Why is a 20% increase followed by a 20% decrease not equal to the original?

Because the second percent applies to the larger amount, not the original. Multiply the multipliers: 1.20 × 0.80 = 0.96, so you end at 96% of the start — a 4% net loss.

Do you divide by the old or the new value for percent change?

Always divide by the old (original) value. Percent change equals (new − old) divided by old, times 100%. Dividing by the new value is the single most common percent-change error.

How do you convert a fraction to a percent quickly?

Divide the numerator by the denominator to get a decimal, then multiply by 100. For instance, 5/8 = 0.625 = 62.5%. Memorizing a few benchmarks like 1/8 = 12.5% makes this instant.

Is "25% more than" the reverse of "25% less than"?

No. The two use different bases. If x is 25% less than y, then x = 0.75y, so y is x divided by 0.75, which is 33⅓% more than x — not 25% more.

CramClub

Fractions, Decimals, and Percents

Move fluently among fractions, decimals, and percents, and master the GRE’s favorite traps: percent of a number, percent increase and decrease, successive percent changes, and "percent more/less than."

GREquantitative reasoningmedium18 min read6 worked examples6 practice

Key Formulas

Percent of a number

$p\% \text{ of } x = \dfrac{p}{100}\,x$

"Percent of" means multiply; "is" means equals.

Percent change

$\dfrac{\text{new} - \text{old}}{\text{old}} \times 100\%$

Always divide by the ORIGINAL value, never the new one.

Successive percent change

Final $= \text{Original} \times (1 \pm r_1)(1 \pm r_2)$

Multiply the multipliers; never add the percents.

"More than / less than"

$A$ is $p\%$ more than $B$ ⇒ $A = \left(1 + \tfrac{p}{100}\right)B$

Not symmetric — reset the base when it switches.

Video Lesson

Explanation

Three Languages for One Idea

A fraction, a decimal, and a percent are three ways of writing the same thing: a part of a whole. The GRE constantly switches among them mid-problem, and the fastest test-takers convert on sight. Fluency here pays off directly, because percent word problems are among the most common Quant items — and among the most heavily trapped.

Converting Between Forms

Fix the reference points and the rest follows:

Fraction	Decimal	Percent
$\tfrac{1}{2}$	$0.5$	$50\%$
$\tfrac{1}{4}$	$0.25$	$25\%$
$\tfrac{1}{3}$	$0.333\ldots$	$33\tfrac{1}{3}\%$
$\tfrac{3}{8}$	$0.375$	$37.5\%$
$\tfrac{5}{6}$	$0.8333\ldots$	$83\tfrac{1}{3}\%$

To turn a fraction into a percent, divide and multiply by $100$ . To turn a percent into a fraction, put it over $100$ and reduce. "Percent" literally means "per hundred," so $37.5\%$ is $\tfrac{37.5}{100} = \tfrac{3}{8}$ .

Percent of a Number

The core translation: "percent of" means multiply, and "is" means equals. So " $40\%$ of $30\%$ of $x$ is $18$ " becomes

$0.40 \times 0.30 \times x = 18 \;\Rightarrow\; 0.12x = 18 \;\Rightarrow\; x = 150.$

Chained percents multiply; they do not add. $40\%$ of $30\%$ is $12\%$ , not $70\%$ .

Percent Change

Percent change always compares the change to the original value:

$\text{percent change} = \frac{\text{new} - \text{old}}{\text{old}} \times 100\%.$

A rise from $250$ to $320$ is $\frac{320 - 250}{250} = \frac{70}{250} = 28\%$ . The single most common error is dividing by the new value instead of the original. A decrease uses the same formula and simply produces a negative result.

Successive Percent Changes

When a quantity changes by one percent and then by another, multiply the factors — never add the percents. A $20\%$ increase followed by a $20\%$ decrease is

$1.20 \times 0.80 = 0.96,$

so the final value is $96\%$ of the original — a net $4\%$ loss, not zero. Represent each change as a multiplier ( $+15\%$ is $\times 1.15$ ; $-30\%$ is $\times 0.70$ ) and string them together in any order — multiplication is commutative, so the order of the two changes never affects the final value.

Percent More Than / Less Than

" $A$ is $25\%$ more than $B$ " means $A = 1.25\,B$ ; " $25\%$ less than" means $\times 0.75$ . These are not symmetric. If $x$ is $25\%$ less than $y$ , then $x = 0.75y$ , so $y = \frac{x}{0.75} = \frac{4}{3}x$ — that is, $y$ is $33\tfrac{1}{3}\%$ more than $x$ , not $25\%$ more. Whenever a problem swaps the base, recompute from the definition rather than reusing the first percent.

A Reliable Game Plan

Convert on sight using the reference table; a clean fraction often beats a messy decimal.
"Percent of" ⇒ multiply; chained percents multiply, they do not add.
Percent change ⇒ divide by the original, never the new value.
Successive changes ⇒ multiply the multipliers ( $1.2 \times 0.8 = 0.96$ ).
"More/less than" is directional — reset the base when it changes.

