Mistake Master
Constant ratio or constant product
Direct proportionality holds a ratio still; inverse proportionality holds a product still; and plenty of relationships hold neither, because a flat fee rides along. The arithmetic is one constant and one multiplication. The points slip away when the two kinds trade places, when a fee gets scaled, and when the constant itself gets reported as the answer.
§1
What this topic is about
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Two quantities can be tied together in more than one way. Direct proportionality holds their ratio constant: double one, double the other. Inverse proportionality holds their product constant: double one, halve the other. And plenty of real relationships are neither, usually because a flat fee or starting amount rides along. The topic's three failures are mixing up the first two, assuming the third away, and reporting the constant instead of the answer.
§2
Direct: the ratio never moves
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Direct proportionality is $y = kx$: the graph is a line through the origin, and $k = \dfrac{y}{x}$ is the same for every pair. To find a new value, find $k$ once and multiply.
- Find the constant first: $k = \dfrac{y}{x}$ from the given pair.
- Scale, never add: if $x$ goes from $4$ to $9$, $y$ multiplies by $\dfrac{9}{4}$; it does not gain $5$.
- Zero check: proportional means zero in, zero out.
Worked example. $y$ is directly proportional to $x$, and $y = 20$ when $x = 4$. Find $y$ when $x = 9$.
$k = \dfrac{20}{4} = 5$, so the rule is $y = 5x$.
$$y = 5 \cdot 9 = 45.$$ Adding the change in $x$ to $y$ gives $25$, the additive slip; and $5$ alone is the constant, not the answer.
§3
Inverse: the product never moves
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Inverse proportionality is $xy = k$: more of one means proportionally less of the other. Crew-size problems live here: more workers, less time, same total work.
- Find the constant first: $k = xy$ from the given pair.
- Divide at the new input: $y = \dfrac{k}{x_{\text{new}}}$.
- Direction check: if $x$ grew and your $y$ grew too, the setup ran direct by mistake.
Worked example. Six machines finish a batch in $10$ hours. How long do $15$ machines take?
$k = 6 \cdot 10 = 60$ machine-hours.
$$\dfrac{60}{15} = 4 \text{ hours}.$$ Scaling the time UP with the crew, $10 \cdot \dfrac{15}{6} = 25$, moves the wrong way: more machines cannot take longer.
§4
Not everything is proportional
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A linear relationship with a starting amount, $y = b + rx$, is not proportional: doubling $x$ does not double $y$, because the $b$ never scales. The SAT loves handing you one price and inviting you to scale it.
- Test the origin: a flat fee, setup charge, or starting amount breaks proportionality.
- Rebuild from parts instead of scaling: fee once, rate times quantity.
- The additive slip is the same disease: proportional changes multiply, they never add.
Worked example. A rental costs a $\$12$ fee plus $\$9$ per hour. A $2$-hour rental costs $\$30$. What does a $6$-hour rental cost?
Tripling the $\$30$ gives $\$90$, but that triples the fee too, and the fee is paid once.
$$12 + 6 \cdot 9 = 66.$$ The $\$90$ overshoots by exactly the two extra copies of the fee.
§5
The constant is the tool, not the answer
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Both kinds of proportionality run through a constant, the unit rate or the fixed product. The constant is almost never what the question asks for, and it is always one of the choices.
Worked example. A graph of a proportional relationship passes through $(8, 36)$. What is $y$ when $x = 14$?
$k = \dfrac{36}{8} = 4.5$.
$$y = 4.5 \cdot 14 = 63.$$ Reporting $4.5$ answers a different question, and re-reporting $36$ answers none.
Decide direct or inverse by how the quantities move together, find the constant, apply it at the asked input, and check that a flat fee has not quietly broken the scaling.
§6
Three patterns that cost real points
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Three patterns recur on proportionality questions. They are the same ones the diagnostic routes on.
Direct and inverse trade places.
Fewer workers finish faster, or travel time shrinks as speed drops. The relationship's direction got read backward, and the answer moved the wrong way.
Fix. Before computing, say which way $y$ moves when $x$ grows. Together: constant ratio, divide to find $k$. Opposite: constant product, multiply to find $k$.
Proportionality gets assumed, or handled by addition.
A total containing a flat fee gets scaled, or a proportional change gets added ($x$ went up $5$, so $y$ goes up $5$). Both moves ignore what actually connects the quantities.
Fix. Test the origin before scaling any total. And when the relationship IS proportional, changes multiply through $k$; they never add.
The constant gets reported.
The rate $4.5$ lands on the answer sheet when the question asked for $y$ at $x = 14$, or the worker-hours constant is reported as a time.
Fix. The constant is the connector between every pair; the question asks for one particular value. Reread which one, then apply the constant instead of reporting it.
Ten quick checks across the patterns: direct versus inverse, spotting proportionality in a table, the flat fee that breaks scaling, and using the constant instead of reporting it. Pick or type your answer, then check. Progress is saved.