Percent means per hundred. Percent questions become much easier when every percentage is translated into a decimal multiplier and the correct comparison base is identified before calculation.
Percent meaning and conversions
- Percent
- A ratio measured per \(100\), so \(p\%=p/100\).
| Form | Convert to | Method | Example |
|---|---|---|---|
| Decimal | Percent | Multiply the decimal by \(100\), then attach the percent sign. | \(0.42=42\%\) |
| Fraction | Percent | Divide the numerator by the denominator, then multiply the decimal by \(100\). | \(3/8=37.5\%\) |
| Percent | Decimal | Divide by \(100\). | \(175\%=1.75\) |
| Percent | Fraction | Write over \(100\), then simplify. | \(35\%=35/100=7/20\) |
Percent increase and percent decrease
Classify the result as an increase when the new value is larger and a decrease when it is smaller.
Measure a decrease
A reservoir level changes from \(640\) million liters to \(544\) million liters. Find the percent decrease.
- Find the change
\(640-544=96\) million liters.
- Use the original base
\(96/640=0.15\).
- Convert to percent
\(0.15=15\%\).
Percent multipliers and reverse change
Here \(r\) is written as a decimal. A \(20\%\) increase uses \(1.20\); a \(20\%\) decrease uses \(0.80\).
Recover an original value
After a \(16\%\) increase, a membership count is \(870\). What was the original count?
- Choose the multiplier
An increase of \(16\%\) uses \(1.16\).
- Build the equation
\(1.16x=870\).
- Reverse the multiplication
\(x=870/1.16=750\).
Successive percent changes multiply
Each change acts on the current value, not automatically on the original. Multiply the change factors in order. An increase by \(r\) followed by a decrease by the same \(r\) produces \((1+r)(1-r)=1-r^2\), which is below \(1\) whenever \(r\ne0\).
Equal-looking changes do not cancel
A quantity increases by \(30\%\) and then decreases by \(30\%\). Find the net percent change.
- Multiply factors
\(1.30(0.70)=0.91\).
- Interpret
The final value is \(91\%\) of the original.
- Find the net change
\(100\%-91\%=9\%\).
Percent greater than and percent less than
The comparison base is the quantity after the word ‘than.’ Use the first form when \(A\) is greater than \(B\) and the second when \(A\) is less than \(B\).
Common percent-change mistakes
- Dividing the change by the new value rather than the original value.
- Treating \(5\%\) as \(5\) instead of \(0.05\).
- Treating \(125\%\) as \(0.125\) instead of \(1.25\).
- Adding successive percent changes instead of multiplying their factors.
- Assuming an increase and equal decrease cancel.
- Using the wrong quantity as the base in ‘greater than’ or ‘less than’ language.
- Confusing a percentage-point difference with a relative percent change.
Check your understanding
A value rises from \(120\) to \(150\). Which calculation gives the percent increase?
- \((150-120)/120\)
- \((150-120)/150\)
- \(150/120\)
- \(120/150\)
Show answer and explanation
Answer: \((150-120)/120\)
The change is divided by the original value \(120\), giving \(30/120=25\%\).
What to remember
- Percent means per hundred, and every percent has an equivalent decimal multiplier.
- Percent change always uses the original value as its denominator.
- Increase factors are greater than \(1\); decrease factors are between \(0\) and \(1\).
- Successive changes multiply and must be interpreted from the combined factor.
- In comparison language, the quantity after ‘than’ is the base.
Put these notes into practice
Apply the ideas with SAT-style questions, then reinforce key details with flashcards.