Worked Examples

— follow the steps, hardest last

Convert a fraction to a percent

easy

Write $\tfrac{5}{8}$ as a percent.

Find the whole from a percent

easy

If $15\%$ of a number is $9$ , what is the number?

Compute a percent increase

medium

A subscription rises from $48 to $60. What is the percent increase?

Successive percent changes

medium

A stock gains $30\%$ one year and loses $20\%$ the next. By what percent has it changed overall?

Reverse a "percent less than"

hard

The sale price of a jacket is $20\%$ less than its list price. The list price is what percent greater than the sale price?

Layered percent-of-a-group

hard

At a conference, $60\%$ of attendees are engineers. Of the engineers, $25\%$ hold a PhD, and of the non-engineers, $10\%$ hold a PhD. What percent of all attendees hold a PhD?

Watch Out For

Adding successive percent changes. Treating a $20\%$ rise then a $20\%$ drop as "no change."
Multiply the multipliers: $1.20 \times 0.80 = 0.96$ , a $4\%$ net loss. Never add the percents.
Dividing the percent change by the new value instead of the original.
Percent change is (new − old) ÷ old. The base is always where you started.
Treating " $25\%$ less than" and " $25\%$ more than" as reversible with the same percent.
They use different bases. If $x$ is $25\%$ less than $y$ , then $y$ is $33\tfrac{1}{3}\%$ more than $x$ . Recompute from the definition.
Reading " $40\%$ of $30\%$ " as $70\%$ by adding the percents.
"Of" means multiply: $0.40 \times 0.30 = 0.12 = 12\%$ .

Practice Problems

A price is first increased by $20\%$ and then the new price is decreased by $20\%$ . The final price is what percent of the original price?

If $40\%$ of $30\%$ of a number is $18$ , what is the number?

A quantity increases from $250$ to $320$ . What is the percent increase?

If $x$ is $25\%$ less than $y$ , then $y$ is what percent greater than $x$ ?

In a company, $60\%$ of the employees are women. Of the women, $25\%$ work remotely, and of the men, $40\%$ work remotely. What percent of all employees work remotely?

$y$ is a positive number.

Quantity A: $y$ increased by $10\%$ and then decreased by $10\%$

Quantity B: $y$ decreased by $10\%$ and then increased by $10\%$

Frequently Asked Questions

Why is a 20% increase followed by a 20% decrease not equal to the original?: Because the second percent applies to the larger amount, not the original. Multiply the multipliers: 1.20 × 0.80 = 0.96, so you end at 96% of the start — a 4% net loss.
Do you divide by the old or the new value for percent change?: Always divide by the old (original) value. Percent change equals (new − old) divided by old, times 100%. Dividing by the new value is the single most common percent-change error.
How do you convert a fraction to a percent quickly?: Divide the numerator by the denominator to get a decimal, then multiply by 100. For instance, 5/8 = 0.625 = 62.5%. Memorizing a few benchmarks like 1/8 = 12.5% makes this instant.
Is "25% more than" the reverse of "25% less than"?: No. The two use different bases. If x is 25% less than y, then x = 0.75y, so y is x divided by 0.75, which is 33⅓% more than x — not 25% more.

Keep Going

Number Properties and Divisibility

Reason about integers the way the GRE wants — parity, primes, factors and multiples, divisibility tests, remainders, GCF/LCM, and consecutive-integer facts — so "must be true" questions become quick.

Ratios, Rates, and Proportions

Set up and solve GRE ratio problems with confidence — part-to-part vs. part-to-whole, combining ratios through a shared term, direct and inverse proportion, unit rates, and combined work and speed.

Quantitative Comparison Strategy

Learn the four fixed answer choices, the "cannot be determined" mindset, and the plug-in discipline that turns GRE Quantitative Comparison into free points.

Ready to test yourself?

Put this skill to work on a full-length adaptive practice test.

Fractions, Decimals, and Percents

GREquantitative reasoningmedium18 min read6 worked examples6 practice

Key Formulas

Percent of a number

$p\% \text{ of } x = \dfrac{p}{100}\,x$

"Percent of" means multiply; "is" means equals.

Percent change

$\dfrac{\text{new} - \text{old}}{\text{old}} \times 100\%$

Always divide by the ORIGINAL value, never the new one.

Successive percent change

Final $= \text{Original} \times (1 \pm r_1)(1 \pm r_2)$

Multiply the multipliers; never add the percents.

"More than / less than"

$A$ is $p\%$ more than $B$ ⇒ $A = \left(1 + \tfrac{p}{100}\right)B$

Not symmetric — reset the base when it switches.

