Mistake Master
Percents live on a base, not on their own
Every percent is a percent of something, and that something is where the points hide. The arithmetic is a multiplication. The points slip away when the percent lands on the wrong base, when it gets added like a plain number, and at the end, when the discount gets reported as the price. Name the base, scale it, then answer the exact question.
§1
What this topic is about
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A percent is a fraction with a loud name: $30\%$ means $30$ per hundred OF something. Every percent question turns on three reads: what the base is, that a percent scales rather than adds, and which member of the percent family, the change, the new value, or the rate itself, the question actually asks for.
§2
A percent scales its base
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Percent of a number multiplies: $30\%$ of $60$ is $0.30 \cdot 60 = 18$. The percent never adds or subtracts directly; $25\%$ off an $\$84$ bag is not $84 - 25$.
- Convert to a decimal and multiply: $p\%$ of $N$ is $\dfrac{p}{100} \cdot N$.
- A ${p}\%$ increase multiplies by $1 + \dfrac{p}{100}$; a decrease, by $1 - \dfrac{p}{100}$.
- Watch the decimal: $8\%$ is $0.08$, one place further than $0.8$.
Worked example. A $\$140$ ticket is discounted $25\%$. What is the new price?
The ticket keeps $75\%$: $$140 \cdot 0.75 = 105.$$
Computing $140 - 25 = 115$ treats the percent as dollars. And $35$, the discount itself, is a correct number that answers a different question.
§3
Find the base before you compute
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Every percent is a percent OF something, and the SAT's favorite trap is quietly switching that base. A price that rose $20\%$ to $\$90$ rose $20\%$ of the ORIGINAL, so the original is not $90$ minus $20\%$ of $90$.
- Reverse a percent change by division: the new value is $(100 + p)\%$ of the original, so original $= \text{new} \div \left(1 + \frac{p}{100}\right)$.
- Percent change divides by the STARTING value, never the ending one.
- When two percents act in sequence, each takes the previous result as its base; they multiply, never add.
Worked example. After a $20\%$ increase, a price is $\$90$. What was the original price?
The new price is $120\%$ of the original: $1.2 \cdot \text{original} = 90$.
$$\text{original} = 90 \div 1.2 = 75.$$ Taking $20\%$ off the $90$ gives $72$, the classic wrong-base answer; check: $75 \cdot 1.2 = 90$, while $72 \cdot 1.2 = 86.4$.
§4
Percent change measures the journey
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Percent change is the change divided by where you started. From $40$ to $56$ the change is $16$, and $\dfrac{16}{40} = 40\%$. The new value, the change, and the percent are three different numbers, and the choices will contain all three.
- Percent change $= \dfrac{\text{new} - \text{old}}{\text{old}} \cdot 100$.
- Percentage POINTS subtract two rates ($25\% - 20\%$ is $5$ points); percent change compares them ($\dfrac{5}{20} = 25\%$).
- Up $10\%$ then down $10\%$ is not zero: $1.1 \cdot 0.9 = 0.99$, a $1\%$ loss.
§5
Report the member of the family that was asked
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One percent computation produces a whole family: the base, the change, the new value, the rate. The question names exactly one. The most reliable point in this topic comes from rereading that name after the math is done.
Worked example. A $\$60$ order gets an $8\%$ delivery fee. How much is paid in total?
The fee is $60 \cdot 0.08 = \$4.80$; the total is $60 + 4.80 = \$64.80$.
The question says TOTAL, so $\$64.80$; the $4.80$ is the fee, one member of the family over from the answer.
Scale the base, never add the percent; name the base before undoing a change; divide change by the start; and report the family member the question names.
§6
Three patterns that cost real points
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Three patterns recur on percent questions. They are the same ones the diagnostic routes on.
The percent lands on the wrong base.
A price that rose to $\$144$ gets "un-raised" by taking the percent off the new price, or a percent change gets divided by the new value. Each move uses the number in front of you instead of the base the change was measured against.
Fix. Say the base out loud before computing: the change was a percent OF THE ORIGINAL. Undo increases by dividing by $1 + \frac{p}{100}$, and divide changes by the starting value.
The percent behaves like a plain number.
$25\%$ off $\$84$ becomes $84 - 25$, two successive increases get added, or the decimal slides a place. The percent got treated as an amount instead of a proportion.
Fix. Convert to a decimal and multiply. When changes stack, multiply their factors one at a time; each acts on a new base.
The right computation, the wrong family member.
The discount is computed perfectly and reported as the price, or the new total is reported when the question asked for the change, or a dollar change gets a percent sign.
Fix. After solving, reread the final line: does it ask for the new value, the change, or the percent? Match your number to that exact name.
Ten quick checks across the patterns: scaling the base, reversing a change by division, percent change against the starting value, stacked percents, and reporting the family member asked. Pick or type your answer, then check. Progress is saved.