MathChapter 7: Percents
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Prompt
What does percent mean?
💡Interpret the percent sign as division by \(100\).
Answer
A rate per hundred.
Example\(38\%=38/100=0.38\).
Prompt
How do you convert a decimal to a percent?
💡Move the decimal point two places right.
Answer
Multiply by \(100\) and attach the percent sign.
Prompt
How do you convert a percent to a decimal?
💡Rates below \(1\%\) need extra care.
Answer
Divide by \(100\).
Prompt
How do you convert a fraction to a percent?
💡Find the fraction's decimal value first.
Answer
Divide the numerator by the denominator, then multiply by \(100\).
Example\(3/8=0.375=37.5\%\).
Prompt
What is the percent-change formula?
💡The original value is the denominator.
Answer
\(\text{percent change}=\dfrac{|\text{new}-\text{original}|}{\text{original}}(100\%)\).
ExampleFrom \(80\) to \(100\): \(20/80=25\%\).
Prompt
Which value is the base in a percent-change problem?
💡Ask what the situation started with.
Answer
The original value, or the value before the change.
ExampleFrom \(50\) to \(65\), divide the change by \(50\).
Prompt
What multiplier represents a \(p\%\) increase?
💡Keep the original \(100\%\) and add the increase.
Answer
\(1+p/100\).
ExampleA \(12\%\) increase uses \(1.12\).
Prompt
What multiplier represents a \(p\%\) decrease?
💡Use the fraction that remains.
Answer
\(1-p/100\).
ExampleA \(35\%\) decrease uses \(0.65\).
Prompt
How do you reverse a known percent increase?
💡Do not subtract the rate from the final value.
Answer
Divide the final value by the increase multiplier.
ExampleAfter a \(20\%\) increase, \(360/1.20=300\).
Prompt
How do you reverse a known percent decrease?
💡The final value is only a fraction of the original.
Answer
Divide the final value by the remaining-value multiplier.
ExampleAfter a \(25\%\) decrease, \(150/0.75=200\).
Prompt
How are successive percent changes combined?
💡Each change acts on the current value.
Answer
Multiply their change factors.
ExampleUp \(20\%\), then down \(10\%\), uses \(1.20(0.90)=1.08\).
Prompt
Do equal percent increases and decreases cancel?
💡The decrease acts on a different base.
Answer
No. For positive \(p\), \((1+p/100)(1-p/100)<1\).
ExampleUp \(25\%\), then down \(25\%\), leaves \(93.75\%\).
Prompt
What base is used in ‘\(A\) is \(p\%\) greater than \(B\)’?
💡The comparison is measured relative to \(B\).
Answer
\(B\), the quantity after ‘than.’
Example\(90\) is \(20\%\) greater than \(75\) because \(15/75=0.20\).
Prompt
What base is used in ‘\(A\) is \(p\%\) less than \(B\)’?
💡Divide the difference by \(B\).
Answer
\(B\), the referenced quantity after ‘than.’
Example\(60\) is \(25\%\) less than \(80\).
Prompt
What is the difference between percentage points and percent change?
💡The two answers have different bases.
Answer
Percentage points subtract rates; percent change divides that difference by the original rate.
ExampleFrom \(40\%\) to \(50\%\) is \(10\) points but a \(25\%\) increase.
Prompt
Can a percent be greater than \(100\%\)?
💡Convert normally by dividing by \(100\).
Answer
Yes; it represents more than the reference whole.
Prompt
Can a percent be between \(0\%\) and \(1\%\)?
💡Dividing by \(100\) adds two decimal places.
Answer
Yes; convert it carefully to a small decimal.
Example\(0.25\%=0.0025\).
Prompt
What is a quick check for an increase multiplier?
💡An increase cannot use a remaining fraction below \(1\).
Answer
It must be greater than \(1\).
ExampleA \(7\%\) increase uses \(1.07\), not \(0.07\).
Prompt
What is a quick check for a decrease multiplier smaller than \(100\%\)?
💡The multiplier represents the fraction remaining.
Answer
It must be between \(0\) and \(1\).
ExampleA \(18\%\) decrease uses \(0.82\).
Prompt
What is the most common denominator trap in percent change?
💡Label the values before subtracting.
Answer
Dividing the change by the new value instead of the original value.
ExampleFrom \(100\) to \(125\), use \(25/100\), not \(25/125\).
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