Video Lesson

Explanation

Three Languages for One Idea

Converting Between Forms

Fix the reference points and the rest follows:

Fraction	Decimal	Percent
$\tfrac{1}{2}$	$0.5$	$50\%$
$\tfrac{1}{4}$	$0.25$	$25\%$
$\tfrac{1}{3}$	$0.333\ldots$	$33\tfrac{1}{3}\%$
$\tfrac{3}{8}$	$0.375$	$37.5\%$
$\tfrac{5}{6}$	$0.8333\ldots$	$83\tfrac{1}{3}\%$

Percent of a Number

The core translation: "percent of" means multiply, and "is" means equals. So " $40\%$ of $30\%$ of $x$ is $18$ " becomes

$0.40 \times 0.30 \times x = 18 \;\Rightarrow\; 0.12x = 18 \;\Rightarrow\; x = 150.$

Chained percents multiply; they do not add. $40\%$ of $30\%$ is $12\%$ , not $70\%$ .

Percent Change

Percent change always compares the change to the original value:

$\text{percent change} = \frac{\text{new} - \text{old}}{\text{old}} \times 100\%.$

Successive Percent Changes

When a quantity changes by one percent and then by another, multiply the factors — never add the percents. A $20\%$ increase followed by a $20\%$ decrease is

$1.20 \times 0.80 = 0.96,$

Percent More Than / Less Than

A Reliable Game Plan

Convert on sight using the reference table; a clean fraction often beats a messy decimal.
"Percent of" ⇒ multiply; chained percents multiply, they do not add.
Percent change ⇒ divide by the original, never the new value.
Successive changes ⇒ multiply the multipliers ( $1.2 \times 0.8 = 0.96$ ).
"More/less than" is directional — reset the base when it changes.

Worked Examples

— follow the steps, hardest last

Convert a fraction to a percent

easy

Write $\tfrac{5}{8}$ as a percent.

Find the whole from a percent

easy

If $15\%$ of a number is $9$ , what is the number?

Compute a percent increase

medium

A subscription rises from $48 to $60. What is the percent increase?

Successive percent changes

medium

A stock gains $30\%$ one year and loses $20\%$ the next. By what percent has it changed overall?

Reverse a "percent less than"

hard

The sale price of a jacket is $20\%$ less than its list price. The list price is what percent greater than the sale price?

Layered percent-of-a-group

hard

At a conference, $60\%$ of attendees are engineers. Of the engineers, $25\%$ hold a PhD, and of the non-engineers, $10\%$ hold a PhD. What percent of all attendees hold a PhD?

Watch Out For

Adding successive percent changes. Treating a $20\%$ rise then a $20\%$ drop as "no change."
Multiply the multipliers: $1.20 \times 0.80 = 0.96$ , a $4\%$ net loss. Never add the percents.
Dividing the percent change by the new value instead of the original.
Percent change is (new − old) ÷ old. The base is always where you started.
Treating " $25\%$ less than" and " $25\%$ more than" as reversible with the same percent.
They use different bases. If $x$ is $25\%$ less than $y$ , then $y$ is $33\tfrac{1}{3}\%$ more than $x$ . Recompute from the definition.
Reading " $40\%$ of $30\%$ " as $70\%$ by adding the percents.
"Of" means multiply: $0.40 \times 0.30 = 0.12 = 12\%$ .

Practice Problems

A price is first increased by $20\%$ and then the new price is decreased by $20\%$ . The final price is what percent of the original price?

If $40\%$ of $30\%$ of a number is $18$ , what is the number?

A quantity increases from $250$ to $320$ . What is the percent increase?

If $x$ is $25\%$ less than $y$ , then $y$ is what percent greater than $x$ ?

In a company, $60\%$ of the employees are women. Of the women, $25\%$ work remotely, and of the men, $40\%$ work remotely. What percent of all employees work remotely?

$y$ is a positive number.

Quantity A: $y$ increased by $10\%$ and then decreased by $10\%$

Quantity B: $y$ decreased by $10\%$ and then increased by $10\%$

Frequently Asked Questions

Why is a 20% increase followed by a 20% decrease not equal to the original?: Because the second percent applies to the larger amount, not the original. Multiply the multipliers: 1.20 × 0.80 = 0.96, so you end at 96% of the start — a 4% net loss.
Do you divide by the old or the new value for percent change?: Always divide by the old (original) value. Percent change equals (new − old) divided by old, times 100%. Dividing by the new value is the single most common percent-change error.
How do you convert a fraction to a percent quickly?: Divide the numerator by the denominator to get a decimal, then multiply by 100. For instance, 5/8 = 0.625 = 62.5%. Memorizing a few benchmarks like 1/8 = 12.5% makes this instant.
Is "25% more than" the reverse of "25% less than"?: No. The two use different bases. If x is 25% less than y, then x = 0.75y, so y is x divided by 0.75, which is 33⅓% more than x — not 25% more.

Keep Going

Number Properties and Divisibility

Ratios, Rates, and Proportions

Set up and solve GRE ratio problems with confidence — part-to-part vs. part-to-whole, combining ratios through a shared term, direct and inverse proportion, unit rates, and combined work and speed.

Quantitative Comparison Strategy

Learn the four fixed answer choices, the "cannot be determined" mindset, and the plug-in discipline that turns GRE Quantitative Comparison into free points.

Ready to test yourself?

Put this skill to work on a full-length adaptive practice test.