Almost every number in this course is a comparison, not a raw count. A survey might report 54 students out of 240. A quality check might report 3 defects per 1,000 units. A poll might report a 5-point jump. Reading those comparisons correctly matters. So does telling a percentage apart from a percentage point. Together, they are the arithmetic floor everything else in the course stands on.
"25% of the class failed" means nothing without the class size. 25% of 8 students is 2 people. 25% of 400 students is 100 people. Same percentage, very different headline. Always ask "percent of what?" before reacting to a number.
A proportion is a part-to-whole ratio \dfrac{\text{part}}{\text{whole}}; a percentage is that same ratio scaled to a base of 100: \text{percent} = \dfrac{\text{part}}{\text{whole}} \times 100. A rate compares two different quantities (defects per 1,000 units, deaths per 100,000 people) rather than a part to its own whole. Naming the base — "54 out of 240," never a bare "54%" with the whole left implicit — is what makes a percentage checkable at all.
Percent of a whole, with the base named
Change the whole and the part and watch which number the percentage is actually being taken OF.
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Worked example
Percent of a whole
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1
A shipment contains 850 units; 34 are defective. What percent are defective?
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2
Divide the part by the whole, then scale to a base of 100: \dfrac{34}{850} \times 100.
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3\boxed{4\%}
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Worked example
Percentage points vs. percent change
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1
A defect rate falls from 20% to 15%. A news report says "defects dropped 5%." Is that the same as a 5-percentage-point drop?
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The percentage-point change is 15 - 20 = -5 points — a real, simple subtraction of two percentages. The percent change is \dfrac{15-20}{20}\times 100 = -25\%: defects fell by a quarter of their original rate, a much bigger-sounding number for the same underlying drop.
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3\boxed{\text{5-point drop} \neq \text{25\% change} - \text{both are true, but they answer different questions}}
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Where this shows up
Quality control
A factory floor might report "defects are down 5%." That claim hides an important detail: is it 5 points, or 5 percent? Left unclear, the number could mean two very different things. The two readings can differ by a factor of 5 or more. Anyone checking a report should ask one question first: "percent of what, and is that points or percent?" Only after that question is answered should the headline be trusted.
Check your understanding (counts toward your progress)
A city's approval rating rises from 40% to 44%. A headline calls this a '10% increase.' Is that accurate?
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That's the right arithmetic for percent CHANGE, but the headline needs to specify which reading it means.
4-percentage-point is correct for the point change, but '10% increase' (percent change) isn't automatically wrong — it's a different, also-valid statistic.
4 doesn't round to 10 under any reasonable rule — check the percent-change formula instead.
Why
(44-40)/40 × 100 = 10% is a legitimate percent change; 44-40 = 4 percentage points is a different, equally legitimate number. The headline is ambiguous, not wrong outright, unless it fails to say which one it means.
Out of 380 units shipped, 72 met some condition. What percent is that?
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Why
Divide the part by the whole and scale to a base of 100: (72/380) \times 100 = 18.9\%.
Correct answer
Also accepted:
Why
Which of these is a RATE rather than a percentage?
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A percentage — part of a whole scaled to a base of 100.
A proportion, not a rate — it's a part-to-whole ratio of the same kind of thing.
Also a proportion — 80 students out of 100 students, same units on top and bottom.
Why
A rate compares two DIFFERENT quantities (accidents vs. vehicle-miles); a percentage or proportion compares a part to its own whole.
A survey of 320 residents finds 96 support a new park. What percent support it?
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Why
96/320 × 100 = 30%.
Correct answer
Also accepted:
Why