The Central Idea
The word percent means “per hundred.”
So
\[ 35\% \]means
\[ \frac{35}{100}. \]It can also be written as the decimal
\[ 0.35. \]These three forms represent the same multiplicative quantity:
\[ 35\% = \frac{35}{100} = 0.35. \]Percentages are therefore another form of proportional reasoning.
Percent as a Ratio
If 18 out of 60 students choose an option, then the fraction is
\[ \frac{18}{60}. \]Convert to a decimal:
\[ \frac{18}{60}=0.3. \]Convert to percent:
\[ 0.3=30\%. \]So 30% of the students chose that option.
Percent, Fraction, and Decimal Forms
| Percent | Fraction | Decimal |
|---|---|---|
| \(25\%\) | \(\frac14\) | \(0.25\) |
| \(50\%\) | \(\frac12\) | \(0.50\) |
| \(75\%\) | \(\frac34\) | \(0.75\) |
| \(100\%\) | \(1\) | \(1.00\) |
| \(150\%\) | \(\frac32\) | \(1.50\) |
Convert a Percent to a Decimal
Divide by \(100\), or move the decimal point two places left.
For example,
\[ 42\% = \frac{42}{100} = 0.42. \]Likewise,
\[ 7.5\%=0.075. \]Convert a Decimal to a Percent
Multiply by \(100\), or move the decimal point two places right.
For example,
\[ 0.68=68\%. \]And
\[ 1.25=125\%. \]Percentages Can Exceed 100%
A percent is a ratio, so it does not have to stay between 0% and 100%.
For example,
\[ 150\%=1.5. \]This means one quantity is one and one-half times the reference quantity.
Percentages greater than \(100\%\) are common in growth, comparisons, and relative-size statements.
The Three Basic Percent Quantities
Most elementary percent problems involve three quantities:
- the part;
- the whole or base;
- the percent.
They are related by
\[ \text{part} = \text{percent as a decimal} \cdot \text{whole}. \]The Basic Percent Equation
Let
\[ P=\text{part}, \] \[ r=\text{percent written as a decimal}, \]and
\[ W=\text{whole}. \]Then
\[ \boxed{P=rW}. \]Finding the Part
What is \(30\%\) of \(80\)?
Convert \(30\%\) to decimal form:
\[ 0.30. \]Then
\[ P=0.30(80)=24. \]Therefore,
\[ \boxed{24}. \]Finding the Percent
What percent of \(60\) is \(18\)?
Use
\[ r=\frac{P}{W}. \]Then
\[ r=\frac{18}{60}=0.30. \]Convert to percent:
\[ \boxed{30\%}. \]Finding the Whole
Suppose \(24\) is \(30\%\) of some number \(W\).
Write
\[ 24=0.30W. \]Then
\[ W=\frac{24}{0.30}=80. \]Therefore,
\[ \boxed{80}. \]Percent as a Proportion
Because percent means “per hundred,” we can also write
\[ \frac{\text{part}}{\text{whole}} = \frac{\text{percent}}{100}. \]For example, if \(x\) is 35% of 240:
\[ \frac{x}{240} = \frac{35}{100}. \]This gives
\[ x=84. \]Choose Between the Equation and Proportion Forms
The two forms
\[ P=rW \]and
\[ \frac{P}{W} = \frac{p}{100} \]represent the same relationship.
The decimal-multiplier form is often convenient for calculations.
The proportion form makes the “per hundred” structure especially visible.
Percent as a Multiplier
To find \(p\%\) of a quantity, multiply by
\[ \frac{p}{100}. \]For example, 18% of \(250\) is
\[ 0.18(250)=45. \]The percent acts as a scale factor.
Common Benchmark Percentages
Useful mental benchmarks include:
- \(1\%=0.01\);
- \(5\%=0.05\);
- \(10\%=0.10\);
- \(25\%=\frac14\);
- \(50\%=\frac12\);
- \(75\%=\frac34\);
- \(100\%=1\).
These can make estimation and mental calculation much faster.
Mental Percent Example
Find 15% of \$80.
