Mistake Master
Same quantity, new units
A conversion never changes the quantity, only its label. The arithmetic is one multiplication or division. The points slip away when the factor runs the wrong way, when an area or volume converts like a length, and at the end, when a correct answer arrives wearing the wrong unit. Let the units drive every step.
§1
What this topic is about
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A unit conversion changes the label on a quantity without changing the quantity itself. Three moves cover every SAT conversion: pick the factor that connects the two units, run it in the direction the units demand, and if the quantity is an area or a volume, convert every dimension. Each move has a classic failure, and the wrong answers are built from them.
§2
Point the factor the right way
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Every factor can multiply or divide, and the units decide which. Converting into a smaller unit means more of them, so the number grows; into a larger unit, it shrinks. Write the units beside the numbers and let the one you are leaving cancel.
- Into the smaller unit, multiply: feet to inches is $\times 12$.
- Into the larger unit, divide: minutes to hours is $\div 60$.
- Check the factor itself: time and customary units are not metric, so $60$ and $12$, not $100$.
Worked example. A meeting runs $150$ minutes. How long is that in hours?
Hours are the larger unit, so the count shrinks: divide. $$150 \div 60 = 2.5 \text{ hours}.$$
Multiplying gives $9{,}000$, a number that grew while the unit got bigger, which is impossible. And $2.5$ hours is $2$ hours $30$ minutes; writing $2$ hours $50$ minutes misreads the decimal.
§3
Areas and volumes convert every dimension
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A square yard is not $3$ square feet. It is a $3$-foot by $3$-foot square, so it holds $9$ square feet. Area converts by the linear factor squared, and volume by the factor cubed, because the factor acts once per dimension.
- Linear: $\times k$. Area: $\times k^2$. Volume: $\times k^3$.
- Convert the sides first if in doubt: a $2$ yd by $2$ yd square is $6$ ft by $6$ ft, which is $36$ square feet.
- The exponent belongs to the factor, not to the quantity: $4$ square yards becomes $4 \cdot 9$, never $4^2 \cdot 9$.
Worked example. A closet floor is $4$ square yards. How many square feet is it?
One yard is $3$ feet, so one square yard is $3 \cdot 3 = 9$ square feet.
$$4 \cdot 9 = 36 \text{ square feet}.$$ Multiplying by $3$ once gives $12$, the most common wrong answer on this pattern; the second dimension still has to convert.
§4
Chain conversions one unit at a time
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Speeds and rates carry two units, and each converts on its own. Kilometers per hour to meters per second multiplies by $1{,}000$ for the distance and divides by $3{,}600$ for the time. Handle them one at a time and no step gets lost.
- Convert the top and bottom units separately; never let one factor stand in for both.
- Each factor still obeys the direction rule: the unit being left must cancel.
- A mixed problem converts BEFORE the arithmetic: match the units, then divide or multiply.
Worked example. A walker covers $5.4$ kilometers at $90$ meters per minute. How many minutes does the walk take?
The rate is in meters, so the distance must be too: $5.4$ km is $5{,}400$ meters.
$$5{,}400 \div 90 = 60 \text{ minutes}.$$ Dividing $5.4$ by $90$ first gives $0.06$, a walk of four seconds; mismatched units always produce a nonsense size.
§5
Finish in the unit the question names
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The most expensive slip in this topic is not a conversion error at all: the work is right, and the answer gets reported in the wrong unit. The question's final line names a unit, and that name is part of the answer.
Worked example. A trip of $30$ kilometers at $60$ kilometers per hour takes how many minutes?
Time is distance over speed: $\dfrac{30}{60} = 0.5$ hours.
$$0.5 \cdot 60 = 30 \text{ minutes}.$$ Gridding $0.5$ answers in hours, a unit the question did not ask for.
Point each factor by the units, square or cube it for areas and volumes, convert rate units one at a time, and report the answer in the unit the question names.
§6
Three patterns that cost real points
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Three patterns recur on unit conversion questions. They are the same ones the diagnostic routes on.
An area or volume converts by the linear factor.
A $6$-square-yard rug becomes $18$ square feet instead of $54$, or a cubic foot becomes $12$ cubic inches instead of $1{,}728$. The factor acted once when the dimensions demanded two or three.
Fix. Convert the unit, not the number: a square yard is $3 \times 3$ feet, a cubic foot is $12 \times 12 \times 12$ inches. Square or cube the factor before it touches the quantity.
The factor runs the wrong way, or the wrong factor runs.
Minutes become hours by multiplying, or feet-and-inches get a metric $10$. The flipped answer is a clean number, so nothing feels wrong.
Fix. Ask which unit is smaller before computing: converting into the smaller unit must make the number grow. Then check the constant against the actual pair of units.
The answer arrives in the wrong unit.
The trip takes $0.75$ hours, the question asks for minutes, and $0.75$ goes on the grid. Every step of the math was right.
Fix. Reread the final line and name its unit. If your number carries a different one, a single conversion still stands between you and the point.
Ten quick checks across the patterns: pointing the factor, squaring and cubing it for areas and volumes, converting rate units, and finishing in the unit the question names. Pick or type your answer, then check. Progress is saved.