at the right way and the wrong way to do this problem.

First, the right way: Sunnydale starts out with a population of 150. In the first year, the population increases by 20%, so we add 30 people (practice Fast Math here: 10% + 10% = 20%, so 15 + 15 = 30).

Our new population is 150 + 30 = 180. This year, Sunnydale will add 10% but, this time, 10% is based on the new population figure of 180, not the old figure 150. This year, we add 18 people, so our population at the end of the year is expected to be 180 + 18 = 198.

Now, the wrong way: If we just add 20% and 10%, for an increase of 30%, we would have said that our population is based on a 30% increase of the 150 figure, or 45 (10% + 10% + 10% = 15 + 15 + 15 = 45).

Our final answer would be 195. You can expect this number to show up in the answer choices, so you would not realize it if you made this mistake.

The reason we must do each step separately in a successive change problem is that the starting point for each step is a different number – it’s based on the number you just calculated in the preceding step. Just remember that you must do these problems step-by-step to get them right. You will never get it right if you just add or subtract the percents and do the math all at once.

Now we will examine unspecified values and interest problems. If you haven’t read the first three installments yet, you may want to do so before you continue with today’s lesson; you can find the first three segments on our Web site (along with an archive of other interesting past topics).

Many medium and difficult percent problems work with unspecified values; that is, the problem may talk about the price of a TV but never give us an actual value for that price.

The GMAT is trying to get you to set up the problem algebraically, with a variable for the unspecified number. Algebra can be challenging, however, because it’s quite easy to make careless mistakes and the test writers actually write the wrong answers based upon the most common mistakes. As a result, we don’t even know we’ve made a mistake because we find our incorrect answer among the answer choices.

We can sidestep the GMAT’s built-in trap by choosing a real number, thus turning an algebra problem into an arithmetic problem!

Whenever a problem keeps talking about the value of something but never supplies an actual number, either in the question or in the answer choices, then we may pick our own number. On problem-solving questions, when we see variables in the answer choices, this is a great clue that we probably weren’t given an actual number and, therefore, may pick our own; problems with percents or ratios as the answers often have unspecified values, too.

But don’t just rely on the answers; check the problem itself! If you can’t find a specified value anywhere for something the problem keeps talking about, then you may pick your own number.

On percent problems, a great number to pick is 100. (Can you guess why?) Try the below problem to see why 100 is a good starting point.

Lucy is very bad at doing laundry and she tosses a sweater that is supposed to be dry clean only into the washing machine. It stretches by 9%. Lucy thinks, “No problem! I’ll put it in the dryer to dry it out and it will shrink back down to its original size.” It shrinks by 22%. Can Lucy ever wear the sweater again? (Actually, the real question is: the final sweater size is approximately what percent of the original sweater size?)

You can do this problem with algebra, but your chances of making a mistake are fairly high – and, of course, on the GMAT, they use the most common mistakes to write the wrong answers, so you don’t even know when you’ve made a mistake.

Instead, let’s say that the original size is 100. (100 what? Square inches? Cubic centimeters? It doesn’t matter.)

First, the sweater stretches by 9%. Fast Math! 10% - 1% = 10 – 1 = 9, so the new size is 109.

Then it shrinks by 22%. 10% + 10% + 1% + 1% = 10.9 + 10.9 + 1.09 + 1.09 = 23.98.

This is the shrinkage, so the new shirt size is 109 – 23.98 = 85.02

Final / original = 85.02 / 100 = approximately 85%of the original size.

It seems that Lucy is not going to get to wear her sweater after all. And now it’s pretty obvious why we wanted to choose 100 for our original number. It makes that last calculation really easy.

By the way, this general technique of picking a number when no number is specified works on many types of problems, not just percent problems. The method is outlined in several chapters found throughout Manhattan GMAT’s five math Strategy Guides.

Remember, picking your own number is a way to turn algebra problems into arithmetic problems, and even the best algebra practitioner in the world is better at arithmetic than algebra!

Everyone needs a basic working knowledge of simple interest on the test; for those aspiring to higher scores, it’s also a good idea to memorize the compound interest formula. Although the need for this formula is fairly rare on the GMAT, it is next to impossible to complete a compound interest problem in two minutes without the formula.

Simple interest is just your principal (the money you start with) multiplied by your interest rate (usually given as an annual rate) multiplied by the time (also annualized).

More simply, I = Prt, where P is the principal, r is the interest rate and t is the number of years the money is invested. You will be given values for three of the four variables in the formula and you will have to calculate the fourth.