Ten percent is
\[ \$8. \]Five percent is half of that:
\[ \$4. \]So 15% is
\[ 8+4=12. \]Therefore,
\[ \boxed{\$12}. \]Percent Change
Percent change compares the amount of change with the original value.
The formula is
\[ \boxed{ \text{percent change} = \frac{\text{new}-\text{original}} {\text{original}} \cdot100\% }. \]Absolute Change Versus Percent Change
Suppose a price rises from \$40 to \$50.
The absolute change is
\[ 50-40=10. \]The percent change is
\[ \frac{10}{40}\cdot100\% = 25\%. \]Absolute change and percent change describe different aspects of the same event.
Why the Original Value Is the Denominator
If a value changes from \(A\) to \(B\), we ask:
How large is the change compared with where we started?
That is why
\[ \frac{B-A}{A} \]uses the original value \(A\) as the reference.
Percent Increase
Suppose a quantity rises from \(80\) to \(100\).
The increase is
\[ 100-80=20. \]Relative to the original value:
\[ \frac{20}{80}=0.25. \]So the percent increase is
\[ \boxed{25\%}. \]Percent Decrease
Suppose a quantity falls from \(80\) to \(60\).
The decrease in magnitude is
\[ 80-60=20. \]Relative to the original:
\[ \frac{20}{80}=0.25. \]So the percent decrease is
\[ \boxed{25\%}. \]Signed Percent Change
Using
\[ \frac{\text{new}-\text{original}}{\text{original}}, \]a decrease automatically produces a negative result.
For \(80\to60\):
\[ \frac{60-80}{80} = -\frac14 = -25\%. \]The negative sign indicates decrease.
Growth Factors
An increase of \(r\) percent can be performed with one multiplier.
If \(r\) is written as a decimal, then
\[ \boxed{\text{new}=(1+r)(\text{original})}. \]The quantity
\[ 1+r \]is the growth factor.
Growth-Factor Example
Increase \$200 by 15%.
The growth factor is
\[ 1+0.15=1.15. \]Then
\[ 200(1.15)=230. \]Therefore,
\[ \boxed{\$230}. \]Decay Factors
A decrease of \(r\) percent uses
\[ \boxed{\text{new}=(1-r)(\text{original})}. \]The multiplier
\[ 1-r \]is the remaining fraction after the decrease.
Decay-Factor Example
Decrease \$200 by 15%.
The decay factor is
\[ 1-0.15=0.85. \]Then
\[ 200(0.85)=170. \]Therefore,
\[ \boxed{\$170}. \]Finding the Original Value from a Percent Increase
Suppose a price after a 20% increase is \$96.
Let the original price be \(P\).
A 20% increase means
\[ 96=1.20P. \]So
\[ P=\frac{96}{1.20}=80. \]Therefore, the original price was
\[ \boxed{\$80}. \]Finding the Original Value from a Percent Decrease
Suppose a sale price after a 25% discount is \$90.
A 25% discount leaves 75% of the original:
\[ 90=0.75P. \]Thus,
\[ P=\frac{90}{0.75}=120. \]The original price was
\[ \boxed{\$120}. \]Reverse Percentage Problems
When the final amount and percent change are known, do not simply add or subtract the percent from the final amount.
Use the multiplier equation:
\[ \text{final} = \text{multiplier}\cdot\text{original}. \]Then divide by the multiplier to recover the original value.
Discounts
A 30% discount means the customer pays
\[ 100\%-30\%=70\% \]of the original price.
So the sale price is
\[ \boxed{\text{sale price}=0.70(\text{original price})}. \]Discount Example
An item costs \$85 and is discounted 20%.
The remaining fraction is
\[ 0.80. \]So
\[ 85(0.80)=68. \]The sale price is
\[ \boxed{\$68}. \]Markup
A markup increases a base cost by a percentage of that base.
If an item costing \$50 is marked up 40%, then
\[ 50(1.40)=70. \]The selling price is
\[ \boxed{\$70}. \]Tax
If a purchase price is \(P\) and the tax rate is \(r\), then the tax is
\[ T=rP. \]The total price is
\[ P+T = P+rP = (1+r)P. \]So tax can be handled directly with a growth factor.
Tax Example
A \$120 purchase has an 8% tax.
