GRE Exam: A Simple Guide To Part-to-Whole Relationships - Percentages

Part Versus Whole
A percent is another way to express a part-to-whole relationship. Percent literally means “per one hundred” and is used to determine the piece that a quantity represents when the whole is 100.

For example, if Bob has 200 pies, and 75% of the pies are blueberry, then the number of blueberry pies is

Note from the preceding example that to convert from a percent to a decimal, you should drop the percent and move the decimal two places to the left.

or example:

The formula to calculate percent is

Example: 62 is what percent of 1,000?

SOLUTION: Use the percent formula: 62 is the part and 100 is the whole, so:

Percent questions can be phrased in a few ways.

Let’s look at a few examples and how they fit into the preceding formula.

Example: 15 is what percent of 50?

SOLUTION: The part is 15 and the whole is 50.

The solution is thus:

Example: What percent of 200 is 350?

SOLUTION: The part is 350 and the whole is 200 (note that the part can be greater than the whole).

The solution is thus:

Example: 60% of what number is 540?

SOLUTION: The part is 540, and the whole is unknown.

Thus:


Percent Change
Some GRE questions will ask you to calculate the percent by which a certain value increases or decreases.

For example:
If the price of a shirt increased from $80 to $100, then by what percent did the price of the shirt increase?

Any time you are asked to solve for percent change, use the following formula:

When using this formula, make sure that the denominator is the original value and that the value in the numerator is positive.

In the given example, the change is 100 – 80 = 20.

The original value is the price of the shirt before the increase, in this case, $80.

Thus the percent change .

In the preceding example, you knew the original value and the new value, and you were asked to solve for the percent change.

What if you are told the percent change and are asked to solve for the original or new value?

Use the following formulas:
- If a quantity increases by p percent, the new quantity will equal (100 + p)% × the original quantity.
- If the price of a $3,000 computer increased by 20%, what is the new price?

SOLUTION: Since the price increases by 20%, the new price will be (100% + 20%) × $3,000 = 120% × 3,000 = 1.2 × $3,000 = $3,600.

Example: After a 20% increase, the price of a computer is $3,000. What was the original price of the computer?

SOLUTION: Let o = the original price.

Thus

In this example, many test-takers make the mistake of subtracting 20% from $3,000 to determine the original price.

This approach is wrong because it confuses which whole the 20% is a piece of: adding 20% to $2,500 is not the same as taking 20% away from $3,000.

If a quantity decreases by p percent, the new quantity will equal (100 – p)% × the original quantity.

Example: After a 25% decrease, the price of a car was $24,000. What was the original price?

SOLUTION: Let p = the original price of the car.

Thus

 

Consecutive Percentages
Certain GRE questions will involve more than one percent change to a value.

For example:
On January 1, the price of a certain stock was $400. On January 2, the price of the stock was 20% greater than it was on January 1. On January 3, the price was 20% less than it was on January 2. What was the price of the stock on January 3?

SOLUTION: To answer these questions, you should use the preceding formulas, but keep the following point in mind: Each successive percent change relates to the quantity that immediately precedes it, not to the starting quantity.
It would be wrong to think that the 20% changes cancel each other out and yield a value equal to the original value. This is so because the value that goes up 20% from January 1 to January 2 is different from the value that goes down from January 2 to January 3.

Since the percents are pieces of different wholes, they do not simply cancel out. So you should instead use the percent change formulas given in the previous section:


Percent Greater Versus Percent Of
If you don’t read the following question carefully, you might fall for a trap:

Example: 150 is what percent greater than 50?
Many test-takers simply use the percent formula for this question:

But this is wrong. Why? Because the question is asking, 150 is what percent greater than 50, not what percent of 50.

To calculate percent greater, use the following formula:

In the previous example, 150 is 300% of 50, so it is (300% – 100%) = 200% greater than 50.

Plugging in Numbers with Percentages
In the guide on fractions and decimals, you read that plugging in numbers was a useful strategy in Discrete Quantitative questions that only give fractions and no specified amounts.

The same applies to percent questions. If you see a Discrete Quantitative question with only percents and no actual amounts, plug in 100 for the total.

Example: 20% of the employees in a company are managers. 10% of these managers have been with the company for at least 10 years. The number of managers who have not been with the company for at least 10 years is what percent of all the employees?

SOLUTION: Let the total number of employees = 100. Thus the number of managers is 20%(100) = 20. The percentage of managers who have not been with the company for 10 years is 100% – 10% = 90%. Thus the number of managers who have not been with the company for 10 years is 90%(20) = 18. Now the question is, 18 is what percent of 100? The answer is 18 (notice how simple the calculations are when you choose 100!).

Quantitative Comparison Strategy: Percentages
Often percent questions in Quantitative Comparison questions will test your ability to differentiate between the original value and new values.

When looking at successive percent changes, always keep in mind that each percent change occurs to the previous value, not the original one.

SOLUTION: It may be tempting to assume that the percent changes will offset each other and that the answer is thus C. But the second percent change is on a bigger value than the original percent change. Thus the amount by which the second price decreases is greater than the amount by which the first price increased. Thus the final price must be less than the original price. The correct answer is A.