The compound interest formula is I = P(1 + r/n)^nt, where P is the principal, r is the interest rate, n is the number of times per year the interest is compounded and t is the number of years the money is invested. Again, you will be given values for all but one of the variables and you will have to calculate the missing number.

Lastly, we will examine some of the most difficult manifestations of percents on the exam.

Mixture problems literally refer to percent problems that deal with chemical or other mixtures (liquid mixtures may also be called solutions).

You’ll have at least two mixtures in the problem with different concentrations of the components that make up the mixture. You will need to create a chart for each separate mixture or solution. Follow the below method and you will be able to do any mixture problem. (If this topic is giving you fits, you may first want to review the less confusing chemical mixture example in chapter 3 of Manhattan GMAT’s Fractions, Decimals, & Percents Strategy Guide.)

Set up a chart with three columns: substance, amount, and percentage. Give each substance its own row in the chart and include a final row for the Total. (Remember, make as many charts as you have separate solutions.)

For example, Nick is developing a new health drink. He begins with 10 ml of mango juice and adds 20 ml of carrot juice. He fills the rest of the glass with extract of wheat germ juice. He decides it isn’t quite right but he doesn’t want to start over from scratch, so he pours the entire mixture into a new glass that is two and a half times the size of his original glass. He fills this new glass to the brim with a mixture of 50% Red Bull and 50% ginseng sorbet. If the carrot juice represents 8% of the final mixture, how much Red Bull is in the solution?

Original mixture:

New Mixture

The items in plain text, in the tables, are numbers or variables we can fill in based on the initial problem. The items in bold are a result of our calculations.

We know that the Red Bull and ginseng sorbet are equal in amount because they each take up 50% of the capacity of the new glass, so we can represent them with the same variable, y.

Since 20 ml of the carrot juice represents 8% of the entire new mixture, we can use Fast Math to calculate 100% of the new mixture (see installment two of this article if you’ve forgotten how to do this). 20 / 8 * 100 = 250.

Since we’re told the capacity of the new glass is 2.5 times the capacity of the old glass, we know that the capacity of the old glass is 250 / 2.5 = 100 ml. We also know that the original 100 ml is comprised of only the mango, carrot and wheat germ, so the remaining 150 ml must be comprised of the two new ingredients, Red Bull and ginseng sorbet.

These two ingredients are present in equal amounts, so each is 75 ml. Therefore, the new mixture contains 75 ml of Red Bull.

This problem can seem even more complicated than it is because we have a lot of things we can figure out but only a few calculations get us to the answer in the end. This is a classic time-waster.

Note that you may not always want or even be able to fill in every square of the charts you create. If you feel like you’re stuck, examine the problem again to make sure that every detail is represented in your charts.

Percents or Logic? Many of the percent problems on the GMAT are difficult more due to confusing wording than complicated math; the GMAT classifies these as percent problems, but solving these relies just as much on logic as it does on math.

If you can just figure out what they’re talking about, you can generally do whatever calculations are required. The key to these problems is to set up an appropriate chart, set of equations, or sequence of steps that account for all of the data you’re given in the problem. Let’s try one final problem:

During one week, Shelley swapped 20 of her own books with 20 members of her Book Exchange Club, one book to each member. Books are only returned after both members of a particular exchange pair have finished reading their loaner books. If Shelley read at least 60% of her loaner books by Saturday and at least 16 of Shelley’s book-swap partners read their respective books by Saturday, what is the smallest percentage of Shelley’s books that can be returned to Shelley on Saturday?

The math in this problem isn’t the hard part; the hard part is figuring out what’s going on!

Our question asks us to minimize the number of books Shelley is permitted to swap back on Saturday. To do so, we need to (a) minimize the number of books that Shelly read and then (b) minimize the number of books that Shelly read that correspond to swap-partners that also read their books.

If Shelley read at least 60% of her 20 loaner books, that means she read a minimum of 20 * 0.6 = 12 books. If she read 12 books, this means that she did NOT read 8 books.

Of the 16 swap partners who have read their respective books, let’s assume that 8 of those are the 8 whose books Shelley has not read. That leaves us 8 swap partner pairs for which both Shelley and the partner have read their loaner books and can now swap back. Shelley can have 8 books out of 20 returned to her on Saturday, or 8/20 = 2 / 5 = 40%.

If you look back, you can see that the math itself is not that complicated, but the train of thought can get pretty convoluted. Don’t panic if you can’t see all the way to the end of the problem!

Remember our overall goal on percent problems: we want to be able to solve all but the hardest in less than two minutes, so we save time for harder problems elsewhere on the test. Practice makes perfect! Practice the techniques we’ve reviewed and you’ll be a Percent Whiz.