Tax:
\[ 0.08(120)=9.60. \]Total:
\[ 120+9.60=129.60. \]Or directly:
\[ 120(1.08)=129.60. \]Thus,
\[ \boxed{\$129.60}. \]Tips
A 20% tip on a \$45 bill is
\[ 0.20(45)=9. \]The total with tip is
\[ 45+9=54. \]Or use the multiplier
\[ 1.20. \]Then
\[ 45(1.20)=54. \]Commission
If a salesperson earns 7% commission on \$12,000 of sales, then
\[ 0.07(12000)=840. \]The commission is
\[ \boxed{\$840}. \]Percent Error
Percent error compares the magnitude of an error with a reference or accepted value.
A common form is
\[ \boxed{ \text{percent error} = \frac{|\text{measured}-\text{accepted}|} {|\text{accepted}|} \cdot100\% }. \]The absolute value makes the error magnitude nonnegative.
Percent Error Example
Suppose an accepted value is \(50\), and a measurement gives \(47\).
Error magnitude:
\[ |47-50|=3. \]Percent error:
\[ \frac{3}{50}\cdot100\%=6\%. \]So the measurement has
\[ \boxed{6\%\text{ error}}. \]Percentage Points
Percent change and percentage-point change are not the same.
Suppose a rate rises from
\[ 20\% \]to
\[ 25\%. \]The percentage-point increase is
\[ 25\%-20\%=5\text{ percentage points}. \]Percent Increase of a Percentage
For the same change from 20% to 25%, the relative percent increase is
\[ \frac{25-20}{20}\cdot100\% = 25\%. \]So the rate rose by:
- 5 percentage points;
- 25% relative to its original level.
These statements are both correct but mean different things.
Why Percentage Points Matter
When comparing percentages themselves, percentage points prevent ambiguity.
For example, an interest rate moving from 4% to 6% is:
- an increase of 2 percentage points;
- a relative increase of 50%.
Successive Percent Changes
Percent changes are multiplicative.
Suppose a value increases 10% and then increases another 10%.
The combined multiplier is
\[ 1.10\cdot1.10=1.21. \]So the total increase is
\[ 21\%, \]not 20%.
Why Repeated Percentages Compound
The second 10% increase is calculated from the already-increased value.
For example, starting from \(100\):
\[ 100(1.10)=110, \]then
\[ 110(1.10)=121. \]The second increase is \(11\), not \(10\).
An Increase and Equal Decrease Do Not Cancel
Suppose a value increases 20% and then decreases 20%.
The combined multiplier is
\[ 1.20(0.80)=0.96. \]So the final value is 96% of the original.
That is a net decrease of
\[ 4\%. \]Why Equal Percentage Changes Do Not Reverse Each Other
The increase and decrease use different bases.
Starting from \(100\):
\[ 100\to120 \]is a 20% increase.
Then a 20% decrease of \(120\) is
\[ 24, \]giving
\[ 96. \]To return from 120 to 100 requires a decrease of
\[ \frac{20}{120} = 16\frac23\%. \]Finding the Reversal Percentage
If a value increases by \(r\), the multiplier is
\[ 1+r. \]To return to the original, multiply by its reciprocal:
\[ \frac{1}{1+r}. \]The required decrease is therefore not generally \(r\).
Repeated Decreases
Two consecutive 20% discounts give the multiplier
\[ 0.80(0.80)=0.64. \]So the final price is 64% of the original.
The total discount is
\[ 36\%, \]not 40%.
Successive Discount Example
An item costs \$200 and is discounted 25%, then an additional 10%.
First discount:
\[ 200(0.75)=150. \]Second discount:
\[ 150(0.90)=135. \]Final price:
\[ \boxed{\$135}. \]The combined multiplier is
\[ 0.75(0.90)=0.675. \]So the combined discount is
\[ 32.5\%. \]Repeated Growth
If a quantity grows by \(r\) each period, repeated growth follows
\[ A=A_0(1+r)^n, \]where:
- \(A_0\) is the initial value;
- \(r\) is the growth rate per period;
- \(n\) is the number of periods.
This is the beginning of exponential growth.
Repeated Decay
If a quantity decreases by \(r\) each period, then
\[ A=A_0(1-r)^n. \]The same multiplier is applied repeatedly.
This is exponential decay.
Percent Change as a Ratio of Differences
The formula
\[ \frac{\text{new}-\text{original}}{\text{original}} \]compares a difference with a baseline.
That makes percent change a combination of additive and multiplicative thinking:
- find the additive change;
- compare it multiplicatively with the original.
Relative Change Makes Different Scales Comparable
An increase of \$10 has different significance depending on the starting value.
From \$20 to \$30:
\[ \frac{10}{20}=50\%. \]From \$200 to \$210:
\[ \frac{10}{200}=5\%. \]The same absolute change can represent very different relative changes.
Percent Change Can Exceed 100%
Suppose a quantity rises from \(20\) to \(50\).
The increase is
\[ 30. \]Relative to the original:
\[ \frac{30}{20}=1.5. \]So the percent increase is
\[ \boxed{150\%}. \]Decrease Is Limited When Values Stay Nonnegative
If a positive quantity falls to zero, the percent decrease is
\[ 100\%. \]A decrease greater than 100% would produce a negative final value.
Whether that is meaningful depends on the quantity being modeled.
Percent Change from Zero Is Undefined
The formula
\[ \frac{\text{new}-\text{original}}{\text{original}} \]cannot be used when the original value is zero because it would require division by zero.
So a statement such as “increased from 0 to 5 by what percent?” does not have an ordinary finite percent-change answer.
Negative Original Values Require Care
Percent change is most straightforward when the baseline is positive.
If the original value is negative, interpretation becomes more subtle because dividing by a negative baseline changes the sign behavior.
In such contexts, it is often better to state the convention being used rather than applying a memorized formula without interpretation.
Comparing Two Quantities by Percent Difference
Sometimes neither value is naturally the “original.”
Then a symmetric measure called percent difference may be used:
\[ \frac{|A-B|} {\frac{|A|+|B|}{2}} \cdot100\%. \]This is different from percent change because it uses the average magnitude as the reference instead of choosing one value as the baseline.
Percent Change Versus Percent Difference
Use percent change when one value is clearly the original or reference.
Use percent difference when comparing two values symmetrically and neither is naturally the baseline.
The denominator communicates the meaning of the comparison.
Worked Example: Find the Part
Find 18% of 350.
\[ 0.18(350)=63. \]Therefore,
\[ \boxed{63}. \]Worked Example: Find the Percent
42 is what percent of 120?
\[ r=\frac{42}{120}=0.35. \]Thus,
\[ \boxed{35\%}. \]Worked Example: Find the Whole
54 is 45% of what number?
\[ 54=0.45W. \]Then
\[ W=\frac{54}{0.45}=120. \]Therefore,
\[ \boxed{120}. \]Worked Example: Percent Increase
A quantity rises from 240 to 300.
Change:
\[ 300-240=60. \]Relative change:
\[ \frac{60}{240}=0.25. \]Percent increase:
\[ \boxed{25\%}. \]Worked Example: Percent Decrease
A quantity falls from 150 to 120.
Decrease:
\[ 150-120=30. \]Relative decrease:
\[ \frac{30}{150}=0.20. \]Percent decrease:
\[ \boxed{20\%}. \]Worked Example: Reverse Discount
An item is on sale for \$72 after a 20% discount. Find the original price.
The customer pays
\[ 80\%=0.80 \]of the original.
So
\[ 72=0.80P. \]Then
\[ P=90. \]The original price was
\[ \boxed{\$90}. \]Worked Example: Percentage Points
A rate rises from 12% to 15%.
Percentage-point increase:
\[ 15\%-12\%=3\text{ points}. \]Relative percent increase:
\[ \frac{15-12}{12}\cdot100\% = 25\%. \]So it rose by 3 percentage points, or 25% relative to its original value.
A Reliable Basic Percent Workflow
- Identify the part, whole, and percent.
- Convert the percent to decimal form when using multiplication.
- Use \(P=rW\).
- Rearrange for the unknown quantity.
- Check whether the result is reasonable relative to the percent.
A Reliable Percent-Change Workflow
- Identify the original value.
- Identify the new value.
- Compute the change: new minus original.
- Divide by the original value.
- Multiply by 100%.
- Interpret the sign as increase or decrease.
- State the reference clearly when ambiguity is possible.
A Reliable Multiplier Workflow
- Convert the percent to decimal form.
- For an increase, use \(1+r\).
- For a decrease, use \(1-r\).
- Multiply the original quantity by the factor.
- For reverse problems, divide the final amount by the factor.
- For repeated changes, multiply the factors rather than adding the percentages.
Common Errors
Using the New Value as the Percent-Change Denominator
Ordinary percent change compares the change with the original value.
Confusing Percent with Percentage Points
A change from 20% to 25% is 5 percentage points but a 25% relative increase.
Adding Successive Percent Changes
Repeated changes compound multiplicatively.
Assuming Equal Increase and Decrease Percentages Cancel
They are applied to different base values.
Subtracting a Discount Percent from the Final Price to Recover the Original
Reverse percentage problems require division by the remaining multiplier.
Forgetting to Convert Percent to Decimal Form
For multiplication,
\[ 8\%=0.08, \]not \(8\).
Using Percent Change from an Original Value of Zero
The standard formula is undefined because the denominator is zero.
Practice
1. Find a Percent of a Number
Find 35% of 240.
Check Answer
\[ 0.35(240)=84. \]Answer:
\[ \boxed{84}. \]2. Find the Percent
18 is what percent of 72?
Check Answer
\[ \frac{18}{72}=0.25. \]Answer:
\[ \boxed{25\%}. \]3. Find the Whole
45 is 30% of what number?
Check Answer
\[ 45=0.30W. \] \[ W=150. \]Answer:
\[ \boxed{150}. \]4. Percent Increase
A value rises from 80 to 92. Find the percent increase.
Check Answer
\[ 92-80=12. \] \[ \frac{12}{80}=0.15. \]Answer:
\[ \boxed{15\%}. \]5. Percent Decrease
A value falls from 250 to 200. Find the percent decrease.
Check Answer
\[ 250-200=50. \] \[ \frac{50}{250}=0.20. \]Answer:
\[ \boxed{20\%}. \]6. Discount
An \$80 item is discounted 15%. Find the sale price.
Check Answer
\[ 80(0.85)=68. \]Answer:
\[ \boxed{\$68}. \]7. Reverse Percentage
After a 25% increase, a value is 150. What was the original value?
Check Answer
\[ 150=1.25P. \] \[ P=120. \]Answer:
\[ \boxed{120}. \]8. Percentage Points
A rate increases from 8% to 10%. Find the change in percentage points and the relative percent increase.
Check Answer
Percentage-point change:
\[ 10\%-8\%=2\text{ points}. \]Relative increase:
\[ \frac{10-8}{8}\cdot100\%=25\%. \]Answer: 2 percentage points and 25% relative increase.
9. Successive Changes
A value increases 20% and then decreases 20%. What percentage of the original remains?
Check Answer
\[ 1.20(0.80)=0.96. \]So
\[ \boxed{96\%} \]of the original remains.
Key Ideas
- Percent means “per hundred.”
- A percent can be written as a fraction over 100 or as a decimal multiplier.
- The basic percent relationship is \(P=rW\).
- Percent problems may ask for the part, whole, or percent.
- Percent change compares the change with the original value.
- Absolute change and relative percent change are different measurements.
- Growth uses the factor \(1+r\); decrease uses \(1-r\).
- Reverse percentage problems are solved by dividing by the multiplier.
- Percentage-point change is different from relative percent change.
- Repeated percent changes compound multiplicatively.
- Equal percent increases and decreases generally do not cancel.
- Percent change from an original value of zero is undefined in the ordinary formula.
- Discounts, markups, taxes, tips, commissions, percent error, growth, and decay all use the same ratio-and-multiplier structure.
Where This Leads
Percentages turn proportional reasoning into a standardized language for comparing quantities against a reference of 100.
The same multiplier structure leads naturally into financial mathematics, compound growth and decay, probability, statistics, error analysis, index numbers, interest, inflation, population change, and many models where relative change matters more than absolute change.